{"id": "GMAT Club Math Book 2024 v8_p1_c1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 1, "page_end": 1, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Club Math Book 2024 v8_p2_c1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 2, "page_end": 2, "topic_guess": "word_problems", "text": "GMAT word problem note: Track mean, median, range, distribution, and how operations change spread.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p3_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 3, "page_end": 3, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. NUMBER THEORY Definition Number Theory is concerned with the properties of numbers in general, and in particular integers.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p3_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 3, "page_end": 3, "topic_guess": "number_theory", "text": "GMAT number properties note: GMAT Number Types GMAT deals with only Real Numbers:. Integers, Fractions and Irrational Numbers.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p3_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 3, "page_end": 3, "topic_guess": "number_theory", "text": "GMAT number properties note: INTEGERS Definition Integers are defined as: all negative natural numbers, zero, and positive natural numbers. Note that integers do not include decimals or fractions - just whole numbers.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p3_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 3, "page_end": 3, "topic_guess": "number_theory", "text": "GMAT number properties note: Even and Odd Numbers An even number is an integer that is \"evenly divisible\" by 2, i.e., divisible by 2 without a remainder. An even number is an integer of the form, where is an integer.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p3_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 3, "page_end": 3, "topic_guess": "number_theory", "text": "GMAT number properties note: An odd number is an integer that is not evenly divisible by 2. An odd number is an integer of the form, where is an integer.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p3_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 3, "page_end": 3, "topic_guess": "number_theory", "text": "GMAT number properties note: Addition / Subtraction: even +/- even = even. Multiplication: even * even = even.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p3_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 3, "page_end": 3, "topic_guess": "number_theory", "text": "GMAT number properties note: Division of two integers can result into an even/odd integer or a fraction.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p4_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 4, "page_end": 4, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Watch divisibility, factors, remainders, parity, and prime structure. An even number is an integer that is \"evenly divisible\" by 2, i.e., divisible by 2 without a remainder and as zero is evenly divisible by 2 then it must be even.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p4_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 4, "page_end": 4, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: 0 is neither positive nor negative integer (the only one of this kind). 0 is divisible by EVERY integer.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p4_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 4, "page_end": 4, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: IRRATIONAL NUMBERS Fractions (also known as rational numbers) can be written as terminating (ending) or repeating decimals terminating, non-repeating decimals are non-rational, so they are called the \"irrationals\". Putting these two major classifications, the rationals and the irrationals, together in one set gives you the \"real\" numbers.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p4_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 4, "page_end": 4, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: POSITIVE AND NEGATIVE NUMBERS A positive number is a real number that is greater than zero. A negative number is a real number that is smaller than zero.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p4_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 4, "page_end": 4, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Zero is not positive, nor negative. Multiplication: positive * positive = positive positive * negative = negative negative * negative = positive Division: positive / positive = positive positive / negative = negative negative / negative = positive 2√ π =.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p5_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 5, "page_end": 5, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Prime Numbers A Prime number is a natural number with exactly two distinct natural number divisors: 1 and itself.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p5_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 5, "page_end": 5, "topic_guess": "number_theory", "text": "GMAT number properties note: Otherwise a number is called a composite number. Therefore, 1 is not a prime, since it only has one divisor, namely 1.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p5_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 5, "page_end": 5, "topic_guess": "number_theory", "text": "GMAT number properties note: A number is prime if it cannot be written as a product of two factors and, both of which are greater than 1: n = ab. There are infinitely many prime numbers.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p5_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 5, "page_end": 5, "topic_guess": "number_theory", "text": "GMAT number properties note: The only even prime number is 2, since any larger even number is divisible by 2. Also 2 is the smallest prime.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p5_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 5, "page_end": 5, "topic_guess": "number_theory", "text": "GMAT number properties note: All prime numbers except 2 and 5 end in 1, 3, 7 or 9, since numbers ending in 0, 2, 4, 6 or 8 are multiples of 2 and numbers ending in 0 or 5 are multiples of 5. Similarly, all prime numbers above 3 are of the form or, because all other numbers are divisible by 2 or 3.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p5_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 5, "page_end": 5, "topic_guess": "number_theory", "text": "GMAT number properties note: Any nonzero natural number can be factored into primes, written as a product of primes or powers of primes. Moreover, this factorization is unique except for a possible reordering of the factors.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p5_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 5, "page_end": 5, "topic_guess": "number_theory", "text": "GMAT number properties note: Prime factorization: every positive integer greater than 1 can be written as a product of one or more prime integers in a way which is unique.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p6_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 6, "page_end": 6, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Factors A divisor of an integer, also called a factor of, is an integer which evenly divides without leaving a remainder.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p6_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 6, "page_end": 6, "topic_guess": "number_theory", "text": "GMAT number properties note: In general, it is said is a factor of, for non-zero integers and, if there exists an integer such that. 1 (and -1) are divisors of every integer.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p6_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 6, "page_end": 6, "topic_guess": "number_theory", "text": "GMAT number properties note: Every integer is a divisor of itself. Every integer is a divisor of 0, except, by convention, 0 itself.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p6_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 6, "page_end": 6, "topic_guess": "number_theory", "text": "GMAT number properties note: Numbers divisible by 2 are called even and numbers not divisible by 2 are called odd. A positive divisor of n which is different from n is called a proper divisor.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p6_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 6, "page_end": 6, "topic_guess": "number_theory", "text": "GMAT number properties note: An integer n > 1 whose only proper divisor is 1 is called a prime number. Equivalently, one would say that a prime number is one which has exactly two factors: 1 and itself.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p6_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 6, "page_end": 6, "topic_guess": "number_theory", "text": "GMAT number properties note: Any positive divisor of n is a product of prime divisors of n raised to some power. If a number equals the sum of its proper divisors, it is said to be a perfect number.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p6_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 6, "page_end": 6, "topic_guess": "number_theory", "text": "GMAT number properties note: The proper divisors of 6 are 1, 2, and 3: 1+2+3 = 6, hence 6 is a perfect number. There are some elementary rules:.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p6_c1_r8", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 6, "page_end": 6, "topic_guess": "number_theory", "text": "GMAT number properties note: If is a factor of and is a factor of, then is a factor of. In fact, is a factor of for all integers and.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p6_c1_r9", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 6, "page_end": 6, "topic_guess": "number_theory", "text": "GMAT number properties note: If is a factor of and is a factor of, then or. If is a factor of, and, then a is a factor of.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p6_c1_r10", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 6, "page_end": 6, "topic_guess": "number_theory", "text": "GMAT number properties note: If is a prime number and is a factor of.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p7_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 7, "page_end": 7, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Power of the common factors).", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p7_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 7, "page_end": 7, "topic_guess": "number_theory", "text": "GMAT number properties note: Every common divisor of a and b is a divisor of gcd(a, b). A*b = gcd(a, b)*lcm(a, b) Lowest Common Multiple - LCM The lowest common multiple or lowest common multiple (lcm) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple both of a and of b.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p7_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 7, "page_end": 7, "topic_guess": "number_theory", "text": "GMAT number properties note: Since it is a multiple, it can be divided by a and b without a remainder. If either a or b is 0, so that there is no such positive integer, then lcm(a, b) is defined to be zero.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p7_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 7, "page_end": 7, "topic_guess": "number_theory", "text": "GMAT number properties note: To find the LCM, you will need to do prime-factorization. Then multiply all the factors (pick the highest power of the common factors).", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p7_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 7, "page_end": 7, "topic_guess": "number_theory", "text": "GMAT number properties note: Perfect Square A perfect square, is an integer that can be written as the square of some other integer. There are some tips about the perfect square:.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p7_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 7, "page_end": 7, "topic_guess": "number_theory", "text": "GMAT number properties note: The number of distinct factors of a perfect square is ALWAYS ODD. The sum of distinct factors of a perfect square is ALWAYS ODD.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p7_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 7, "page_end": 7, "topic_guess": "number_theory", "text": "GMAT number properties note: A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. Perfect square always has even number of powers of prime factors.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p7_c1_r8", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 7, "page_end": 7, "topic_guess": "number_theory", "text": "GMAT number properties note: Divisibility Rules 2 - If the last digit is even, the number is divisible by 2. 3 - If the sum of the digits is divisible by 3, the number is also.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p7_c1_r9", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 7, "page_end": 7, "topic_guess": "number_theory", "text": "GMAT number properties note: 4 - If the last two digits form.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p8_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 8, "page_end": 8, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Note: factorial of negative numbers is undefined.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p8_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 8, "page_end": 8, "topic_guess": "number_theory", "text": "GMAT number properties note: Trailing z eros are a sequence of 0's in the decimal representation (or more generally, in any positional representation) of a number, after which no other digits follow. The number of trailing zeros in the decimal representation of n!, the factorial of a non-negative integer, can be determined with this formula: , where k must be chosen such that.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p8_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 8, "page_end": 8, "topic_guess": "number_theory", "text": "GMAT number properties note: How many zeros are in the end (after which no other digits follow) of? (denominator must be less than 32, is less) Hence, there are 7 zeros in the end of 32!", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p8_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 8, "page_end": 8, "topic_guess": "number_theory", "text": "GMAT number properties note: The formula actually counts the number of factors 5 in n!, but since there are at least as many factors 2, this is equivalent to the number of factors 10, each of which gives one more trailing zero. Finding the number of powers of a prime number, in the.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p8_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 8, "page_end": 8, "topic_guess": "number_theory", "text": "GMAT number properties note: Till What is the power of 2 in 25!? Finding the power of non-prime in n!:.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p8_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 8, "page_end": 8, "topic_guess": "number_theory", "text": "GMAT number properties note: How many powers of 900 are in 50! Make the prime factorization of the number:, then find the powers of these prime numbers in the n!.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p8_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 8, "page_end": 8, "topic_guess": "number_theory", "text": "GMAT number properties note: Find the power of 5: n Factorials Factorial of a non-negative integer, denoted by n!, is the product of all positive in.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p9_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 9, "page_end": 9, "topic_guess": "number_theory", "text": "GMAT number properties note: Track mean, median, range, distribution, and how operations change spread. We need all the prime {2,3,5} to be represented twice in 900, 5 can provide us with only 6 pairs, thus there is 900 in the power of 6 in 50!.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p9_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 9, "page_end": 9, "topic_guess": "number_theory", "text": "GMAT number properties note: Consecutive Integers Consecutive integers are integers that follow one another, without skipping any integers. 7, 8, 9, and -2, -1, 0, 1, are consecutive integers.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p9_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 9, "page_end": 9, "topic_guess": "number_theory", "text": "GMAT number properties note: Sum of consecutive integers equals the mean multiplied by the number of terms,. Given consecutive integers,, (mean equals to the average of the first and last terms), so the sum equals to.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p9_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 9, "page_end": 9, "topic_guess": "number_theory", "text": "GMAT number properties note: If n is odd, the sum of consecutive integers is always divisible by n. Given, we have consecutive integers.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p9_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 9, "page_end": 9, "topic_guess": "number_theory", "text": "GMAT number properties note: The sum of 9+10+11 = 30, therefore, is divisible by 3. If n is even, the sum of consecutive integers is never divisible by n.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p9_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 9, "page_end": 9, "topic_guess": "number_theory", "text": "GMAT number properties note: The sum of 9+10+11+12 = 42, therefore, is not divisible by 4. The product of consecutive integers is always divisible by.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p9_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 9, "page_end": 9, "topic_guess": "number_theory", "text": "GMAT number properties note: Given consecutive integers:. The product of 3*4*5*6 is 360, which is divisible by 4!", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p9_c1_r8", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 9, "page_end": 9, "topic_guess": "number_theory", "text": "GMAT number properties note: Evenly Spaced Set Evenly spaced set or an arithmetic progression is a sequence of numbers such that the difference of any two successive members of the sequence is a constant.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p10_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 10, "page_end": 10, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Watch divisibility, factors, remainders, parity, and prime structure. Middle term multiplied by number of terms.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p10_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 10, "page_end": 10, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: There are five terms in the set {1, 7, 13, 19, 25}, middle term is 13, so the sum is 13*5 = 65. FRACTIONS Definition Fractional numbers are ratios (divisions) of integers.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p10_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 10, "page_end": 10, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: In other words, a fraction is formed by dividing one integer by another integer. Set of Fraction is a subset of the set of Rational Numbers.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p10_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 10, "page_end": 10, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Fraction can be expressed in two forms fractional representation and decimal representation. Fractional representation Fractional representation is a way to express numbers that fall in between integers (note that integers can also be expressed in fractional form).", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p10_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 10, "page_end": 10, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: A fraction expresses a part-to-whole relationship in terms of a numerator (the part) and a denominator (the whole). The number on top of the fraction is called numerator or nominator.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p10_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 10, "page_end": 10, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: The number on bottom of the fraction is called denominator. In the fraction,, 9 is the numerator and 7 is denominator.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p10_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 10, "page_end": 10, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Fractions that have a value between 0 and 1 are called proper fraction. The numerator is always smaller than the denominator.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p10_c1_r8", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 10, "page_end": 10, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Fractions that are greater than 1 are called improper fraction. Improper fraction can also be written as a mixed number.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p10_c1_r9", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 10, "page_end": 10, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: An integer combined with a proper fraction is called mixed number.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p11_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 11, "page_end": 11, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Reciprocal Reciprocal for a number, denoted by or, is a number which when multiplied by yields.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p11_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 11, "page_end": 11, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: The reciprocal of a fraction is. To get the reciprocal of a number, divide 1 by the number.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p11_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 11, "page_end": 11, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Operation on Fractions Adding/Subtracting fractions:. To add/subtract fractions with the same denominator, add the numerators and place that sum over the common denominator.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p11_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 11, "page_end": 11, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: To add/subtract fractions with the different denominator, find the Least Common Denominator (LCD) of the fractions, rename the fractions to have the LCD and add/subtract the numerators of the fractions Multiplying fractions:. To multiply fractions just place the product of the numerators over the product of the denominators.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p11_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 11, "page_end": 11, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Change the divisor into its reciprocal and then multiply. Given, take the reciprocal of.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p11_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 11, "page_end": 11, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Decimal Representation The decimals has ten as its base. Decimals can be terminating (ending) (such as 0.78, 0.2) or repeating Reduced fraction (meaning that fraction is already reduced to its lowest term) can be expressed as terminating decimal if and only (denominator) is of the form, where and are non-negative integers.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p12_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 12, "page_end": 12, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. 2:, the number is of length so we have added two nines.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p12_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 12, "page_end": 12, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Reducing it to lowest terms:. To convert a mixed-recurring decimal to fraction: 1.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p12_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 12, "page_end": 12, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Write down the number consisting with non-repeating digits and r epeating digits. Subtract non-repeating number from above.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p12_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 12, "page_end": 12, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Divide 1-2 by the number with 9's and 0's: for every repeating digit write down a 9, and for every non- repeating digit write down a zero after 9's. The number consisting with non-repeating digits and repeating digits is 2512.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p12_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 12, "page_end": 12, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Subtract 25 (non-repeating number) from above: 2512-25 = 2487. Divide 2487 by 9900 (two 9's as there are two digits in 12 and 2 zeros as there are two digits in 25): 2487/9900 = 829/3300.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p12_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 12, "page_end": 12, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Rounding Rounding is simplifying a number to a certain place value. To round the decimal drop the extra decimal places, and if the first dropped digit is 5 or greater, round up the last digit that you keep.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p12_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 12, "page_end": 12, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: If the first dropped digit is 4 or smaller, round down (keep the same) the last digit that you keep. 3485 rounded to the nearest hundredth = 5.35, since the dropped 8 is greater than 5.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p12_c1_r8", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 12, "page_end": 12, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: 3485 rounded to the nearest thousandth = 5.349, since the dropped.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p13_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 13, "page_end": 13, "topic_guess": "word_problems", "text": "GMAT word problem note: Watch divisibility, factors, remainders, parity, and prime structure. EXPONENTS Exponents are a \"shortcut\" method of showing a number that was multiplied by itself several times.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p13_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 13, "page_end": 13, "topic_guess": "word_problems", "text": "GMAT word problem note: For instance, number multiplied times can be written as, where represents the base, the number that is multiplied by itself times and represents the exponent. The exponent indicates how many times to multiple the base,, by itself.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p13_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 13, "page_end": 13, "topic_guess": "word_problems", "text": "GMAT word problem note: Any nonzero number to the power of 0 is 1. Any number to the power 1 is itself.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p13_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 13, "page_end": 13, "topic_guess": "word_problems", "text": "GMAT word problem note: If the exponent is positive, the power of zero is zero:, where. If the exponent is negative, the power of zero (, where) is undefined, because division by zero is implied.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p13_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 13, "page_end": 13, "topic_guess": "word_problems", "text": "GMAT word problem note: The integer powers of one are one. If n is an even integer, then.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p13_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 13, "page_end": 13, "topic_guess": "word_problems", "text": "GMAT word problem note: If n is an odd integer, then. Operations involving the same exponents:.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p13_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 13, "page_end": 13, "topic_guess": "word_problems", "text": "GMAT word problem note: Keep the exponent, multiply or divide the bases and not Operations involving the same bases:. Keep the base, add or subtract the exponent (add for multiplication, subtract for division) Fraction as power:.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p13_c1_r8", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 13, "page_end": 13, "topic_guess": "word_problems", "text": "GMAT word problem note: When solving equations with even exponents, we must consider both positive and negative possibilities for the solutions. A n an a n n a = 1a0 = 150 (−3 = 1)0 = aa1 = 00n n > 0 0n n < 0.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p14_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 14, "page_end": 14, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. For instance, the two possible solutions are and.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p14_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 14, "page_end": 14, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: When solving equations with odd exponents, we'll have only one solution. For instance for, solution is and for, solution is.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p14_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 14, "page_end": 14, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Exponents and divisibility: is ALWAYS divisible by. Is divisible by if is odd, and not divisible by a+b if n is even.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p14_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 14, "page_end": 14, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: LAST DIGIT OF A PRODUCT Last digits of a product of integers are last digits of the product of last digits of these integers. The last digit of 85945*89*58307 = 5*9*7 = 45*7 = 35 = 5?", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p14_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 14, "page_end": 14, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: LAST DIGIT OF A POWER Determining the last digit of: 1. Last digit of is the same as that of.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p14_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 14, "page_end": 14, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Determine the cyclicity number of. Find the remainder when divided by the cyclisity.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p14_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 14, "page_end": 14, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: When, then last digit of is the same as that of and when, then last digit of is the same as that of, where is the cyclisity number. Integer ending with 0, 1, 5 or 6, in the integer power k>0, has the same last digit as the base.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p14_c1_r8", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 14, "page_end": 14, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Integers ending with 2, 3, 7 and 8 have a cyclicity of 4. Integers ending with 4 (eg.) have a cyclisity of 2.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p14_c1_r9", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 14, "page_end": 14, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: When n is odd will end with 4 and when n is even will end with 6. Integers ending with 9 (eg.) have a cyclisity of 2.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p15_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 15, "page_end": 15, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. , when, then and when, then When the GMAT provides the square root sign for an even root, such as or, then the only accepted answer is the positive root.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p15_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 15, "page_end": 15, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: In contrast, the equation has TWO solutions, +5 and -5. Even roots have only a positive value on the GMAT.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p15_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 15, "page_end": 15, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Odd roots will have the same sign as the base of the root. For GMAT it's good to memorize following values:.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p15_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 15, "page_end": 15, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: PERCENTS A percentage is a way of expressing a number as a fraction of 100 (per cent meaning \"per hundred\"). It is often denoted using the percent sign, \"%\", or the abbreviation \"pct\".", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p15_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 15, "page_end": 15, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Since a percent is an amount per 100, percents can be represented as fractions with a denominator of 100. A percent can be represented as a decimal.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p15_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 15, "page_end": 15, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: The following relationship characterizes how percents and decimals interact. What is 2% represented as a decimal?", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p15_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 15, "page_end": 15, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent Form / 100 = Decimal Form: 2%/100 = 0.02 Percent change General formula for percent increase or decrease, (percent change):. A company received $2 million in royalties on the first $10 million in sales and then $8 million in royalties on the next $100 million in sales.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p16_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 16, "page_end": 16, "topic_guess": "word_problems", "text": "GMAT word problem note: Percent problems usually relate part, whole, rate, and percent change. So the royalties decreased by 60%.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p16_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 16, "page_end": 16, "topic_guess": "word_problems", "text": "GMAT word problem note: If $15,000 is invested at 10% simple annual interest, how much interest is earned after 9 months? If C = 1, meaning that interest is compounded once a year, then the formula will be: , where time is number of years.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p16_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 16, "page_end": 16, "topic_guess": "word_problems", "text": "GMAT word problem note: If $20,000 is invested at 12% annual interest, compounded quarterly, what is the balance after 2 year? ORDER OF OPERATIONS - PEMDAS Perform the operations inside a Parenthesis first (absolute value signs also fall into this category), then Exponents, then Multiplication and Division, from left to right, then Addition and Subtraction, from left to right - PEMDAS.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p16_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 16, "page_end": 16, "topic_guess": "word_problems", "text": "GMAT word problem note: An exclamation mark indicates that one should compute the factorial of the term immediately to its left, before computing any of the lower-precedence operations, unless grouping symbols dictate otherwise. A factorial in an exponent applies to the exponent, while a factorial not in the exponent applies to the entire power.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p16_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 16, "page_end": 16, "topic_guess": "word_problems", "text": "GMAT word problem note: If exponentiation is indicated by stacked symbols, the rule is to work from the top down, thus: and not.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p16_c2_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 16, "page_end": 16, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. N the exponent applies to the entire power.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p16_c2_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 16, "page_end": 16, "topic_guess": "word_problems", "text": "GMAT word problem note: If exponentiation is indicated by stacked symbols, the rule is to work from the top down, thus: and not Percent = ∗ 100 = Change Original = ∗ 100 = 60 Balance(final) = principal∗ (1 + interest C)time∗C Balance(final) = principal∗ (1 + interest)time Balance = 20, 000 ∗ (1 + = 20, 000 ∗ (1.03 = $25, 335.40.12 ()!", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p17_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 17, "page_end": 17, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Https://gmatclub.com/forum/root-80-root-139868.html Medium: 2.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p17_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 17, "page_end": 17, "topic_guess": "number_theory", "text": "GMAT number properties note: Https://gmatclub.com/forum/topic-143744.html 6. Https://gmatclub.com/forum/what-is-63908.html Hard.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p17_c2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 17, "page_end": 17, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Club Math Book 2024 v8_p18_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 18, "page_end": 18, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. GMAT Club Forum Remainders https://gmatclub.com/forum/remainders-144665.html REMAINDERS created by:.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p18_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 18, "page_end": 18, "topic_guess": "number_theory", "text": "GMAT number properties note: This formula can also be written as. Properties When is divided by the remainder is 0 if is a multiple of.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p18_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 18, "page_end": 18, "topic_guess": "number_theory", "text": "GMAT number properties note: When a smaller integer is divided by a larger integer, the quotient is 0 and the remainder is the smaller integer. X y q r y = divisor∗ quotient+ remainder = xq+ r 0 ≤ r < x 15 = 6 ∗ 2 + 3 0 ≤ r < x = q+y x r x y x y x 12 = 3 ∗ 4 + 0 7 = 11 ∗ 0 + 7 y x x− 1 y.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p19_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 19, "page_end": 19, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If a number is divided by 10, its remainder is the last digit of that number.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p19_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 19, "page_end": 19, "topic_guess": "number_theory", "text": "GMAT number properties note: If it is divided by 100 then the remainder is the last two digits and so on. 4 When positive integer n is dived by 18 the remainder is 7:.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p19_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 19, "page_end": 19, "topic_guess": "number_theory", "text": "GMAT number properties note: Now, since the first term (18q) is divisible by 6, then the remainder will only be from the second term, which is 7. 7 divided by 6 yields the remainder of 1.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p19_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 19, "page_end": 19, "topic_guess": "number_theory", "text": "GMAT number properties note: 5 There are several algebraic ways to solve this question, but the easiest way is as follows: since we cannot have two correct answers just pick a prime greater than 3, square it and see what would be the remainder upon division of it by 12. The remainder upon division 25 by 12 is 1.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p20_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 20, "page_end": 20, "topic_guess": "statistics", "text": "GMAT statistics note: Watch divisibility, factors, remainders, parity, and prime structure. (1) x divided by 100 has a remainder of 30.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p20_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 20, "page_end": 20, "topic_guess": "statistics", "text": "GMAT statistics note: (2) x divided by 110 has a remainder of 30. We have that: 30, 130, 230, (2) x divided by 110 has a remainder of 30.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p20_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 20, "page_end": 20, "topic_guess": "statistics", "text": "GMAT statistics note: We have that: 30, 140, 250, Answer:. (1) n is multiple of 5 (2) n is a multiple of 12 (1) n is multiple of 5.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p20_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 20, "page_end": 20, "topic_guess": "statistics", "text": "GMAT statistics note: If n = 5, then n yields the remainder of 5 when divided by 6 but if n = 10, then n yields the remainder of 4 when divided by 6. We already have two different answers, which means that this statement is not sufficient.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p20_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 20, "page_end": 20, "topic_guess": "statistics", "text": "GMAT statistics note: Every multiple of 12 is also a multiple of 6, thus n divided by 6 yields the remainder of 0. 45 divided by yields the remainder of can always be expressed as: (which is the same as), where is the quotient and is the remainder.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p20_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 20, "page_end": 20, "topic_guess": "statistics", "text": "GMAT statistics note: Given that, so according to the above, which means that must be a multiple of 3. Only option E offers answer which.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p21_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 21, "page_end": 21, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If n is greater than 30, what is the remainder that n leaves after division by 30?", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p21_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 21, "page_end": 21, "topic_guess": "number_theory", "text": "GMAT number properties note: 28 Positive integer n leaves a remainder of 4 after division by 6:. Thus n could Positive integer n leaves a remainder of 3 after division by 5:.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p21_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 21, "page_end": 21, "topic_guess": "number_theory", "text": "GMAT number properties note: Thus n could There is a way to derive general formula for (of a type, where is a divisor and is a remainder) based on above two statements:. Divisor would be the least common multiple of above two divisors 5 and 6, hence Remainder would be the first common integer in above two patterns, hence.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p21_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 21, "page_end": 21, "topic_guess": "number_theory", "text": "GMAT number properties note: Therefore general formula based on both statements is. Hence the remainder when positive integer n is divided by 30 is 28.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p21_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 21, "page_end": 21, "topic_guess": "number_theory", "text": "GMAT number properties note: (1) When 3x is divided by 2, there is a remainder. (2) x = 4y + 1, where y is an integer.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p21_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 21, "page_end": 21, "topic_guess": "number_theory", "text": "GMAT number properties note: , notice that we have the product of three consecutive integers. Now, notice that if, then and are consecutive even integers, thus one of them will also be divisible by 4, which will make divisible by 2*4 = 8 (basically if then will be divisible by 8*3 = 24).", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p22_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 22, "page_end": 22, "topic_guess": "algebra", "text": "GMAT algebra note: Watch divisibility, factors, remainders, parity, and prime structure. 0 , now if we expand this expression all terms but the last one will have in them, thus will leave no remainder upon division by 13, the last term will be.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p22_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 22, "page_end": 22, "topic_guess": "algebra", "text": "GMAT algebra note: Thus the question becomes: what is the remainder upon division -1 by 13? The answer to this question is 12:.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p22_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 22, "page_end": 22, "topic_guess": "algebra", "text": "GMAT algebra note: And when x is divided by 7, the remainder is 4. When positive integer y is divided by 5, the remainder is 3.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p22_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 22, "page_end": 22, "topic_guess": "algebra", "text": "GMAT algebra note: And when y is divided by 7, the remainder is 4. If x > y, which of the following must be a factor of x - y?", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p22_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 22, "page_end": 22, "topic_guess": "algebra", "text": "GMAT algebra note: 35 When the positive integer x is divided by 5 and 7, the remainder is 3 and 4, respectively:. Divisor will be the least common multiple of above two divisors 5 and 7, hence 35.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p22_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 22, "page_end": 22, "topic_guess": "algebra", "text": "GMAT algebra note: Remainder will be the first common integer in above two patterns, hence 18. So, to satisfy The same for y (as the same info is given about y):.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p22_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 22, "page_end": 22, "topic_guess": "algebra", "text": "GMAT algebra note: Thus must be a multiple of 35.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p23_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 23, "page_end": 23, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Since given that, then --> So,.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p23_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 23, "page_end": 23, "topic_guess": "number_theory", "text": "GMAT number properties note: Now, if then and if then , so in any case --> --> in order to be multiple of 4 must be multiple of 16 but as we see it's not, so is not multiple of 4. (2) x – y = 3 --> --> but not sufficient to say whether it's multiple of 4.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p23_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 23, "page_end": 23, "topic_guess": "number_theory", "text": "GMAT number properties note: Is the remainder of bigger than the remainder of? (2) The remainder of is 2 First of all any positive integer can yield only three remainders upon division by 3: 0, 1, or Since, the sum of the digits of and is always 1 then the remainders of and are only dependent on the value of the number added to and.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p23_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 23, "page_end": 23, "topic_guess": "number_theory", "text": "GMAT number properties note: There are 3 cases: the digits of and will be 1 more than a multiple of 3). Of the digits of and will be 2 more than a multiple of 3).", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p23_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 23, "page_end": 23, "topic_guess": "number_theory", "text": "GMAT number properties note: Of the digits of and will be a multiple of 3). P = 8 q+ 5 = +x2 y2 y = odd = 2 k+ 1 8q+ 5 = + (2 k+ 1x2)2 = 8 q+ 4 − 4 − 4 k = 4(2 q+ 1 − − k)x2 k2 k2 = 4(2 q+ 1 − − k)x2 k2 k = odd 2q+ 1 − − k = even+ odd− odd− o.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p24_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 24, "page_end": 24, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Means that the remainder of is 0.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p24_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 24, "page_end": 24, "topic_guess": "number_theory", "text": "GMAT number properties note: Now, the question asks whether the remainder of, which is 0, greater than the reminder of, which is 0, 1, or 2. Obviously it cannot be greater, it can be less than or equal to.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p24_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 24, "page_end": 24, "topic_guess": "number_theory", "text": "GMAT number properties note: So, the answer to the question is NO. Check more PS questions on remainders here n 3 2 n +n10m +n10m +m10n.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p25_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 25, "page_end": 25, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p25_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 25, "page_end": 25, "topic_guess": "algebra", "text": "GMAT algebra note: Algebra https://gmatclub.com/forum/algebra-101576.html Algebra 101 Scope Manipulation of various algebraic expressions Equations in 1 & more variables Dealing with non-linear equations Algebraic identities Notation & Assumptions In this document, lower case roman alphabets will be used to denote variables such as a,b,c,x,y,z,w In general it is assumed that the GMAT will only deal with real numbers () or subsets of such as Integers (), rational numbers () etc Concept of variables A variable is a place holder, which can be used in mathematical expressions.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p25_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 25, "page_end": 25, "topic_guess": "algebra", "text": "GMAT algebra note: To represent unknown quantities in known relationships. These are generalized relationships such as, which says for any number, if you square it and take the root, you get the absolute value back.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p25_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 25, "page_end": 25, "topic_guess": "algebra", "text": "GMAT algebra note: So the variable acts like a true placeholder, which may be replaced by any number. Basic rules of manipulation A.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p25_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 25, "page_end": 25, "topic_guess": "algebra", "text": "GMAT algebra note: When switching terms from one side to the other in an algebraic expression + becomes - and vice versa. When switching terms from one side to the other in an algebraic.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p26_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 26, "page_end": 26, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Degree of an expression The degree of an algebraic expression is defined as the highest power of the variables present in the expression.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p26_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 26, "page_end": 26, "topic_guess": "algebra", "text": "GMAT algebra note: Bi-quadratic Egs: the degree is 1 the degree is 3 the degree of x is 3, degree of z is 5, degree of the expression is 5 Solving equations of degree 1:. LINEAR Degree 1 equations or linear equations are equations in one or more variable such that degree of each variable is one.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p26_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 26, "page_end": 26, "topic_guess": "algebra", "text": "GMAT algebra note: One variable Such equations will always have a solution. General form is and solution is One equation in Two variables This is not enough to determine x and y uniquely.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p26_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 26, "page_end": 26, "topic_guess": "algebra", "text": "GMAT algebra note: There can be infinitely many solutions. Two equations in Two variables If you have a linear equation in 2 variables, you need at least 2 equations to solve for both variables.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p26_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 26, "page_end": 26, "topic_guess": "algebra", "text": "GMAT algebra note: If then there are infinite solutions. Any point satisfying one equation will always satisfy the second If then there is no such x and y which will satisfy both equations.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p26_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 26, "page_end": 26, "topic_guess": "algebra", "text": "GMAT algebra note: No solution In all other cases, solving the equations is straight forward, multiply eq (2) by a/d and subtract from (1). More than two equations in Two variables Pick any 2 equations and try to solve them:.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p27_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 27, "page_end": 27, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Solving equations of degree 2:.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p27_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 27, "page_end": 27, "topic_guess": "algebra", "text": "GMAT algebra note: QUADRATIC The general form of a quadratic equation is The equation has no solution if The equation has exactly one solution if This equation has 2 solutions given by if The sum of roots is The product of roots is If the roots are and, the equation can be written as A quick way to solve a quadratic, without the above formula is to factorize it:.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p27_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 27, "page_end": 27, "topic_guess": "algebra", "text": "GMAT algebra note: Step 1> Divide throughout by coeff of x^2 to put it in the form Step 2> Sum of roots = -d and Product = e. Search for 2 numbers which satisfy this criteria, let them be f,g Step 3> The equation may be re-written as (x-f)(x-g) = 0.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p27_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 27, "page_end": 27, "topic_guess": "algebra", "text": "GMAT algebra note: And the solutions are f,g Eg. The sum is -11 and the product is 30.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p27_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 27, "page_end": 27, "topic_guess": "algebra", "text": "GMAT algebra note: So numbers are -5,-6 Solvingequations with DEGREE>2 You will never be asked to solved higher degree equations, except in some cases where using simple tricks these equations can either be factorized or be reduced to a lower degree or both. What you need to note is that an equation of degree n has at most n unique solutions.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p27_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 27, "page_end": 27, "topic_guess": "algebra", "text": "GMAT algebra note: Factorization This is the easiest approach to solving higher degree equations. Though there is no general rule to do this, generally a knowledge of algebraic identities helps.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p27_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 27, "page_end": 27, "topic_guess": "algebra", "text": "GMAT algebra note: The basic idea is that if you can write an equation in the form where each of A,B,C are algebraic expressions.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p28_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 28, "page_end": 28, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. So the solution is y = 1 or 2 or x^3 = 1 or 2 or x = 1 or Other tricks Sometimes we are given conditions such as the variables being integers which make the solutions much easier to find.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p28_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 28, "page_end": 28, "topic_guess": "number_theory", "text": "GMAT number properties note: When we know that the solutions are integral, often times solutions are easy to find using just brute force.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p29_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 29, "page_end": 29, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. ABSOLUTE VALUE (Modulus) Definition The absolute value (or modulus) of a real number x is x's numerical value without regard to its sign.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p29_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 29, "page_end": 29, "topic_guess": "algebra", "text": "GMAT algebra note: How to approach equations with moduli It's not easy to manipulate with moduli in equations. There are two basic approaches that will help you out.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p29_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 29, "page_end": 29, "topic_guess": "algebra", "text": "GMAT algebra note: Both of them are based on two ways of representing modulus as an algebraic expression. This approach might be helpful if an equation has × and /.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p29_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 29, "page_end": 29, "topic_guess": "algebra", "text": "GMAT algebra note: 2) |x| equals x if x> = 0 or -x if x<0. It looks a bit complicated but it's very powerful in dealing with moduli and the most popular approach too (see below).", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p29_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 29, "page_end": 29, "topic_guess": "algebra", "text": "GMAT algebra note: General approach to solving equalities and inequalities with absolute value: 1. Open modulus and set conditions.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p29_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 29, "page_end": 29, "topic_guess": "algebra", "text": "GMAT algebra note: To solve/open a modulus, you need to consider 2 situations to find all roots:. Positive (or rather non-negative) Negative |x| |3| = 3 | − 12| = 12 | − 1.3| = 1.3 |x| ≥ 0.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p30_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 30, "page_end": 30, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. We can also think about conditions like graphics.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p30_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 30, "page_end": 30, "topic_guess": "algebra", "text": "GMAT algebra note: Is a key point in which the expression under modulus equals zero. All points right are the first condition and all points left are second condition.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p30_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 30, "page_end": 30, "topic_guess": "algebra", "text": "GMAT algebra note: Check conditions for each solution: a) has to satisfy initial condition.. Otherwise, we would have to reject x = 5.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p30_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 30, "page_end": 30, "topic_guess": "algebra", "text": "GMAT algebra note: B) has to satisfy initial condition.. Otherwise, we would have to reject x = -3.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p30_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 30, "page_end": 30, "topic_guess": "algebra", "text": "GMAT algebra note: How many solutions does the equation have? There are 3 key points here: -8, -3, 4.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p30_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 30, "page_end": 30, "topic_guess": "algebra", "text": "GMAT algebra note: So we have 4 conditions: a). We reject the solution because our condition is not satisfied (-1 is not less than -8) b).", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p30_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 30, "page_end": 30, "topic_guess": "algebra", "text": "GMAT algebra note: We reject the solution because our condition is not satisfied (-15 is not within (-8,-3) interval.) c). We reject the solution because our condition is not satisfied (9 is not within (-3,4) interval.) d).", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p30_c1_r8", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 30, "page_end": 30, "topic_guess": "algebra", "text": "GMAT algebra note: We reject the solution because our condition is not satisfied (-1 is not more than 4) (Optional) The following illustration may help you understand how to open modulus at different conditions.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p31_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 31, "page_end": 31, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. X e {, } and both solutions satisfy the condition.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p31_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 31, "page_end": 31, "topic_guess": "word_problems", "text": "GMAT word problem note: (Optional) The following illustration may help you understand how to open modulus at different conditions. At the same time, often there are shortcuts and tricks that allow you to solve absolute value problems in 10-20 sec.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p31_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 31, "page_end": 31, "topic_guess": "word_problems", "text": "GMAT word problem note: Thinking of inequality with modulus as a segment at the number line. What inequality represents this condition?", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p31_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 31, "page_end": 31, "topic_guess": "word_problems", "text": "GMAT word problem note: Traditional 3-steps method is too time-consume technique. First of all we find length (9-1) = 8 and center (1+8/2 = 5) of the segment represented by 1<x<9.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p31_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 31, "page_end": 31, "topic_guess": "word_problems", "text": "GMAT word problem note: Now, let’s look at our options. Only B and D has 8/2 = 4 on the right side and D had left site 0 at x = 5.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p31_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 31, "page_end": 31, "topic_guess": "word_problems", "text": "GMAT word problem note: Converting inequalities with modulus into a range expression. In many cases, especially in DS problems, it helps avoid silly mistakes.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p31_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 31, "page_end": 31, "topic_guess": "word_problems", "text": "GMAT word problem note: |x+3|>3 is equal to x e (-inf,-6)&(0,+inf) III. Thinking about absolute values as the distance between points at the number line.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p31_c1_r8", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 31, "page_end": 31, "topic_guess": "word_problems", "text": "GMAT word problem note: We can think of absolute values here as the distance between points.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p32_c1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 32, "page_end": 32, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Https://gmatclub.com/forum/4-272003.html Medium:.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p33_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 33, "page_end": 33, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. It is an ordered list of objects.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p33_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 33, "page_end": 33, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: It can be finite or infinite. The elements may repeat themselves more than once in the sequence, and their ordering is important unlike a set Arithmetic Progressions Definition It is a special type of sequence in which the difference between successive terms is constant.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p33_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 33, "page_end": 33, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: General Term is the ith term is the common difference is the first term Defining Properties Each of the following is necessary & sufficient for a sequence to be an AP:. Constant If you pick any 3 consecutive terms, the middle one is the mean of the other two For all i,j > k > = 1:.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p33_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 33, "page_end": 33, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Summation The sum of an infinite AP can never be finite except if & The general sum of a n term AP with common difference d is given by The sum formula may be re-written as = + d = + (n− 1)dan an−1 a1 ai d a1 − = ai ai−1 −ai ak i−k −aj ak j−k = 0a1 d = 0 (2a+ (n− 1)d)n n∗ Avg(,) = ∗ (FirstTerm + LastTerm)a1 an n.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p34_c1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 34, "page_end": 34, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Constant If you pick any 3 consecutive terms, the middle one is the geometric mean of the other two For all i,j > k > = 1:.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p35_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 35, "page_end": 35, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Track mean, median, range, distribution, and how operations change spread. Important Properties Of any three consecutive terms of a HP, the middle one is always the harmonic mean of the other two, where the harmonic mean (HM) is defined as:.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p35_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 35, "page_end": 35, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Linkage Each progression provides us a definition of \"mean\":. OR For all non-negative real numbers:.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p35_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 35, "page_end": 35, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: AM > = GM > = HM In particular for 2 numbers:.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p35_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 35, "page_end": 35, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The HM = (2*50*2)/(52) = 3.85 AM > GM > HM AM*HM = 100 = GM^2 Misc Notes A subsequence (any set of consequutive terms) of an AP is an AP A subsequence (any set of consequutive terms) of a GP is a GP A subsequence (any set of consequutive terms) of a HP is a HP If given an AP, and I pick out a subsequence from that AP, consisting of the terms such that are in AP then the new subsequence will also be an AP 1) = 2n-1 Pick out the subsequence of terms If given a GP, and I pick out a subsequence from that GP, consisting of the terms such that are in AP then the new subsequence will also be a GP a b HM(a,b) HM(a, b) = 2ab a+b a+b a1+..+an n ab− −√ (a1∗..", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p35_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 35, "page_end": 35, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: ∗an) 1n 2ab a+b n a1 an ,,,...ai1 ai2 ai3 i1, i2, i3 = 1, d = 2a1 ,,,...a5 a10 a15 = 9a1 d = 10 ,,,...bi1 bi2 bi3 i1, i2, i3.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p36_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 36, "page_end": 36, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Pick out the subsequence of terms The special sequence in which each term is the sum of previous two terms is known as the fibonacci sequence.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p36_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 36, "page_end": 36, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: It is neither an AP nor a GP. In a finite AP, the mean of all the terms is equal to the mean of the middle two terms if n is even and the middle term if n is odd.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p36_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 36, "page_end": 36, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Solution We know that P(H) = P(T) = 0.5 This is just a finite GP, with first term = 0.5, n = 5 and ratio = 0.5. (1) a1 = 8 (2) a12 = 24 Solution (1) a1 = 8, does not tell us anything about the common difference, so impossible to say how many terms are greater than 24 (2) a12 = 24, and we know common difference is non-zero.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p36_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 36, "page_end": 36, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: So either all the terms below a12 are greater than 24 and the terms above it less than 24 or the other way around. In either case, there are exactly 11 terms either side of a12.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p37_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 37, "page_end": 37, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. If T is the sum of the first 10 terms in the sequence then T is a)greater than 2 b)between 1 and 2 c)between 1/2 and 1 d)between 1/4 and 1/2 e)less than 1/4.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p37_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 37, "page_end": 37, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Solution The sequence given has first term 1/2 and each subsequent term can be obtained by multiplying with -1/2. What is the sum of the first 15 terms of the arithmetic progression?", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p37_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 37, "page_end": 37, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: 270 Solution Now we need the sum of first 15 terms, which is given by:.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p38_c1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 38, "page_end": 38, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p39_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 39, "page_end": 39, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Algebra Functions & Coordinate Geometry https://gmatclub.com/forum/algebra-functions-87652.html FUNCTIONS & COORDINATE GEOMETRY Definition Coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p39_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 39, "page_end": 39, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: The Coordinate Plane In coordinate geometry, points are placed on the \"coordinate plane\" as shown below. The coordinate plane is a two-dimensional surface on which we can plot points, lines and curves.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p39_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 39, "page_end": 39, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: It has two scales, called the x-axis and y-axis, at right angles to each other. The plural of axis is 'axes' (pronounced \"AXE-ease\").", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p39_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 39, "page_end": 39, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: A point's location on the plane is given by two numbers, one that tells where it is on the x-axis and another which tells where it is on the y-axis. Together, they define a single, unique position on the plane.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p39_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 39, "page_end": 39, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: These are the coordinates of the point A, sometimes referred to as its.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p40_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 40, "page_end": 40, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: X axis The horizontal scale is called the x-axis and is usually drawn with the zero point in the middle. Values to the right are positive and those to the left are negative.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p40_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 40, "page_end": 40, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Y axis The vertical scale is called the y-axis and is also usually drawn with the zero point in the middle. Values above the origin are positive and those below are negative.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p40_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 40, "page_end": 40, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Origin The point where the two axes cross (at zero on both scales) is called the origin. Quadrants When the origin is in the center of the plane, they divide it into four areas called quadrants.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p40_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 40, "page_end": 40, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: The first quadrant, by convention, is the top right, and then they go around counter- clockwise. It is conventional to label them with numerals but we talk about them as \"first, second, third, and fourth quadrant\".", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p40_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 40, "page_end": 40, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Point (x,y) The coordinates are written as an \"ordered pair\". The letter P is simply the name of the point and is used to distinguish it from others.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p40_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 40, "page_end": 40, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: The two numbers in parentheses are the x and y coordinate of the point. The first number (x) specifies how far along the x (horizontal) axis the point is.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p40_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 40, "page_end": 40, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: The second is the y coordinate and specifies how far up or down the y axis to go. It is called an ordered pair because the order of the two numbe.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p41_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 41, "page_end": 41, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Negative numbers mean to go left (x) or down (y). Distance between two points Given coordinates of two points, distance D between two points is given by: (where is the difference between the x-coordinates and is the difference between the y-coordinates of the points) As you can see, the distance formula on the plane is derived from the Pythagorean theorem.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p41_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 41, "page_end": 41, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Above formula can be written in the following way for given two points and Vertical and horizontal lines If the line segment is exactly vertical or horizontal, the formula above will still work fine, but there is an easier way. For a horizontal line, its length is the difference between the x- coordinates.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p41_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 41, "page_end": 41, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: For a vertical line its length is the difference between the y-coordinates. Distance between the point A (x,y) and the origin As the one point is origin with coordinate O (0,0) the formula can be simplified to:.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p41_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 41, "page_end": 41, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Find the distance between the point A (3,-1) and B (-1,2) Solution:. Substituting values in the equation we'll get D = d + dx2 y2− − − − − − − −√ dx dy (,)x1 y1 (,)x2 y2 D = (− + (−x2 x1)2 y2 y1)2− −− − − − − − − − − − − − − − − − − D = +x2 y2− − − − − −√ D = (− + (−x2 x1)2 y2 y1)2− −− − − − − − − − − − − − − − − − −√.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p42_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 42, "page_end": 42, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Midpoint of a Line Segment A line segment on the coordinate plane is defined by two endpoints whose coordinates are known. The midpoint of this line is exactly halfway between these endpoints and it's location can be found using the Midpoint Theorem, which states:.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p42_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 42, "page_end": 42, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: The x-coordinate of the midpoint is the average of the x-coordinates of the two endpoints. Likewise, the y-coordinate is the average of the y-coordinates of the endpoints.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p42_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 42, "page_end": 42, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Coordinates of the midpoint of the line segment AB, (and ) are and Lines in Coordinate Geometry In Euclidean geometry, a line is a straight curve. In coordinate geometry, lines in a Cartesian plane can be described algebraically by linear equations and linear functions.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p42_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 42, "page_end": 42, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Every straight line in the plane can represented by a first degree equation with two variables.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p43_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 43, "page_end": 43, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: There are several approaches commonly used in coordinate geometry. It does not matter whether we are talking about a line, ray or line segment.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p43_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 43, "page_end": 43, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: In all cases any of the below methods will provide enough information to define the line exactly. The general form of the equation of a straight line is Where, and are arbitrary constants.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p43_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 43, "page_end": 43, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: This form includes all other forms as special cases. For an equation in this form the slope is and the y intercept is.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p43_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 43, "page_end": 43, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Where: is the slope of the line. Is the y-intercept of the line.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p43_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 43, "page_end": 43, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Is the independent variable of the function. Using two points In figure below, a line is defined by the two points A and B.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p43_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 43, "page_end": 43, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: By providing the coordinates of the two points, we can draw a line. No other line could pass through both these points and so the line they define is unique.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p43_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 43, "page_end": 43, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Ax+ by+ c = 0 a b c − a b − c b y = mx+ b m b x y.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p44_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 44, "page_end": 44, "topic_guess": "algebra", "text": "GMAT algebra note: The equation of a straight line passing through points and is:. Find the equation of a line passing through the points A (17,4) and B (9,9).", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p44_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 44, "page_end": 44, "topic_guess": "algebra", "text": "GMAT algebra note: Substituting the values in equation we'll get: --> --> OR if we want to write the equation in the slope-intercept form: 4. Using one point and the slope Sometimes on the GMAT you will be given a point on the line and its slope and from this information you will need to find the equation or check if this line goes through another point.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p44_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 44, "page_end": 44, "topic_guess": "algebra", "text": "GMAT algebra note: You can think of the slope as the direction of the line. So once you know that a line goes through a certain point, and which direction it is pointing, you have defined one unique line.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p44_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 44, "page_end": 44, "topic_guess": "algebra", "text": "GMAT algebra note: In figure below, we see a line passing through the point A at (14,23). We also see that it's slope is +2 (which means it goes 2 up for every one across).", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p44_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 44, "page_end": 44, "topic_guess": "algebra", "text": "GMAT algebra note: With these two facts we can establish a unique line. (,)P1 x1 y1 (,)P2 x2 y2 y−y1 x−x1 −y1 y2 −x1 x2 y−y1 x−x1 −y1 y2 −x1 x2 = y−4 x−17 = y−4 x−17 8 8y− 32 = −5 x+ 85 8y+ 5x− 117 = 0 y = − x+5.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p45_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 45, "page_end": 45, "topic_guess": "algebra", "text": "GMAT algebra note: The equation of a straight line that passes through a point with a slope m is:. Find the equation of a line passing through the point A (14,23) and the slope 2.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p45_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 45, "page_end": 45, "topic_guess": "algebra", "text": "GMAT algebra note: Substituting the values in equation we'll get 4. The equation of a straight line whose x and y intercepts are a and b, respectively, is:.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p45_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 45, "page_end": 45, "topic_guess": "algebra", "text": "GMAT algebra note: Find the equation of a line whose x intercept is 5 and y intercept is 2. Substituting the values in equation we'll get --> OR if we want to write the equation in the slope-intercept form:.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p45_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 45, "page_end": 45, "topic_guess": "algebra", "text": "GMAT algebra note: Slope of a Line The slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p45_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 45, "page_end": 45, "topic_guess": "algebra", "text": "GMAT algebra note: The slope is defined as the ratio of the \"rise\" divided by the \"run\" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two points on the line. (,)P1 x1 y1 y− = m(x−)y1 x1 y− = m(x−)y1 x1 y− 23 = 2(x− 14) y = 2 x− 5 + = 1x a y b + = 1x a y b + = 1x y 5y+ 2x− 10 = 0 y = − x+ 22.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p46_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 46, "page_end": 46, "topic_guess": "algebra", "text": "GMAT algebra note: Given two points and on a line, the slope of the line is:. If the equation of the line is given in the Point-intercept form:, then is the slope.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p46_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 46, "page_end": 46, "topic_guess": "algebra", "text": "GMAT algebra note: This form of a line's equation is called the slope-intercept form, because can be interpreted as the y-intercept of the line, the y-coordinate where the line intersects the y-axis. If the equation of the line is given in the General form:, then the slope is and the y intercept is.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p46_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 46, "page_end": 46, "topic_guess": "algebra", "text": "GMAT algebra note: SLOPE DIRECTION The slope of a line can be positive, negative, zero or undefined. (,)x1 y1 (,)x2 y2 m m = −y2 y1 −x2 x1 y = mx+ b m b ax+ by+ c = 0 − a b − c b.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p47_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 47, "page_end": 47, "topic_guess": "algebra", "text": "GMAT algebra note: Positive slope Here, y increases as x increases, so the line slopes upwards to the right. The slope will be a positive number.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p47_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 47, "page_end": 47, "topic_guess": "algebra", "text": "GMAT algebra note: The line below has a slope of about +0.3, it goes up about 0.3 for every step of 1 along the x-axis. Negative slope Here, y decreases as x increases, so the line slopes downwards to the right.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p47_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 47, "page_end": 47, "topic_guess": "algebra", "text": "GMAT algebra note: The slope will be a negative number. The line below has a slope of about -0.3, it goes down about 0.3 for every step of 1 along the x-axis.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p47_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 47, "page_end": 47, "topic_guess": "algebra", "text": "GMAT algebra note: Zero slope Here, y does not change as x increases, so the line in exactly horizontal. The slope of any horizontal line is always zero.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p47_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 47, "page_end": 47, "topic_guess": "algebra", "text": "GMAT algebra note: The line below goes neither up nor down as x increases, so its slope is zero. Undefined slope When the line is exactly vertical, it does not have a defined slope.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p47_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 47, "page_end": 47, "topic_guess": "algebra", "text": "GMAT algebra note: The two x coordinates are the same, so the difference is zero. The slope calculation is then something like When you divide anything by zero the result has no meaning.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p47_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 47, "page_end": 47, "topic_guess": "algebra", "text": "GMAT algebra note: The line above is exactly vertical, so it has no defined slope. If the slope of a line is negative, the line WILL intersect quadrants II and IV.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p47_c1_r8", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 47, "page_end": 47, "topic_guess": "algebra", "text": "GMAT algebra note: X and Y intercepts of the line with negative slope have the same sign. Therefore if X and Y intersects are positive, the line intersects quadrant I.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p48_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 48, "page_end": 48, "topic_guess": "algebra", "text": "GMAT algebra note: If the slope is 1 the angle formed by the line is degrees. Given a point and slope, equation of a line can be found.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p48_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 48, "page_end": 48, "topic_guess": "algebra", "text": "GMAT algebra note: The equation of a straight line that passes through a point with a slope is:. Vertical and horizontal lines A vertical line is parallel to the y-axis of the coordinate plane.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p48_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 48, "page_end": 48, "topic_guess": "algebra", "text": "GMAT algebra note: All points on the line will have the same x-coordinate. A vertical line has no slope.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p48_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 48, "page_end": 48, "topic_guess": "algebra", "text": "GMAT algebra note: Or put another way, for a vertical line the slope is undefined. The equation of a vertical line is:.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p48_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 48, "page_end": 48, "topic_guess": "algebra", "text": "GMAT algebra note: Where: x is the coordinate of any point on the line. A is where the line crosses the x-axis (x intercept).", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p48_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 48, "page_end": 48, "topic_guess": "algebra", "text": "GMAT algebra note: Notice that the equation is independent of y. Any point on the vertical line satisfies the equation.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p48_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 48, "page_end": 48, "topic_guess": "algebra", "text": "GMAT algebra note: A horizontal line is parallel to the x-axis of the coordinate plane. All points on the line will have the same y-coordinate.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p48_c1_r8", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 48, "page_end": 48, "topic_guess": "algebra", "text": "GMAT algebra note: (,)x1 y1 m y− = m(x−)y1 x1 x = a.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p49_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 49, "page_end": 49, "topic_guess": "algebra", "text": "GMAT algebra note: A horizontal line has a slope of zero. The equation of a horizontal line is:.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p49_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 49, "page_end": 49, "topic_guess": "algebra", "text": "GMAT algebra note: Where: y is the coordinate of any point on the line. B is where the line crosses the y-axis (y intercept).", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p49_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 49, "page_end": 49, "topic_guess": "algebra", "text": "GMAT algebra note: Notice that the equation is independent of x. Any point on the horizontal line satisfies the equation.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p49_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 49, "page_end": 49, "topic_guess": "algebra", "text": "GMAT algebra note: Parallel lines Parallel lines have the same slope. The slope can be found using any method that is convenient to you: y = b.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p50_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 50, "page_end": 50, "topic_guess": "algebra", "text": "GMAT algebra note: From two given points on the line. From the equation of the line in slope-intercept form From the equation of the line in point-slope form The equation of a line through the point and parallel to line is:.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p50_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 50, "page_end": 50, "topic_guess": "algebra", "text": "GMAT algebra note: Distance between two parallel lines and can be found by the formula:. One line is defined by two points at (5,5) and (25,15).", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p50_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 50, "page_end": 50, "topic_guess": "algebra", "text": "GMAT algebra note: The other is defined by an equation in slope-intercept form form y = 0.52x - 2.5. For the top line, the slope is found using the coordinates of the two points that define the line.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p50_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 50, "page_end": 50, "topic_guess": "algebra", "text": "GMAT algebra note: For the lower line, the slope is taken directly from the formula. Recall that the slope intercept formula is y = mx + b, where m is the slope.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p50_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 50, "page_end": 50, "topic_guess": "algebra", "text": "GMAT algebra note: So looking at the formula we see that the slope is 0.52. So, the top one has a slope of 0.5, the lower slope is 0.52, which are not equal.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p50_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 50, "page_end": 50, "topic_guess": "algebra", "text": "GMAT algebra note: Therefore, the lines are not parallel. Define a line through a point C parallel to a line passes through the points A and B.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p50_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 50, "page_end": 50, "topic_guess": "algebra", "text": "GMAT algebra note: (,)P1 x1 y1 ax+ by+ c = 0 a(x−) + b(y−) = 0x1 y1 y = mx+ b y = mx+ c D = |b−c| Slope = = 0.515−5.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p51_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 51, "page_end": 51, "topic_guess": "algebra", "text": "GMAT algebra note: For the line to be parallel to AB it will have the same slope, and will pass through a given point, C(12,10). We therefore have enough information to define the line by it's equation in point-slope form form:.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p51_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 51, "page_end": 51, "topic_guess": "algebra", "text": "GMAT algebra note: Perpendicular lines For one line to be perpendicular to another, the relationship between their slopes has to be negative reciprocal. In other words, the two lines are perpendicular if and only if the product of their slopes is.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p51_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 51, "page_end": 51, "topic_guess": "algebra", "text": "GMAT algebra note: SlopeAB = = −0.5220−7 y = −0.52(x− 12) + 10 y = −0.52x + 16.24 m.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p52_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 52, "page_end": 52, "topic_guess": "algebra", "text": "GMAT algebra note: The two lines and are perpendicular if The equation of a line passing through the point) and perpendicular to line is:. Are the two lines below perpendicular?", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p52_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 52, "page_end": 52, "topic_guess": "algebra", "text": "GMAT algebra note: To answer, we must find the slope of each line and then check to see if one slope is the negative reciprocal of the other or if their product equals to -1. X+ y+ = 0a1 b1 c1 x+ y+ = 0a2 b2 c2 + = 0a1a2 b1b2 (,P1 x1 y1 ax+ by+ c = 0 b(x−) − a(y−) = 0x1 y1 SlopeAB = = = 0.3585−19.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p53_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 53, "page_end": 53, "topic_guess": "algebra", "text": "GMAT algebra note: If the lines are perpendicular, each will be the negative reciprocal of the other. It doesn't matter which line we start with, so we will pick AB:.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p53_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 53, "page_end": 53, "topic_guess": "algebra", "text": "GMAT algebra note: Negative reciprocal of 0.358 is So, the slope of CD is -2.22, and the negative reciprocal of the slope of AB is -2.79. These are not the same, so the lines are not perpendicular, even though they may look as though they are.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p53_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 53, "page_end": 53, "topic_guess": "algebra", "text": "GMAT algebra note: Define a line passing through the point E and perpendicular to a line passing through the points C and D on the graph above. The point E is on the y-axis and so is the y-intercept of the desired line.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p53_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 53, "page_end": 53, "topic_guess": "algebra", "text": "GMAT algebra note: Once we know the slope of the line, we can express it using its equation in slope-intercept form y = mx+b, where m is the slope and b is the y-intercept. First find the slope of line CD:.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p53_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 53, "page_end": 53, "topic_guess": "algebra", "text": "GMAT algebra note: The line we seek will have a slope which is the negative reciprocal of:. Since E is on the Y-axis, we know that the intercept is 10.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p53_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 53, "page_end": 53, "topic_guess": "algebra", "text": "GMAT algebra note: Plugging these values into the line equation, the line we need is described by the equation This is one of the ways a line can be defined and so we have solved the problem. If we wanted to plot the line, we would find anoth.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p54_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 54, "page_end": 54, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. To find the intersection of two straight lines: 1.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p54_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 54, "page_end": 54, "topic_guess": "algebra", "text": "GMAT algebra note: First we need their equations 2. Then, since at the point of intersection, the two equations will share a point and thus have the same values of x and y, we set the two equations equal to each other.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p54_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 54, "page_end": 54, "topic_guess": "algebra", "text": "GMAT algebra note: This gives an equation that we can solve for x 3. We substitute the x value in one of the line equations (it doesn't matter which) and solve it for y.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p54_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 54, "page_end": 54, "topic_guess": "algebra", "text": "GMAT algebra note: This gives us the x and y coordinates of the intersection. Find the point of intersection of two lines that have the following equations (in slope- intercept form):.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p54_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 54, "page_end": 54, "topic_guess": "algebra", "text": "GMAT algebra note: At the point of intersection they will both have the same y-coordinate value, so we set the equations equal to each other:. This gives us an equation in one unknown (x) which we can solve:.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p54_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 54, "page_end": 54, "topic_guess": "algebra", "text": "GMAT algebra note: To find y, simply set x equal to 10 in the equation of either line and solve for y:. Equation for a line (Either line will do) Set x equal to 10:.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p55_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 55, "page_end": 55, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Intercept form): and (A vertical line) Solution:. When one of the lines is vertical, it has no defined slope.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p55_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 55, "page_end": 55, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: We find the intersection slightly differently. On the vertical line, all points on it have an x-coordinate of 12 (the definition of a vertical line), so we simply set x equal to 12 in the first equation and solve it for y.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p55_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 55, "page_end": 55, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Equation for a line Set x equal to 12 So the intersection point is at (12,33). If both lines are vertical or horizontal, they are parallel and have no intersection Distance from a point to a line The distance from a point to a line is the shortest distance between them - the length of a perpendicular line segment from the line to the point.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p55_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 55, "page_end": 55, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: The distance from a point to a line is given by the formula:. When the line is horizontal the formula transforms to:.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p55_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 55, "page_end": 55, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Where: is the y-coordinate of the given point P. Is the y-coordinate of any point on the given vertical line L.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p55_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 55, "page_end": 55, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: | | the vertical bars mean \"absolute value\" - make it positive even if it calculates to a negative. When the line is vertical the formula transforms to:.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p55_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 55, "page_end": 55, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Where: is the x-coordinate of the given point P. Is the x-coordinate of any point on the given vertical line L.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p56_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 56, "page_end": 56, "topic_guess": "geometry", "text": "GMAT geometry note: If the circle is centered at the origin (0, 0), then the equation simplifies to:. Number line A number line is a picture of a straight line on which every point corresponds to a real number and every real number to a point.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p56_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 56, "page_end": 56, "topic_guess": "geometry", "text": "GMAT geometry note: On the GMAT we can often see such statement: is halfway between and on the number line. Remember this statement can ALWAYS be expressed as:.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p56_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 56, "page_end": 56, "topic_guess": "geometry", "text": "GMAT geometry note: Also on the GMAT we can often see another statement:. The distance between and on the number line is the same as the distance between and.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p56_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 56, "page_end": 56, "topic_guess": "geometry", "text": "GMAT geometry note: Parabola A parabola is the graph associated with a quadratic function, i.e. A function of the form + = x2 y2 r2 k m n = km+n p m p n |p− m| = |p − n|.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p57_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 57, "page_end": 57, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Focus on equations, expressions, variables, and step-by-step manipulation. The general or standard form of a quadratic function is, or in function form,, where is the independent variable, is the dependent variable, and,, and are constants.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p57_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 57, "page_end": 57, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: The larger the absolute value of, the steeper (or thinner) the parabola is, since the value of y is increased more quickly. If is positive, the parabola opens upward, if negative, the parabola opens downward.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p57_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 57, "page_end": 57, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: The x-intercepts, if any, are also called the roots of the function. The x- intercepts are the solutions to the equation and can be calculated by the formula: and Expression is called discriminant:.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p57_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 57, "page_end": 57, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: If discriminant is positive parabola has two intercepts with x-axis. If discriminant is negative parabola has no intercepts with x-axis.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p57_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 57, "page_end": 57, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: If discriminant is zero parabola has one intercept with x-axis (tangent point). Given, the y-intercept is, as y intercept means the value of y when x = 0.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p57_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 57, "page_end": 57, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: The vertex represents the maximum (or minimum) value of the function, and is very important in calculus. Y = a + bx+ cx2 y = a + bx+ cx2 f(x) = a + bx+ cx2 x y a b c a a 0 = a + bx+ cx2 = x1 2a = x2 2a − 4acb2 y = a + bx+ cx2 c.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p58_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 58, "page_end": 58, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: The vertex of the parabola is located at point. Note: typically just is calculated and plugged in for x to find y.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p58_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 58, "page_end": 58, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Hard: (−,b 2a c−)b2 4a −,b 2a.", "topic": "geometry"}
{"id": "GMAT Club Math Book 2024 v8_p59_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 59, "page_end": 59, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. It deals primarily with the translation of word problems into equations.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p59_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 59, "page_end": 59, "topic_guess": "word_problems", "text": "GMAT word problem note: Discussions relating to specific types of word problems will be dealt with separately (see end of post). The Following Points Outline a General Approach to Word Problems: 1) Read the entire question carefully and get a feel for what is happening.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p59_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 59, "page_end": 59, "topic_guess": "word_problems", "text": "GMAT word problem note: Identify what kind of word problem you're up against. 2) Make a note of exactly what is being asked.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p59_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 59, "page_end": 59, "topic_guess": "word_problems", "text": "GMAT word problem note: 3) Simplify the problem - this is what is usually meant by 'translating the English to Math'. Sometimes a simple illustration makes the problem much easier to approach.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p59_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 59, "page_end": 59, "topic_guess": "word_problems", "text": "GMAT word problem note: 4) It is not always necessary to start from the first line. Invariably, you will find it easier to define what you have been asked for and then work backwards to get the information that is needed to obtain the answer.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p59_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 59, "page_end": 59, "topic_guess": "word_problems", "text": "GMAT word problem note: 5) Use variables (a, b, x, y, etc.) or numbers (100 in case of percentages, any common multiple in case of fractions, etc.) depending on the situation. Think for a moment and choose the best possible value that would help.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p60_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 60, "page_end": 60, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. 7) Once you have the equations written down it's time to do the math!", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p60_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 60, "page_end": 60, "topic_guess": "word_problems", "text": "GMAT word problem note: This is usually quite simple. Be very careful so as not to make any silly mistakes in calculations.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p60_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 60, "page_end": 60, "topic_guess": "word_problems", "text": "GMAT word problem note: 8) Lastly, after solving, cross check to see that the answer you have obtained corresponds to what was asked. The makers of these GMAT questions love to trick students who don’t pay careful attention to what is being asked.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p60_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 60, "page_end": 60, "topic_guess": "word_problems", "text": "GMAT word problem note: My recommendation is to refer to them only in the initial phases of study. With practice, decoding a word problem should come naturally.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p60_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 60, "page_end": 60, "topic_guess": "word_problems", "text": "GMAT word problem note: If, on test day, you still have to try and remember what the math translations to some English term is, you haven’t practiced enough! Plus SUBTRACTION: decreased by.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p60_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 60, "page_end": 60, "topic_guess": "word_problems", "text": "GMAT word problem note: Subtracted from MULTIPLICATION: of. Increased/decreased by a factor of (this type can.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p61_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 61, "page_end": 61, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. The ‘how much is left’ construction Sometimes, the question will give you a total amount that is made up of a number of smaller amounts of unspecified sizes.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p61_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 61, "page_end": 61, "topic_guess": "word_problems", "text": "GMAT word problem note: In this case, just assign a variable to the unknown amounts and the remaining amount will be what is left after deducting this named amount from the total. A hundred-pound order of animal feed was filled by mixing products from Bins A, B and C, and that twice as much was added from Bin C as from Bin A.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p61_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 61, "page_end": 61, "topic_guess": "word_problems", "text": "GMAT word problem note: Then the amount from Bin C was \"2a\", and the amount taken from Bin B was the remaining portion of the hundred pounds: 100 – a – 2a. The length of a rectangular garden is 2 meters more than its width.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p61_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 61, "page_end": 61, "topic_guess": "word_problems", "text": "GMAT word problem note: Express its length in terms of its width. Key words: more than (implies addition).", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p61_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 61, "page_end": 61, "topic_guess": "word_problems", "text": "GMAT word problem note: Is (implies equal to) Thus, t.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p62_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 62, "page_end": 62, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Focus on equations, expressions, variables, and step-by-step manipulation. The ratio of the length of a rectangular garden to its width is 2.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p62_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 62, "page_end": 62, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Express its length in terms of its width. Key words: ratio of (implies division).", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p62_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 62, "page_end": 62, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Is (implies equal to) Thus, the phrase ‘ratio of length to width is 2’ becomes:. The length of a rectangular garden surrounded by a walkway is twice its width.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p62_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 62, "page_end": 62, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: If difference between the length and width of just the rectangular garden is 10 meters, what will be the width of the walkway if just the garden has width 6 meters? Key words: and (implies addition).", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p62_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 62, "page_end": 62, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Twice (implies multiplication). Difference between (implies subtraction where order is important).", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p62_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 62, "page_end": 62, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Is, will be (imply equal to) Since this is a slightly more complicated problem, let us first define what we want. 'What will be the width of the walkway' implies that we should assign a variable for width of the walkway and find its value.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p62_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 62, "page_end": 62, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Thus, let width of the walkway be ‘x’. Now, in order to find the width of walkway, we need to have some relation between the total length/width of the rectangular garden + walkway and the length/width of just the gard.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p63_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 63, "page_end": 63, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Now let’s see what the question gives us.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p63_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 63, "page_end": 63, "topic_guess": "algebra", "text": "GMAT algebra note: ‘Garden with width 6 meters’ translates to:. ‘Length of a rectangular garden surrounded by walkway is twice its width’ translates to:.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p63_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 63, "page_end": 63, "topic_guess": "algebra", "text": "GMAT algebra note: Length of garden + length of walkway = 2*(width of garden + width of walkway) L = 2*W ‘Difference between the length and width of just the rectangular garden is 10 meters’ translates to:. Length of garden – width of garden = 10 (L – 2x) – (W – 2x) = 10 L – W = 10 Now, since we have two equations and two variables (L and W), we can find their values.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p63_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 63, "page_end": 63, "topic_guess": "algebra", "text": "GMAT algebra note: With practice, writing out word problems in the form of equations will become second nature. How much you need to practice depends on your own individual ability.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p63_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 63, "page_end": 63, "topic_guess": "algebra", "text": "GMAT algebra note: It could be 10 questions or it could be 100. But once you’re able to effortlessly translate word problems into equations, more than half your battle will already be won.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p64_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 64, "page_end": 64, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. In case you are not familiar with translating word problems into equations please go through THIS POST FIRST.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p64_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 64, "page_end": 64, "topic_guess": "word_problems", "text": "GMAT word problem note: What is a ‘Work’ Word Problem? It involves a number of people or machines working together to complete a task.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p64_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 64, "page_end": 64, "topic_guess": "word_problems", "text": "GMAT word problem note: We are usually given individual rates of completion. We are asked to find out how long it would take if they work together.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p64_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 64, "page_end": 64, "topic_guess": "word_problems", "text": "GMAT word problem note: Sounds simple enough doesn’t it? There is just one simple concept you need to understand in order to solve any ‘work’ related word problem.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p64_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 64, "page_end": 64, "topic_guess": "word_problems", "text": "GMAT word problem note: The ‘Work’ Problem Concept STEP 1:. Calculate how much work each person/machine does in one unit of time (could be days, hours, minutes, etc).", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p64_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 64, "page_end": 64, "topic_guess": "word_problems", "text": "GMAT word problem note: If we are given that A completes a certain amount of work in X hours, simply reciprocate the number of hours to get the per hour work. Thus in one hour, A would complete of the work.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p64_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 64, "page_end": 64, "topic_guess": "word_problems", "text": "GMAT word problem note: But what is the logic behind this? Assume we are given that Jack paints a wall in 5 hours.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p64_c1_r8", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 64, "page_end": 64, "topic_guess": "word_problems", "text": "GMAT word problem note: This means that in every hour, he completes a fraction of the work so that at the end of 5 hours, the fraction of work he has completed will becom.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p65_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 65, "page_end": 65, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Completed will be (1*1)/5 STEP 2:.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p65_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 65, "page_end": 65, "topic_guess": "word_problems", "text": "GMAT word problem note: Add up the amount of work done by each person/machine in that one unit of time. This would give us the total amount of work completed by both of them in one hour.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p65_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 65, "page_end": 65, "topic_guess": "word_problems", "text": "GMAT word problem note: Calculate total amount of time taken for work to be completed when all persons/machines are working together. The logic is similar to one we used in STEP 1, the only difference being that we use it in reverse order.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p65_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 65, "page_end": 65, "topic_guess": "word_problems", "text": "GMAT word problem note: This means that in one hour, A and B working together will complete of the work. Therefore, working together, they will complete the work in Z hours.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p65_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 65, "page_end": 65, "topic_guess": "word_problems", "text": "GMAT word problem note: DON'T go about these problems trying to remember some formula. Once you understand the logic underlying the above steps, you will have all the information you need to solve any ‘work’ related word problem.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p65_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 65, "page_end": 65, "topic_guess": "word_problems", "text": "GMAT word problem note: (You will see that the formula you might have come across can be very easily and logically deduced from this concept). Now, lets go through a few problems so that the above-mentioned concept becomes crystal clear.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p65_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 65, "page_end": 65, "topic_guess": "word_problems", "text": "GMAT word problem note: Lets start off with a simple one:. Jack can paint a wall in 3 hours.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p65_c1_r8", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 65, "page_end": 65, "topic_guess": "word_problems", "text": "GMAT word problem note: John can do the same job in 5 hours.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p66_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 66, "page_end": 66, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Working, independently X takes 12 hours to finish a certain work.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p66_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 66, "page_end": 66, "topic_guess": "word_problems", "text": "GMAT word problem note: He finishes 2/3 of the work. The rest of the work is finished by Y whose rate is 1/10 of X.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p66_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 66, "page_end": 66, "topic_guess": "word_problems", "text": "GMAT word problem note: In how much time does Y finish his work? Now the only reason this is trickier than the first problem is because the sequence of events are slightly more complicated.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p66_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 66, "page_end": 66, "topic_guess": "word_problems", "text": "GMAT word problem note: The concept however is the same. So if our understanding of the concept is clear, we should have no trouble at all dealing with this.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p66_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 66, "page_end": 66, "topic_guess": "word_problems", "text": "GMAT word problem note: ‘Working, independently X takes 12 hours to finish a certain work’ This statement tells us that in one hour, X will finish of the work. ‘He finishes 2/3 of the work’ This tells us that of the work still remains.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p66_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 66, "page_end": 66, "topic_guess": "word_problems", "text": "GMAT word problem note: ‘The rest of the work is finished by Y whose rate is (1/10) of X’ Y has to complete of the work. ‘Y's rate is (1/10) that of X‘.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p66_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 66, "page_end": 66, "topic_guess": "word_problems", "text": "GMAT word problem note: We have already calculated rate at which X works to be. Therefore, rate at which Y works is.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p66_c1_r8", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 66, "page_end": 66, "topic_guess": "word_problems", "text": "GMAT word problem note: ‘In how much time does Y finish his work?’ If Y completes of the work in 1 hour, then he will complete of the work in 40 hours. So as you can see, even though the question might have been a little difficult to follow at first reading, the solution was in fact quite simple.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p66_c1_r9", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 66, "page_end": 66, "topic_guess": "word_problems", "text": "GMAT word problem note: We didn’t use any new concepts. All we did was apply our knowledge of the concept we learnt earlier to.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p67_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 67, "page_end": 67, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Calculate the rate at which B works:.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p67_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 67, "page_end": 67, "topic_guess": "word_problems", "text": "GMAT word problem note: ‘B prints 5 pages a minute more than printer A’ This means that the difference between the amount of work B and A complete in one minute corresponds to 5 pages. So, let us calculate that difference.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p67_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 67, "page_end": 67, "topic_guess": "word_problems", "text": "GMAT word problem note: It will be ‘How many pages does the task contain?’ If of the job consists of 5 pages, then the 1 job will consist of pages. Machine A and Machine B are used to manufacture 660 sprockets.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p67_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 67, "page_end": 67, "topic_guess": "word_problems", "text": "GMAT word problem note: It takes machine A ten hours longer to produce 660 sprockets than machine B. Machine B produces 10% more sprockets per hour than machine A.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p67_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 67, "page_end": 67, "topic_guess": "word_problems", "text": "GMAT word problem note: How many sprockets per hour does machine A produce? The rate of A is sprockets per hour.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p67_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 67, "page_end": 67, "topic_guess": "word_problems", "text": "GMAT word problem note: The rate of B is sprockets per hour. We are told that B produces 10% more sprockets per hour than A, thus --> --> the rate of A is sprockets per hour.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p67_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 67, "page_end": 67, "topic_guess": "word_problems", "text": "GMAT word problem note: As you can see, the main reason the 'tough' problems are 'tough' is because they test a number of other concepts apart from just the ‘work’ concept. However, once you manage to form the equations, they are really not all that tough.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p67_c1_r8", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 67, "page_end": 67, "topic_guess": "word_problems", "text": "GMAT word problem note: And as far as the concept of ‘work’ word problems is concerned – it is always the same!", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p67_c2_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 67, "page_end": 67, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Age to form the equations, they are really not all that tough.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p67_c2_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 67, "page_end": 67, "topic_guess": "algebra", "text": "GMAT algebra note: And as far as the concept of ‘work’ word problems is concerned – it is always the same! T+10 t t+10 t t = 100 = 6660 t+10.", "topic": "algebra"}
{"id": "GMAT Club Math Book 2024 v8_p68_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 68, "page_end": 68, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. In case you are not familiar with translating word problems into equations please go through this post first: https://gmatclub.com/forum/word-problem...", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p68_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 68, "page_end": 68, "topic_guess": "word_problems", "text": "GMAT word problem note: Html What is a ‘D/S/T’ Word Problem? Usually involve something/someone moving at a constant or average speed.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p68_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 68, "page_end": 68, "topic_guess": "word_problems", "text": "GMAT word problem note: Out of the three quantities (speed/distance/time), we are required to find one. Information regarding the other two will be provided in the question stem.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p68_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 68, "page_end": 68, "topic_guess": "word_problems", "text": "GMAT word problem note: Distance = Speed x Time I’m sure most of you are already familiar with the above formula (or some variant of it). But how many of you truly understand what it signifies?", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p68_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 68, "page_end": 68, "topic_guess": "word_problems", "text": "GMAT word problem note: When you see a ‘D/S/T’ question, do you blindly start plugging values into the formula without really understanding the logic behind it? If then answer to that question is yes, then you would probably have noticed that your accuracy isn’t quite where you’d want it to be.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p68_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 68, "page_end": 68, "topic_guess": "word_problems", "text": "GMAT word problem note: My advice here, as usual, is to make sure you understand the concept behind the formula rather than just using it blindly. The Distance = Speed x Time formula is just a way of s.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p69_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 69, "page_end": 69, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Travel for 2 hours at that speed, you would have traveled 100 miles.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p69_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 69, "page_end": 69, "topic_guess": "word_problems", "text": "GMAT word problem note: 3 hours would be 150 miles, etc. If you were to double the speed, then you would have traveled 100 miles in the first hour and 200 miles at the end of the second hour.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p69_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 69, "page_end": 69, "topic_guess": "word_problems", "text": "GMAT word problem note: We can figure out any one of the components by knowing the other two. Similarly, if a friend visits you from 100 miles away and tells you that it took him 4 hours to reach, you will know that he AVERAGED 25 mph.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p69_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 69, "page_end": 69, "topic_guess": "word_problems", "text": "GMAT word problem note: All calculations depend on AVERAGE SPEED. Supposing your friend told you that he was stuck in traffic along the way and that he traveled at 50 mph whenever he could move.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p69_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 69, "page_end": 69, "topic_guess": "word_problems", "text": "GMAT word problem note: Therefore, although practically he never really traveled at 25 mph, you can see how the standstills due to traffic caused his average to reduce. Now, if you think about it, from the information given, you can actually tell how long he was driving and how long he was stuck due to traffic (assuming.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p69_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 69, "page_end": 69, "topic_guess": "word_problems", "text": "GMAT word problem note: What is false but what they never worry about in these problems. That he was either traveling at 50 mph or 0 mph).", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p69_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 69, "page_end": 69, "topic_guess": "word_problems", "text": "GMAT word problem note: If he was traveling constantly at 50 mph, he should have reached in 2 hours.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p70_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 70, "page_end": 70, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. About how to use the formula.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p70_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 70, "page_end": 70, "topic_guess": "word_problems", "text": "GMAT word problem note: So the trick is to be able to understand EXACTLY what they are giving you and EXACTLY what it is that is missing, but you do that from thinking, not from the formula, because the formula only works for the COMPONENTS of any trip where you are going an average speed for a certain amount of time.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p70_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 70, "page_end": 70, "topic_guess": "word_problems", "text": "GMAT word problem note: ONCE the conditions deal with different speeds or different times, you have to look at each of those components and how they go together. And that can be very difficult if you are not methodical in how you think about the components and how they go together.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p70_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 70, "page_end": 70, "topic_guess": "word_problems", "text": "GMAT word problem note: The formula doesn't tell you which components you need to look at and how they go together. For that, you need to think, and the thinking is not always as easy or straightforward as it seems like it ought to be.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p70_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 70, "page_end": 70, "topic_guess": "word_problems", "text": "GMAT word problem note: In the case of your friend above, if we call the time he spent driving 50 mph, T1. Then the time he spent standing still is (4 - T1) hours, since the whole trip took 4 hours.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p70_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 70, "page_end": 70, "topic_guess": "word_problems", "text": "GMAT word problem note: So we have 100 miles = (50 mph x T1) + (0 mph x [4 - T1]) which is equivalent then to: 100 miles = 50 mph x T1 So, T1 will equal 2 hours. And, since the time he spent going zero is (4 - 2), it also turns out to be 2 hours.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p70_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 70, "page_end": 70, "topic_guess": "word_problems", "text": "GMAT word problem note: Sometimes the right answers will seem counter-intuitive, so it is really important to think ab.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p71_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 71, "page_end": 71, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. In the next section we will learn how to do just that.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p71_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 71, "page_end": 71, "topic_guess": "word_problems", "text": "GMAT word problem note: Resolving the Components When you first start out with these problems, the best way to approach them is by organizing the data in a tabular form. Use a separate column each for distance, speed and time and a separate row for the different components involved (2 parts of a journey, different moving objects, etc.).", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p71_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 71, "page_end": 71, "topic_guess": "word_problems", "text": "GMAT word problem note: The last row should represent total distance, total time and average speed for these values (although there might be no need to calculate these values if the question does not require them). Assign a variable for any unknown quantity.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p71_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 71, "page_end": 71, "topic_guess": "word_problems", "text": "GMAT word problem note: If there is more than one unknown quantity, do not blindly assign another variable to it. Look for ways in which you can express that quantity in terms of the quantities already present.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p71_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 71, "page_end": 71, "topic_guess": "word_problems", "text": "GMAT word problem note: Assign another variable to it only if this is not possible. In each row, the quantities of distance, speed and time will always satisfy d = s*t.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p71_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 71, "page_end": 71, "topic_guess": "word_problems", "text": "GMAT word problem note: The distance and time column can be added to give you the values of total distance and total time but you CANNOT add the speeds. If you drive 20 mph on one street, and 40 mph on another street, does that mean you averaged 60 mph?", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p71_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 71, "page_end": 71, "topic_guess": "word_problems", "text": "GMAT word problem note: Once the table is ready, form the equations and solve for what has been asked!", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p72_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 72, "page_end": 72, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Make sure that the units for time and distance agree with the units for the rate.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p72_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 72, "page_end": 72, "topic_guess": "word_problems", "text": "GMAT word problem note: For instance, if they give you a rate of feet per second, then your time must be in seconds and your distance must be in feet. Sometimes they try to trick you by using the wrong units, and you have to catch this and convert to the correct units.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p72_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 72, "page_end": 72, "topic_guess": "word_problems", "text": "GMAT word problem note: A Few More Points to Note Motion in Same Direction (Overtaking):. The first thing that should strike you here is that at the time of overtaking, the distances traveled by both will be the same.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p72_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 72, "page_end": 72, "topic_guess": "word_problems", "text": "GMAT word problem note: Motion in Opposite Direction (Meeting):. The first thing that should strike you here is that if they start at the same time (which they usually do), then at the point at which they meet, the time will be the same.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p72_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 72, "page_end": 72, "topic_guess": "word_problems", "text": "GMAT word problem note: In addition, the total distance traveled by the two objects under consideration will be equal to the sum of their individual distances traveled. The key thing here is that the distance going and coming back is the same.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p72_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 72, "page_end": 72, "topic_guess": "word_problems", "text": "GMAT word problem note: All values in black have been given in the question stem. All values in blue have been calculated.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p72_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 72, "page_end": 72, "topic_guess": "word_problems", "text": "GMAT word problem note: To qualify for a race, you need to average 60 mph driving two laps around a 1.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p73_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 73, "page_end": 73, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. In the first row, we are given the distance and the speed.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p73_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 73, "page_end": 73, "topic_guess": "word_problems", "text": "GMAT word problem note: Thus it is possible to calculate the time. Time(1) = Distance(1)/Speed(1) = 1/30 In the second row, we are given just the distance.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p73_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 73, "page_end": 73, "topic_guess": "word_problems", "text": "GMAT word problem note: Since we have to calculate speed, let us give it a variable 'x'. Now, by using the 'D/S/T' relationship, time can also be expressed in terms of 'x'.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p73_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 73, "page_end": 73, "topic_guess": "word_problems", "text": "GMAT word problem note: Time(2) = Distance(2)/Speed(2) = 1/x In the third row, we know that the total distance is 2 miles (by taking the sum of the distances in row 1 and 2) and that the average speed should be 60 mph. Thus we can calculate the total time that the two laps should take.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p73_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 73, "page_end": 73, "topic_guess": "word_problems", "text": "GMAT word problem note: Time(3) = Distance(3)/Speed(3) = 2/60 = 1/30 Now, we know that the total time should be the sum of the times in row 1 and 2. Thus we can form the following equation:.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p73_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 73, "page_end": 73, "topic_guess": "word_problems", "text": "GMAT word problem note: Time(3) = Time(1) + Time(2) ---> 1/30 = 1/30 + 1/x From this, it becomes clear that '1/x' must be 0. Since 'x' is the reciprocal of 0, which does not exist, there can be no speed for which the average can be made up in the second lap.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p73_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 73, "page_end": 73, "topic_guess": "word_problems", "text": "GMAT word problem note: An executive drove from home at an average speed of 30 mph to an airport where a helicopter was waiting. The executive boarded the helicopter and flew to the corporate offices at an average speed of 60 mph.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p73_c1_r8", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 73, "page_end": 73, "topic_guess": "word_problems", "text": "GMAT word problem note: The entire distance was 150 miles. The entire trip took three hours.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p74_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 74, "page_end": 74, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Thus, the distance he spent driving will be '150 - x' Now, in the first row, we have the distance in terms of 'x' and we have been given the speed.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p74_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 74, "page_end": 74, "topic_guess": "word_problems", "text": "GMAT word problem note: Thus we can calculate the time he spent driving in terms of 'x'. Time(1) = Distance(1)/Speed(1) = (150 - x)/30 Similarly, in the second row, we again have the distance in terms of 'x' and we have been given the speed.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p74_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 74, "page_end": 74, "topic_guess": "word_problems", "text": "GMAT word problem note: Thus we can calculate the time he spent flying in terms of 'x'. Time(2) = Distance(2)/Speed(2) = x/60 Now, notice that we have both the times in terms of 'x'.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p74_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 74, "page_end": 74, "topic_guess": "word_problems", "text": "GMAT word problem note: Also, we know the total time for the trip. Thus, summing the individual times spent driving and flying and equating it to the total time, we can solve for 'x'.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p74_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 74, "page_end": 74, "topic_guess": "word_problems", "text": "GMAT word problem note: Time(1) + Time(2) = Time(3) --> (150 - x)/30 + x/60 = 3 --> x = 120 miles Answer: 120 miles Note:. In this problem, we did not calculate average speed for row 3 since we did not need it.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p74_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 74, "page_end": 74, "topic_guess": "word_problems", "text": "GMAT word problem note: Remember not to waste time in useless calculations! A passenger train leaves the train depot 2 hours after a freight train left the same depot.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p74_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 74, "page_end": 74, "topic_guess": "word_problems", "text": "GMAT word problem note: The freight train is traveling 20 mph slower than the passenger train. Find the speed of the passenger train, if it overtakes the freight train in three hours.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p74_c1_r8", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 74, "page_end": 74, "topic_guess": "word_problems", "text": "GMAT word problem note: Since this is an 'overtaking' problem, the first thing that should strike us is that the dist.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p75_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 75, "page_end": 75, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. From the second row, we can form the following equation:.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p75_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 75, "page_end": 75, "topic_guess": "word_problems", "text": "GMAT word problem note: Distance(2) = Speed(2) * Time(2) = (x - 20)*5 Now, equating the distances because they are equal we get the following equation: 3*x = 5*(x - 20) --> x = 50 mph. Two cyclists start at the same time from opposite ends of a course that is 45 miles long.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p75_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 75, "page_end": 75, "topic_guess": "word_problems", "text": "GMAT word problem note: One cyclist is riding at 14 mph and the second cyclist is riding at 16 mph. How long after they begin will they meet?", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p75_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 75, "page_end": 75, "topic_guess": "word_problems", "text": "GMAT word problem note: Since this is a 'meeting' problem, there are two things that should strike you. First, since they are starting at the same time, when they meet, the time for which both will have been cycling will be the same.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p75_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 75, "page_end": 75, "topic_guess": "word_problems", "text": "GMAT word problem note: Second, the total distance traveled by the will be equal to the sum of their individual distances. Since we are asked to find the time, let us assign it as a variable 't'.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p75_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 75, "page_end": 75, "topic_guess": "word_problems", "text": "GMAT word problem note: (which is same for both cyclists) In the first row, we know the speed and we have the time in terms of 't'. Thus we can get the following equation:.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p75_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 75, "page_end": 75, "topic_guess": "word_problems", "text": "GMAT word problem note: Distance(1) = Speed(1) * Time(1) = 14*t In the second row, we know the speed and again we have the time in terms of 't'. Distance(2) = Speed(2) * Time(2) = 16*t Now we know that the total distance traveled is 45 miles and it is equal to the sum of the two.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p76_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 76, "page_end": 76, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. In the first row, we have the speed of the boat in terms of 'b' and we are given the time.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p76_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 76, "page_end": 76, "topic_guess": "word_problems", "text": "GMAT word problem note: Thus we can get the following equation:. Distance(1) = Speed(1) * Time(1) = (b + 3)*3 In the second row, we again have the speed in terms of 'b' and we are given the time.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p76_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 76, "page_end": 76, "topic_guess": "word_problems", "text": "GMAT word problem note: Distance(2) = Speed(2) * Time(2) = (b - 3)*4 Since the two distances are equal, we can equate them and solve for 'b'. Distance(1) = Distance(2) --> (b + 3)*3 = (b - 3)*4 --> b = 21 mph.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p77_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 77, "page_end": 77, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Overlapping Sets https://gmatclub.com/forum/overlapping-sets-144260.html ADVANCED OVERLAPPING SETS Some hard GMAT quantitative questions will require you to know and understand the formulas for set theory, presenting three sets and asking various questions about them.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p77_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 77, "page_end": 77, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: There are two main formulas to solve questions involving three overlapping sets. FIRST FORMULA Let's see how this formula is derived.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p77_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 77, "page_end": 77, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: When we add three groups A, B, and C some sections are counted more than once. For instance: sections d, e, and f are counted twice and section g thrice.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p77_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 77, "page_end": 77, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Hence we need to subtract sections d, e, and f ONCE (to count section g only once) and subtract section g TWICE (again to count section g only once). In the formula above,, where AnB means intersection of A and B (sections d, and g), AnC means intersection of A and C (sections e, and g), and BnC means intersection of B and C (sections f, and g).", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p77_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 77, "page_end": 77, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Now, when we subtract (d, and g), (e, and g), and (f, and g) from, we are subtract sections d, e, and f ONCE BUT section g THREE TIMES (and we need to subtract section g only twice), therefor we should add only section g, which is intersection of A, B and C (AnBnC) again to get. SECOND FORMULA Notice that EXACTLY (only) 2-group overlaps is not the same as 2-group ov.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p78_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 78, "page_end": 78, "topic_guess": "geometry", "text": "GMAT geometry note: Translate the wording into variables, relationships, and equations before calculating. Again: when we add three groups A, B, and C some sections are counted more than once.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p78_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 78, "page_end": 78, "topic_guess": "geometry", "text": "GMAT geometry note: For instance: sections d, e, and f are counted twice and section g thrice. Hence we need to subtract sections d, e, and f ONCE (to count section g only once) and subtract section g TWICE (again to count section g only once).", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p78_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 78, "page_end": 78, "topic_guess": "geometry", "text": "GMAT geometry note: When we subtract from A+B+C we subtract sections d, e, and f once (fine) and next we need to subtract ONLY section g () twice. Now, how this concept can be represented in GMAT problem?", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p78_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 78, "page_end": 78, "topic_guess": "geometry", "text": "GMAT geometry note: Workers are grouped by their areas of expertise, and are placed on at least one team. 20 are on the marketing team, 30 are on the Sales team, and 40 are on the Vision team.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p78_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 78, "page_end": 78, "topic_guess": "geometry", "text": "GMAT geometry note: 5 workers are on both the Marketing and Sales teams, 6 workers are on both the Sales and Vision teams, 9 workers are on both the Marketing and Vision teams, and 4 workers are on all three teams. How many workers are there in total?", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p78_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 78, "page_end": 78, "topic_guess": "geometry", "text": "GMAT geometry note: Translating: \"are placed on at least one team\": members of none = 0. \"20 are on the marketing team\":.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p78_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 78, "page_end": 78, "topic_guess": "geometry", "text": "GMAT geometry note: \"40 are on the Vision team\":. \"5 workers are on both the Marketing and Sales teams\":.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p79_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 79, "page_end": 79, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Of 20 Adults, 5 belong to A, 7 belong to B, and 9 belong to C.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p79_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 79, "page_end": 79, "topic_guess": "statistics", "text": "GMAT statistics note: If 2 belong to all three organizations and 3 belong to exactly 2 organizations, how many belong to none of these organizations? \"5 belong to A, 7 belong to B, and 9 belong to C\":.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p79_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 79, "page_end": 79, "topic_guess": "statistics", "text": "GMAT statistics note: \"2 belong to all three organizations\":. \"3 belong to exactly 2 organizations\": (sum of EXACTLY 2-group overlaps) = 3, so the sum of sections d, e, and f is given to be 3, (among these 3 adults there are no one who is a member of ALL 3 clubs) Question::.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p79_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 79, "page_end": 79, "topic_guess": "statistics", "text": "GMAT statistics note: Apply second formula: --> Answer: 6. This semester, each of the 90 students in a certain class took at least one course from A, B, and C.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p79_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 79, "page_end": 79, "topic_guess": "statistics", "text": "GMAT statistics note: If 60 students took A, 40 students took B, 20 students took C, and 5 students took all the three, how many students took exactly two courses? \"of the 90 students in a certain class took at least one course from A, B, and C\":.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p79_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 79, "page_end": 79, "topic_guess": "statistics", "text": "GMAT statistics note: \"60 students took A, 40 students took B, 20 students took C\":. \"5 students took all the three courses\":.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p79_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 79, "page_end": 79, "topic_guess": "statistics", "text": "GMAT statistics note: Apply second formula: --> Answer: 20.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p80_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 80, "page_end": 80, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. \"4 names were on all 3 rosters\":.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p80_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 80, "page_end": 80, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Apply first formula: --> Answer: 50. There are 50 employees in the office of ABC Company.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p80_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 80, "page_end": 80, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Of these, 22 have taken an accounting course, 15 have taken a course in finance and 14 have taken a marketing course. Nine of the employees have taken exactly two of the courses and 1 employee has taken all three of the courses.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p80_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 80, "page_end": 80, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: How many of the 50 employees have taken none of the courses? Translating: \"There are 50 employees in the office of ABC Company\":.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p80_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 80, "page_end": 80, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: \"22 have taken an accounting course, 15 have taken a course in finance and 14 have taken a marketing course\". \"Nine of the employees have taken exactly two of the courses\": (sum of EXACTLY 2-group overlaps) = 9.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p80_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 80, "page_end": 80, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: \"1 employee has taken all three of the courses\":. Apply second formula: --> Answer: 10.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p80_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 80, "page_end": 80, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: In a consumer survey, 85% of those surveyed liked at least one of three products: 1, 2, and 3. 50% of those asked liked product 1, 30% liked product 2, and 20% liked product 3.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p80_c1_r8", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 80, "page_end": 80, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: If 5% of the people in the survey liked all three of the products, what percentage of the survey participants liked more than one of the three products?", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p81_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 81, "page_end": 81, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Translate the wording into variables, relationships, and equations before calculating. Apply first formula: {Total} = {Hockey}+{Cricket}+{Football}-{HC+CH+HF}+{All three}+{Neither} 50 = 20+15+11-(7+4+5)+{All three}+18 --> {All three} = 2.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p81_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 81, "page_end": 81, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Those who play ONLY Hockey and Cricket are 7-2 = 5. Those who play ONLY Cricket and Football are 4-2 = 2.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p81_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 81, "page_end": 81, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Those who play ONLY Hockey and Football are 5-2 = 3. Hence, 5+2+3 = 10 students play exactly two of these sports.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p81_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 81, "page_end": 81, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: A student has decided to take GMAT and TOEFL examinations, for which he has allocated a certain number of days for preparation. On any given day, he does not prepare for both GMAT and TOEFL.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p81_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 81, "page_end": 81, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: How many days did he allocate for the preparation? (1) He did not prepare for GMAT on 10 days and for TOEFL on 12 days.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p81_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 81, "page_end": 81, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: (2) He prepared for either GMAT or TOEFL on 14 days We have: {Total} = {GMAT } + {TOEFL} - {Both} + {Neither}. Since we are told that \"on any given day, he does not prepare for both GMAT and TOEFL\", then {Both} = 0, so {Total} = {GMAT } + {TOEFL} + {Neither}.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p81_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 81, "page_end": 81, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: We need to find {Total} (1) He did not prepare for GMAT on 10 days and for TOEFL on 12 days --> {Total} - {GMAT } = 10 and {Total} - {TOEFL} = 12.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p82_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 82, "page_end": 82, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Notice that max range for the original question is not limited when the max # of people in all 3 groups for revised question is 17+28+35 (in case there is 0 overlap between the 3 groups).", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p82_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 82, "page_end": 82, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: For more questions on overlapping sets check our Question Banks Problem Solving Questions on Overlapping Sets.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p83_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 83, "page_end": 83, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Probability https://gmatclub.com/forum/probability-87244.html PROBABILITY Definition A number expressing the probability (p) that a specific event will occur, expressed as the ratio of the number of actual occurrences (n) to the number of possible occurrences (N).", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p83_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 83, "page_end": 83, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: A number expressing the probability (q) that a specific event will not occur:. Therefore, the probability of getting head is 50% or and the probability of getting tail is 50% or.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p83_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 83, "page_end": 83, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: All possibilities: {H,T} Dice p = n N q = = 1 − p(N−n) N.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p84_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 84, "page_end": 84, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. There are 6 equally possible outcomes when we roll a die.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p84_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 84, "page_end": 84, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: The probability of getting any number out of 1-6 is. All possibilities: {1,2,3,4,5,6} Marbles, Balls, Cards...", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p84_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 84, "page_end": 84, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Let's assume we have a jar with 10 green and 90 white marbles. If we randomly choose a marble, what is the probability of getting a green marble?", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p84_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 84, "page_end": 84, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: N = 10 + 90 = 100 The number of green marbles: n = 10 Probability of getting a green marble:. There is one important concept in problems with marbles/cards/balls.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p84_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 84, "page_end": 84, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: When the first marble is removed from a jar and not replaced, the probability for the second marble differs (vs.). Whereas in case of a coin or dice the probabilities are always the same (and).", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p84_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 84, "page_end": 84, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Usually, a problem explicitly states: it is a problem with replacement or without replacement. Independent events Two events are independent if occurrence of one event does not influence occurrence of other events.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p85_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 85, "page_end": 85, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Watch divisibility, factors, remainders, parity, and prime structure. After one flip and one toss, what is the probability of getting heads and a \"4\"?", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p85_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 85, "page_end": 85, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Tossing a coin and rolling a die are independent events. The probability of getting heads is and probability of getting a \"4\" is.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p85_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 85, "page_end": 85, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Therefore, the probability of getting heads and a \"4\" is:. If there is a 20% chance of rain, what is the probability that it will rain on the first day but not on the second?", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p85_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 85, "page_end": 85, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: The probability of rain is 0.2. Therefore probability of sunshine is q = 1 - 0.2 = 0.8.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p85_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 85, "page_end": 85, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: This yields that the probability of rain on the first day and sunshine on the second day is:. If Robert chooses randomly one integer from the first set and one integer from the second set, what is the probability of getting two odd integers?", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p85_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 85, "page_end": 85, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: There is a total of 5 integers in the first set and 3 of them are odd: {1, 3, 7}. Therefore, the probability of getting odd integer out of first set is.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p85_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 85, "page_end": 85, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: There are 3 integers in the second set and 2 of them are odd: {3, 5}. Therefore, the probability of getting an odd integer out of second set is.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p85_c1_r8", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 85, "page_end": 85, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Finally, the probability of of getting two odd integers is:. Mutually exclusive events Shakespeare's phrase \"To be,.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p86_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 86, "page_end": 86, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Combination of independent and mutually exclusive events Many probability problems contain combination of both independent and mutually exclusive events.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p86_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 86, "page_end": 86, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: To solve those problems it is important to identify all events and their types. One of the typical problems can be presented in a following general form:.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p86_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 86, "page_end": 86, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: If the probability of a certain event is p, what is the probability of it occurring k times in n-time sequence? (Or in English, what is the probability of getting 3 heads while tossing a coin 8 times?) Solution:.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p86_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 86, "page_end": 86, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: But it isn't the right answer. It would be right if we specified exactly each position for events in the sequence.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p86_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 86, "page_end": 86, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: So, we need to take into account that there are more than one outcomes. HHHTTTTT and HHTTTTTH are different mutually exclusive outcomes but they both have 3 heads and 5 tails.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p86_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 86, "page_end": 86, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Therefore, we need to include all combinations of heads and tails. In our general question, probability of occurring event k times in n-time sequence could be expressed as:.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p87_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 87, "page_end": 87, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. The total number of possible committees is.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p87_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 87, "page_end": 87, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: The number of possible committee that includes both Bob and Rachel is. 2) reversal combinatorial approach:.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p87_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 87, "page_end": 87, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Instead of counting probability of occurrence of certain event, sometimes it is better to calculate the probability of the opposite and then use formula p = 1 - q. The number of possible committee that does not includes both Bob and Rachel is: where, - the number of committees formed from 6 other people.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p87_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 87, "page_end": 87, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: The number of committees formed from Rob or Rachel and one out of 6 other people. The probability of choosing Bob or Rachel as a first person in committee is 2/8.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p87_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 87, "page_end": 87, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: The probability of choosing Rachel or Bob as a second person when first person is already chosen is 1/7. The probability that the committee includes both Bob and Rachel is.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p87_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 87, "page_end": 87, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: 4) reversal probability approach:. We can choose any first person.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p87_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 87, "page_end": 87, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Then, if we have Rachel or Bob as first choice, we can choose any other person out of 6 people. If we have neither Rachel nor Bob as first choice, we can choose any person out of remaining 7 people.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p87_c1_r8", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 87, "page_end": 87, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Given that there are 5 married couples.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p88_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 88, "page_end": 88, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. We choose 1 couple out of 5 couples.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p88_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 88, "page_end": 88, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: We chose one person out of remaining 8 people. The total number of combinations to choose 3 people out of 10 people.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p88_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 88, "page_end": 88, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: 3) probability approach: 1st person: - we choose any person out of 10. 2nd person: - we choose any person out of 8 = 10-2(one couple from previous choice) 3rd person: - we choose any person out of 6 = 10-4(two couples from previous choices).", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p88_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 88, "page_end": 88, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Probability tree Sometimes, at 700+ level you may see complex probability problems that include conditions or restrictions. For such problems it could be helpful to draw a probability tree that include all possible outcomes and their probabilities.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p88_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 88, "page_end": 88, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Julia and Brian play a game in which Julia takes a ball and if it is green, she wins. If the first ball is not green, she takes the second ball (without replacing first) and she wins if the two balls are white or if the first ball is gray and the second ball is white.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p88_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 88, "page_end": 88, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: What is the probability of Julia winning if the jar contains 1 gray, 2 white and 4 green balls? Let's draw all possible outcomes and calculate all probabilities.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p88_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 88, "page_end": 88, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Now, It is pretty obvious that the probability of Julia's win is:. C5 C8 C10 p = 1 − = 1 − = ∗C5 1 C8 C10 p = 1 ∗ ∗ = 8 P = + ∗ + ∗ = 4.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p89_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 89, "page_end": 89, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Symmetry Symmetry sometimes lets you solve seemingly complex probability problem in a few seconds.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p89_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 89, "page_end": 89, "topic_guess": "word_problems", "text": "GMAT word problem note: Bob and Rachel want to sit such that Bob is always left to Rachel. How many ways it can be done?", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p89_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 89, "page_end": 89, "topic_guess": "word_problems", "text": "GMAT word problem note: Because of symmetry, the number of ways that Bob is left to Rachel is exactly 1/2 of all possible ways:.", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p90_c1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 90, "page_end": 90, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p91_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 91, "page_end": 91, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Enumeration Enumeration is a method of counting all possible ways to arrange elements.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p91_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 91, "page_end": 91, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Although it is the simplest method, it is often the fastest method to solve hard GMAT problems and is a pivotal principle for any other combinatorial method. In fact, combination and permutation is shortcuts for enumeration.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p91_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 91, "page_end": 91, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: The main idea of enumeration is writing down all possible ways and then count them. There are three marbles: 1 blue, 1 gray and 1 green.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p91_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 91, "page_end": 91, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: In how many ways is it possible to arrange marbles in a row? Let's write out all possible ways:.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p91_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 91, "page_end": 91, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: In how many ways is it possible.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p92_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 92, "page_end": 92, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. To arrange marbles in a row if blue and green marbles have to be next to each other?", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p92_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 92, "page_end": 92, "topic_guess": "statistics", "text": "GMAT statistics note: Let's write out all possible ways to arrange marbles in a row and then find only arrangements that satisfy question's condition:. There are three marbles: 1 blue, 1 gray and 1 green.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p92_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 92, "page_end": 92, "topic_guess": "statistics", "text": "GMAT statistics note: In how many ways is it possible to arrange marbles in a row if gray marble have to be left to blue marble? Arrangements of n different objects Enumeration is a great way to count a small number of arrangements.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p92_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 92, "page_end": 92, "topic_guess": "statistics", "text": "GMAT statistics note: But when the total number of arrangements is large, enumeration can't be very useful, especially taking into account GMAT time restriction. Fortunately, there are some methods that can speed up counting of all arrangements.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p92_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 92, "page_end": 92, "topic_guess": "statistics", "text": "GMAT statistics note: The number of arrangements of n different objects in a row is a typical problem that can be solve this way: 1. How many objects we can put at 1st place?", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p92_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 92, "page_end": 92, "topic_guess": "statistics", "text": "GMAT statistics note: How many objects we can put at 2nd place? We can't put the object that already placed at 1st place.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p92_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 92, "page_end": 92, "topic_guess": "statistics", "text": "GMAT statistics note: How nany objects we can put at n-th place? Therefore, the total number of arrangements of n different objects in a.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p93_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 93, "page_end": 93, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Combination A combination is an unordered collection of k objects taken from a set of n distinct objects.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p93_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 93, "page_end": 93, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: The number of ways how we can choose k objects out of n distinct objects is denoted as: knowing how to find the number of arrangements of n distinct objects we can easily find formula for combination: 1. The total number of arrangements of n distinct objects is n!", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p93_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 93, "page_end": 93, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Now we have to exclude all arrangements of k objects (k!) and remaining (n-k) objects ((n-k)!) as the order of chosen k objects and remained (n-k) objects doesn't matter. Permutation A permutation is an ordered collection of k objects taken from a set of n distinct objects.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p93_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 93, "page_end": 93, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Now we have to exclude all arrangements of remaining (n-k) objects ((n-k)!) as the order of remained (n-k) objects doesn't matter. If we exclude order of chosen objects from permutation formula, we will get combination formula:.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p94_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 94, "page_end": 94, "topic_guess": "statistics", "text": "GMAT statistics note: Circular arrangement note: for n distinct objects arranged in a circle, the number of different relative arrangements is (n - 1)!. Unlike a row, rotating every object by one position does not create a new circular arrangement because the relative order stays the same.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p94_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 94, "page_end": 94, "topic_guess": "statistics", "text": "GMAT statistics note: On GMAT counting questions, define what counts as a different arrangement before choosing a permutations or combinations formula.", "topic": "probability"}
{"id": "GMAT Club Math Book 2024 v8_p95_c1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 95, "page_end": 95, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure.", "topic": "number_properties"}
{"id": "GMAT Club Math Book 2024 v8_p95_c2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 95, "page_end": 95, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Html of 1 All times are UTC - 8 hours [DST].", "topic": "word_problems"}
{"id": "GMAT Club Math Book 2024 v8_p96_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 96, "page_end": 96, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p96_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 96, "page_end": 96, "topic_guess": "statistics", "text": "GMAT statistics note: Standard Deviation https://gmatclub.com/forum/standard-deviation-87905.html STANDARD DEVIATION Definition Standard Deviation (SD, or STD or) - a measure of the dispersion or variation in a distribution, equal to the square root of variance or the arithmetic mean (average) of squares of deviations from the arithmetic mean.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p96_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 96, "page_end": 96, "topic_guess": "statistics", "text": "GMAT statistics note: In simple terms, it shows how much variation there is from the \"average\" (mean). It may be thought of as the average difference from the mean of distribution, how far data points are away from the mean.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p96_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 96, "page_end": 96, "topic_guess": "statistics", "text": "GMAT statistics note: A low standard deviation indicates that data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p97_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 97, "page_end": 97, "topic_guess": "statistics", "text": "GMAT statistics note: Standard deviation measures the spread of a set around its mean. Key GMAT properties: standard deviation is always non-negative, and it equals 0 only when all values in the set are equal.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p97_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 97, "page_end": 97, "topic_guess": "statistics", "text": "GMAT statistics note: If every value in a set is multiplied by the same constant, the standard deviation changes by that same factor in absolute value. If the same constant is added to or subtracted from every value, the standard deviation does not change because the spread stays the same.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p98_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 98, "page_end": 98, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. If a new element is added to set and standard deviation of a new set is , then: 1) if 2) if 3) if 4) is the lowest if Tips and Tricks GMAC in majority of problems doesn't ask you to calculate standard deviation.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p98_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 98, "page_end": 98, "topic_guess": "statistics", "text": "GMAT statistics note: Instead it tests your intuitive understanding of the concept. In 90% cases it is a faster way to use just average of instead of true formula for standard deviation, and treat standard deviation as \" average difference between elements and mean\".", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p98_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 98, "page_end": 98, "topic_guess": "statistics", "text": "GMAT statistics note: Therefore, before trying to calculate standard deviation, maybe you can solve a problem much faster by using just your intuition. Not all points contribute equally to standard deviation.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p98_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 98, "page_end": 98, "topic_guess": "statistics", "text": "GMAT statistics note: Taking into account that standard deviation uses sum of squares of deviations from mean, the most remote points will essentially contribute to standard deviation. The point 10 gives in sum of squares but point 6 gives only.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p98_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 98, "page_end": 98, "topic_guess": "statistics", "text": "GMAT statistics note: So, when you need to find what set has the largest standard deviation, always look for set with the largest range because remote points have a very significant contribution to standard deviation. If we create a new set that consists of all elements of the initial set but decreased by 17%, what is the chan.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p99_c1_r1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 99, "page_end": 99, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Means that all elements strictly related to each other.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p99_c1_r2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 99, "page_end": 99, "topic_guess": "statistics", "text": "GMAT statistics note: If we shift the set by adding or subtracting any integer, does it change standard deviation (average deviation of elements from the mean)? One thing we should know is the number of elements in the set, because the more elements we have the broader they are distributed relative to the mean.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p99_c1_r3", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 99, "page_end": 99, "topic_guess": "statistics", "text": "GMAT statistics note: Now, look at DS statements, all we need it is just first statement. Standard deviation of set is 18.3.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p99_c1_r4", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 99, "page_end": 99, "topic_guess": "statistics", "text": "GMAT statistics note: How many elements are 1 standard deviation above the mean? Now, we need to count all numbers greater than 42+18.3 = 60.3.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p99_c1_r5", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 99, "page_end": 99, "topic_guess": "statistics", "text": "GMAT statistics note: There is a set A of 19 integers with mean 4 and standard deviation of 3. Now we form a new set B by adding 2 more elements to the set A.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p99_c1_r6", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 99, "page_end": 99, "topic_guess": "statistics", "text": "GMAT statistics note: What two elements will decrease the standard deviation the most? A) 9 and 3 B) -3 and 3 C) 6 and 1 D) 4 and 5 E) 5 and 5 Solution:.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p99_c1_r7", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 99, "page_end": 99, "topic_guess": "statistics", "text": "GMAT statistics note: The closer to the mean, the greater decrease in standard deviation. D has 4 (equal our mean) and 5 (differs from mean only by 1).", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p99_c1_r8", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 99, "page_end": 99, "topic_guess": "statistics", "text": "GMAT statistics note: All other options have larger deviation from mean. Normal distribution It is a more advance concept that you will never see in GMAT but understanding statistic properties of standard deviation can help you to be m.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p100_c1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 100, "page_end": 100, "topic_guess": "general", "text": "GMAT quant note: Percent problems usually relate part, whole, rate, and percent change.", "topic": "percent"}
{"id": "GMAT Club Math Book 2024 v8_p101_c1", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 101, "page_end": 101, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread.", "topic": "statistics"}
{"id": "GMAT Club Math Book 2024 v8_p101_c2", "source_name": "GMAT Club Math Book 2024 v8", "source_file": "GMAT Club Math Book 2024 v8.pdf", "source_type": "pdf", "page_start": 101, "page_end": 101, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Html of 1 All times are UTC - 8 hours [DST].", "topic": "word_problems"}
{"id": "book 2_p3_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 3, "page_end": 3, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. MANHATTAN PREP GMAT Advanced Quant GMAT STRATEGY GUIDE This supplemental guide provides in-depth and comprehensive explanations of the advanced math skills necessary for the highest-level performance on the GMAT.", "topic": "word_problems"}
{"id": "book 2_p3_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 3, "page_end": 3, "topic_guess": "word_problems", "text": "GMAT word problem note: GMAT® is a registered trademark of the Graduate Management Admissions CouncilTM. Manhattan Prep is neither endorsed by nor affiliated with GMAC.", "topic": "word_problems"}
{"id": "book 2_p4_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 4, "page_end": 4, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "book 2_p5_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 5, "page_end": 5, "topic_guess": "strategy", "text": "GMAT strategy note: Translate the wording into variables, relationships, and equations before calculating. A Qualified Welcome Who Should Use This Book Try It Yourself The Purpose of This Book An Illustration Learning How to Think Plan of This Book Solutions to Try-It Problems Part 1:.", "topic": "word_problems"}
{"id": "book 2_p5_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 5, "page_end": 5, "topic_guess": "strategy", "text": "GMAT strategy note: Advanced Principles In This Chapter... Advanced Principles Principle #1:.", "topic": "word_problems"}
{"id": "book 2_p5_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 5, "page_end": 5, "topic_guess": "strategy", "text": "GMAT strategy note: Understand the Basics Principle #2:. Solve—and Put Pen to Paper Principle #4:.", "topic": "word_problems"}
{"id": "book 2_p5_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 5, "page_end": 5, "topic_guess": "strategy", "text": "GMAT strategy note: Strategies & Tactics In This Chapter... Principles In This Chapter...", "topic": "word_problems"}
{"id": "book 2_p6_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 6, "page_end": 6, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Never Rephrase Yes/No as Value Principle #3:.", "topic": "algebra"}
{"id": "book 2_p6_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 6, "page_end": 6, "topic_guess": "algebra", "text": "GMAT algebra note: Work from Facts to Question Principle #4:. Be a Contrarian Principle #5:.", "topic": "algebra"}
{"id": "book 2_p6_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 6, "page_end": 6, "topic_guess": "algebra", "text": "GMAT algebra note: Strategies & Tactics In This Chapter... Strategies & Tactics Advanced Strategies Advanced Guessing Tactics Summary Common Wrong Answers Problem Set Solutions Part 2:.", "topic": "algebra"}
{"id": "book 2_p6_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 6, "page_end": 6, "topic_guess": "algebra", "text": "GMAT algebra note: Common Terms and Quadratic Templates In This Chapter... Visual Solutions In This Chapter...", "topic": "algebra"}
{"id": "book 2_p7_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 7, "page_end": 7, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Translate the wording into variables, relationships, and equations before calculating. Hybrid Problems In This Chapter...", "topic": "word_problems"}
{"id": "book 2_p7_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 7, "page_end": 7, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Hybrid Problems Identify and Sequence the Parts Where to Start Minor Hybrids Problem Set Solutions Part 3:.", "topic": "word_problems"}
{"id": "book 2_p7_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 7, "page_end": 7, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Workout Set 1 Workout Set 1 Answer Key Workout Set 1 Solutions Workout Set 2 Workout Set 2 Answer Key Workout Set 2 Solutions Workout Set 3 Workout Set 3 Answer Key Workout Set 3 Solutions Workout Set 4 Workout Set 4 Answer Key Workout Set 4 Solutions Workout Set 5 Workout Set 5 Answer Key Workout Set 5 Solutions Workout Set 6 Workout Set 6:.", "topic": "word_problems"}
{"id": "book 2_p7_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 7, "page_end": 7, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Answer Key Workout Set 6 Solutions Workout Set 7 Workout Set 7 Answer Key.", "topic": "word_problems"}
{"id": "book 2_p8_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 8, "page_end": 8, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating. Workout Set 7 Solutions Workout Set 8 Workout Set 8 Answer Key Workout Set 8 Solutions Workout Set 9 Workout Set 9:.", "topic": "word_problems"}
{"id": "book 2_p8_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 8, "page_end": 8, "topic_guess": "general", "text": "GMAT quant note: Answer Key Workout Set 9 Solutions Workout Set 10 Workout Set 10 Answer Key Workout Set 10 Solutions Workout Set 11 Workout Set 11 Answer Key Workout Set 11 Solutions Workout Set 12 Workout Set 12 Answer Key Workout Set 12 Solutions Workout Set 13 Workout Set 13 Answer Key Workout Set 13 Solutions Workout Set 14 Workout Set 14 Answer Key Workout Set 14 Solutions Workout Set 15 Workout Set 15 Answer Key Workout Set 15 Solutions Workout Set 16 Workout Set 16 Answer Key Workout Set 16:.", "topic": "word_problems"}
{"id": "book 2_p8_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 8, "page_end": 8, "topic_guess": "general", "text": "GMAT quant note: Answers and Explanations mba Mission mba Mission Go Beyond Books.", "topic": "word_problems"}
{"id": "book 2_p9_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 9, "page_end": 9, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Our Manhattan Prep resources are based on the continuing experiences of our instructors and students. The overall vision for this edition was developed by Chelsey Cooley, who determined what new areas to cover and who wrote all of the problems that are new to this edition.", "topic": "geometry"}
{"id": "book 2_p9_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 9, "page_end": 9, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Chelsey served as the primary author of this edition and Emily Meredith Sledge was the primary editor. Emily also served as the primary author of the first edition of this guide.", "topic": "geometry"}
{"id": "book 2_p9_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 9, "page_end": 9, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Mario Gambino managed production for the many—and quite complicated—images that appear in this guide. Matthew Callan coordinated the production work for this guide.", "topic": "geometry"}
{"id": "book 2_p9_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 9, "page_end": 9, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Once the manuscript was done, Naomi Beesen and Ben Ku edited and Cheryl Duckler and Stacey Koprince proofread the entire guide from start to finish. Carly Schnur designed the covers.", "topic": "geometry"}
{"id": "book 2_p9_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 9, "page_end": 9, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: No part of this work may be reproduced or used in any form or by any means—graphic, electronic, or mechanical, including photocopying, recording, taping, web distribution—without the prior written permissio.", "topic": "geometry"}
{"id": "book 2_p10_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 10, "page_end": 10, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. GMAT® STRATEGY GUIDES GMAT All the Quant GMAT All the Verbal GMAT Integrated Reasoning and Essay STRATEGY GUIDE SUPPLEMENTS Math Verbal GMAT Foundations of Math GMAT Advanced Quant GMAT Foundations of Verbal.", "topic": "word_problems"}
{"id": "book 2_p11_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 11, "page_end": 11, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. I hope this book provides just the guidance you need to get the most out of your GMAT studies.", "topic": "word_problems"}
{"id": "book 2_p11_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 11, "page_end": 11, "topic_guess": "word_problems", "text": "GMAT word problem note: At Manhattan Prep, we continually aspire to provide the best instructors and resources possible. If you have any questions or feedback, please do not hesitate to contact us.", "topic": "word_problems"}
{"id": "book 2_p11_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 11, "page_end": 11, "topic_guess": "word_problems", "text": "GMAT word problem note: Email our Student Services team at gmat@manhattanprep.com or give us a shout at 212-721-7400 (or 800-576-4628 in the United States or Canada). We try to keep all our books free of errors, but if you think we’ve goofed, please visit manhattanprep.com/GMAT/errata.", "topic": "word_problems"}
{"id": "book 2_p11_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 11, "page_end": 11, "topic_guess": "word_problems", "text": "GMAT word problem note: Our Manhattan Prep Strategy Guides are based on the continuing experiences of both our instructors and our students. The primary author of the this edition of the Advanced Quant guide was Chelsey Cooley and the primary editor was Emily Meredith Sledge.", "topic": "word_problems"}
{"id": "book 2_p11_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 11, "page_end": 11, "topic_guess": "word_problems", "text": "GMAT word problem note: Project management and design were led by Matthew Callan and Mario Gambino.", "topic": "word_problems"}
{"id": "book 2_p12_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 12, "page_end": 12, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Particular thanks to instructors Stacey Koprince and Ben Ku for their content contributions.", "topic": "word_problems"}
{"id": "book 2_p12_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 12, "page_end": 12, "topic_guess": "word_problems", "text": "GMAT word problem note: Finally, we are indebted to all of the Manhattan Prep students who have given us excellent feedback over the years. This book wouldn’t be half of what it is without their voice.", "topic": "word_problems"}
{"id": "book 2_p12_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 12, "page_end": 12, "topic_guess": "word_problems", "text": "GMAT word problem note: And now that you are one of our students too, please chime in! I look forward to hearing from you.", "topic": "word_problems"}
{"id": "book 2_p12_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 12, "page_end": 12, "topic_guess": "word_problems", "text": "GMAT word problem note: Thanks again and best of luck preparing for the GMAT! Sincerely, Chris Ryan Executive Director, Product Strategy www.manhattanprep.com/gmat 138 West 25th Street, 7th Floor, New York, NY 10001 Tel: 212-721-7400 Fax: 646-514-7425.", "topic": "word_problems"}
{"id": "book 2_p13_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 13, "page_end": 13, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "book 2_p14_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 14, "page_end": 14, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. A Qualified Welcome Who Should Use This Book Try It Yourself The Purpose of This Book An Illustration Learning How to Think Plan of This Book Solutions to Try-It Problems.", "topic": "percent"}
{"id": "book 2_p15_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 15, "page_end": 15, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Introduction A Qualified Welcome Welcome to GMAT Advanced Quant! A er all, the point (0, 0) in the coordinate plane is called the origin, isn’t it?", "topic": "geometry"}
{"id": "book 2_p15_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 15, "page_end": 15, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: (That’s the first and last math joke in this book.) Unfortunately, we have to qualify our welcome right away, because this book isn’t for everyone. At least, it’s not for everyone right away.", "topic": "geometry"}
{"id": "book 2_p16_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 16, "page_end": 16, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Who Should Use This Book You should use this book if you meet the following conditions:.", "topic": "percent"}
{"id": "book 2_p16_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 16, "page_end": 16, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: If you match this description, then please turn the page. You have achieved a scaled score of at least 47 (out of 51) on the Quant section of either the Manhattan Prep practice test or the official practice computer-adaptive test (CAT).", "topic": "percent"}
{"id": "book 2_p16_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 16, "page_end": 16, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: You have worked through the Manhattan Prep All the Quant guide, which covers all of the topics and strategies you need for the Quant section, or you have worked through similar material from another company. This material should include the following:.", "topic": "percent"}
{"id": "book 2_p16_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 16, "page_end": 16, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Algebra Fractions, Decimals, Percents, and Ratios Geometry Number Properties Word Problems You are already comfortable with the core principles in these topics. You want to raise your performance to a scaled score of 49 or higher.", "topic": "percent"}
{"id": "book 2_p16_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 16, "page_end": 16, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: You want to become a significantly smarter test-taker.", "topic": "percent"}
{"id": "book 2_p17_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 17, "page_end": 17, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. If you don’t match this description, then you will probably find this book too difficult at this stage of your preparation.", "topic": "percent"}
{"id": "book 2_p17_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 17, "page_end": 17, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: For now, you are better off working on topic-focused material, as found in the All the Quant guide, and ensuring that you have mastered that material before you return to this book.", "topic": "percent"}
{"id": "book 2_p18_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 18, "page_end": 18, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. Try It Yourself Throughout the chapters of this guide, you’ll see Try-It problems— problems designed to test your skills on certain aspects of GMAT problems.", "topic": "number_properties"}
{"id": "book 2_p18_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 18, "page_end": 18, "topic_guess": "general", "text": "GMAT quant note: Take a look at the following three Try-It problems, which are very difficult. They are at least as hard as any real GMAT problem—probably even harder.", "topic": "number_properties"}
{"id": "book 2_p18_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 18, "page_end": 18, "topic_guess": "general", "text": "GMAT quant note: Go ahead and give these problems a try. You should not expect to solve any of them in two minutes.", "topic": "number_properties"}
{"id": "book 2_p18_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 18, "page_end": 18, "topic_guess": "general", "text": "GMAT quant note: In fact, you might find yourself completely stuck. If that’s the case, switch gears.", "topic": "number_properties"}
{"id": "book 2_p18_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 18, "page_end": 18, "topic_guess": "general", "text": "GMAT quant note: Try-It #0-1 A jar is filled with red, white, and blue tokens that are equivalent except for their color. The chance of randomly selecting a red token, replacing it, then randomly selecting a white token is the same as the chance of randomly selecting a blue token.", "topic": "number_properties"}
{"id": "book 2_p18_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 18, "page_end": 18, "topic_guess": "general", "text": "GMAT quant note: If the number of tokens of every color is a multiple of 3, what is the smallest possible total number of tokens in the jar?", "topic": "number_properties"}
{"id": "book 2_p19_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 19, "page_end": 19, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Try-It #0-2 Arrow , which is a line segment exactly 5 units long with an arrowhead at A, is constructed in the xy-plane. The x- and y- coordinates of A and B are integers that satisfy the inequalities 0 ≤ x ≤ 9 and 0 ≤ y ≤ 9.", "topic": "geometry"}
{"id": "book 2_p19_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 19, "page_end": 19, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: How many different arrows with these - properties can be constructed?", "topic": "geometry"}
{"id": "book 2_p20_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 20, "page_end": 20, "topic_guess": "general", "text": "GMAT quant note: This problem does not require any non-GMAT math, such as trigonometry.).", "topic": "general"}
{"id": "book 2_p21_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 21, "page_end": 21, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The Purpose of This Book This book is designed to prepare you for the most difficult math problems on the GMAT.", "topic": "number_properties"}
{"id": "book 2_p21_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 21, "page_end": 21, "topic_guess": "number_theory", "text": "GMAT number properties note: So...what is a difficult math problem, from the point of view of the GMAT? A difficult math problem is one that most GMAT test-takers get wrong under exam conditions.", "topic": "number_properties"}
{"id": "book 2_p21_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 21, "page_end": 21, "topic_guess": "number_theory", "text": "GMAT number properties note: In fact, this is essentially how the GMAT measures difficulty: by the percent of test-takers who get the problem wrong. So what kinds of math questions do most test-takers get wrong?", "topic": "number_properties"}
{"id": "book 2_p21_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 21, "page_end": 21, "topic_guess": "number_theory", "text": "GMAT number properties note: What characterizes these problems? There are two kinds of features:.", "topic": "number_properties"}
{"id": "book 2_p21_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 21, "page_end": 21, "topic_guess": "number_theory", "text": "GMAT number properties note: Topical nuances or obscure principles These topical nuances are largely covered in the Extra sections of the Manhattan Prep All the Quant guide. This book includes many problems that involve topical nuances.", "topic": "number_properties"}
{"id": "book 2_p21_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 21, "page_end": 21, "topic_guess": "number_theory", "text": "GMAT number properties note: However, the exhaustive theory of divisibility and primes, for instance, is not repeated here. Connected to a particular topic Inherently hard to grasp or unfamiliar Easy to mix up.", "topic": "number_properties"}
{"id": "book 2_p22_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 22, "page_end": 22, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Complex structures Complex structures are essentially disguises for simpler content.", "topic": "word_problems"}
{"id": "book 2_p22_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 22, "page_end": 22, "topic_guess": "word_problems", "text": "GMAT word problem note: These disguises may be difficult to pierce. The path to the answer is twisted or clouded somehow.", "topic": "word_problems"}
{"id": "book 2_p22_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 22, "page_end": 22, "topic_guess": "word_problems", "text": "GMAT word problem note: To solve problems that have simple content but complex structures, you need approaches that are both more general and more creative. This book focuses on these more general and more creative approaches.", "topic": "word_problems"}
{"id": "book 2_p22_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 22, "page_end": 22, "topic_guess": "word_problems", "text": "GMAT word problem note: The three problems on the previous page have complex structures. The solutions are a bit later in this chapter.", "topic": "word_problems"}
{"id": "book 2_p22_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 22, "page_end": 22, "topic_guess": "word_problems", "text": "GMAT word problem note: In the meantime, take a look at another problem. May use simple principles in ways that aren’t obvious May require multiple steps May make you consider many cases May combine more than one topic May need a flash of real insight to complete May make you change direction or switch strategies along the way.", "topic": "word_problems"}
{"id": "book 2_p23_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 23, "page_end": 23, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. An Illustration Give this problem a whirl.", "topic": "algebra"}
{"id": "book 2_p23_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 23, "page_end": 23, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Don’t go on until you have spent a few minutes on it—or until you have figured it out. Try-It #0-4 What should the next number in this sequence be?", "topic": "algebra"}
{"id": "book 2_p23_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 23, "page_end": 23, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: This problem is not exactly GMAT-like, because there is no mathematically definite rule. However, you’ll know when you’ve solved the problem.", "topic": "algebra"}
{"id": "book 2_p23_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 23, "page_end": 23, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: This problem has very simple content but a complex structure. Researchers in cognitive science have used sequence-completion problems such as this one to develop realistic models of human thought.", "topic": "algebra"}
{"id": "book 2_p23_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 23, "page_end": 23, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Here is one such model, simplified but practical. Top-Down Brain and Bottom-Up Brain To solve the sequence-completion problem above, you need two kinds of thinking.", "topic": "algebra"}
{"id": "book 2_p24_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 24, "page_end": 24, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. You might even say that you need two types of brain.", "topic": "percent"}
{"id": "book 2_p24_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 24, "page_end": 24, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: The top-down brain is your conscious self. In cognitive science, the top-down brain is called the executive function.", "topic": "percent"}
{"id": "book 2_p24_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 24, "page_end": 24, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Top-down thinking and planning is indispensable to any problem-solving process. But the corporation in your head is a big place.", "topic": "percent"}
{"id": "book 2_p24_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 24, "page_end": 24, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: For one thing, how does information get to the CEO? And how preprocessed is that information?", "topic": "percent"}
{"id": "book 2_p24_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 24, "page_end": 24, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: The bottom-up brain is your preconscious processor. A er raw sensory input arrives, your bottom-up brain processes that input extensively before it reaches your top-down brain.", "topic": "percent"}
{"id": "book 2_p24_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 24, "page_end": 24, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: For instance, to your optic nerve, every word on this page is just a lot of black squiggles. Your bottom-up brain immediately turns these squiggles into letters, joins the letters into words, summons relevant images and concepts, and finally serves these images and concepts to your top-down brain.", "topic": "percent"}
{"id": "book 2_p24_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 24, "page_end": 24, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: This all happens automatically and swi ly. In fact, it takes effort to interrupt this process.", "topic": "percent"}
{"id": "book 2_p24_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 24, "page_end": 24, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Also, unlike your top-down brain, which does things one at a time, your bottom-up brain can easily do many things at once. How does all this relate to solving the sequence problem above?", "topic": "percent"}
{"id": "book 2_p25_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 25, "page_end": 25, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Each of your brains needs the other one to solve difficult problems.", "topic": "algebra"}
{"id": "book 2_p25_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 25, "page_end": 25, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Your top-down brain needs your bottom-up brain to notice patterns, sniff out valuable leads, and make quick, intuitive leaps and connections. But your bottom-up brain is inarticulate and distractible.", "topic": "algebra"}
{"id": "book 2_p25_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 25, "page_end": 25, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Only your top- down brain can build plans, pose explicit questions, follow procedures, and state findings. Imagine that you are trying to solve a tough murder case.", "topic": "algebra"}
{"id": "book 2_p25_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 25, "page_end": 25, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: To find all the clues in the woods, you need both a savvy detective and a sharp-nosed bloodhound. Your top-down brain is the detective.", "topic": "algebra"}
{"id": "book 2_p25_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 25, "page_end": 25, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Be organized, fast, and flexible to crack the case. Your bottom-up brain is the bloodhound.", "topic": "algebra"}
{"id": "book 2_p25_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 25, "page_end": 25, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: To solve difficult GMAT problems, try to harmonize the activity of your two brains by following an organized, fast, and flexible problem-solving process. Organized You need a general step-by-step approach to guide you.", "topic": "algebra"}
{"id": "book 2_p25_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 25, "page_end": 25, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: One such approach, inspired by expert mathematician George Pólya, is Understand, Plan, Solve (UPS).", "topic": "algebra"}
{"id": "book 2_p26_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 26, "page_end": 26, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. You may never have thought explicitly about steps 1 and 2 before.", "topic": "word_problems"}
{"id": "book 2_p26_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 26, "page_end": 26, "topic_guess": "word_problems", "text": "GMAT word problem note: It may have been easy or even automatic for you to Understand easier problems and to Plan your approach to them. As a result, you may tend to dive right into the Solve stage.", "topic": "word_problems"}
{"id": "book 2_p26_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 26, "page_end": 26, "topic_guess": "word_problems", "text": "GMAT word problem note: Mathematicians know that the real math on hard problems is not Solve. The real math is Understand and Plan.", "topic": "word_problems"}
{"id": "book 2_p26_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 26, "page_end": 26, "topic_guess": "word_problems", "text": "GMAT word problem note: Fast Speed is important for its own sake on the GMAT, of course. What you may not have thought as much about is that being fast can also lower your stress level and promote good process.", "topic": "word_problems"}
{"id": "book 2_p26_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 26, "page_end": 26, "topic_guess": "word_problems", "text": "GMAT word problem note: If you know you can solve quickly, then you can take more time to comprehend the question, consider the given information, and select a strategy. To this end, make sure that you can complete calculations and manipulations fairly rapidly so that you can afford to spend some time on the Understand and Plan stages of your problem-solving process.", "topic": "word_problems"}
{"id": "book 2_p26_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 26, "page_end": 26, "topic_guess": "word_problems", "text": "GMAT word problem note: A little extra time invested up front can pay off handsomely later. Flexible To succeed against difficult problems, you sometimes have to “unstick” yourself.", "topic": "word_problems"}
{"id": "book 2_p26_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 26, "page_end": 26, "topic_guess": "word_problems", "text": "GMAT word problem note: Expect to run into brick walls and encounter dead ends. Returning to first principles and to the general process (e.g., making sure that you fully Understand the problem) can help you back up out of the mud.", "topic": "word_problems"}
{"id": "book 2_p27_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 27, "page_end": 27, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The top-down brain is labeled TD.", "topic": "number_properties"}
{"id": "book 2_p27_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 27, "page_end": 27, "topic_guess": "number_theory", "text": "GMAT number properties note: The bottom-up brain is labeled BU. 1 2 9 64 __ TD: “Okay, let’s Understand this thing.", "topic": "number_properties"}
{"id": "book 2_p27_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 27, "page_end": 27, "topic_guess": "number_theory", "text": "GMAT number properties note: At a glance, they’ve given me an increasing list of numbers, and they want me to find the number that “should” go in the blank, whatever “should” means. Stare at the numbers given?” BU notices that 9 = 32 and 64 = 82.", "topic": "number_properties"}
{"id": "book 2_p27_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 27, "page_end": 27, "topic_guess": "number_theory", "text": "GMAT number properties note: 1 2 32 82 __ TD: “Write in the two squares.” 1 2 32 82 __ BU notices that 1 is a square, too. Sq no sq sq __ TD: “Are they all perfect squares?", "topic": "number_properties"}
{"id": "book 2_p27_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 27, "page_end": 27, "topic_guess": "number_theory", "text": "GMAT number properties note: No, since 2 isn’t.” BU doesn’t like this break in the pattern. 1 2 32 26 — TD: “Wait, back up.", "topic": "number_properties"}
{"id": "book 2_p27_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 27, "page_end": 27, "topic_guess": "number_theory", "text": "GMAT number properties note: What about primes, factoring all the way. 8 = 23, and so 82 = (23)2 = 26.” BU notices 6 = 2 × 3, but so what?", "topic": "number_properties"}
{"id": "book 2_p27_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 27, "page_end": 27, "topic_guess": "number_theory", "text": "GMAT number properties note: 1 2 32 (23)2 __ TD: “Let’s write 26 as (23)2. Anything there?” BU notices lots of 2’s and 3’s, but so what?", "topic": "number_properties"}
{"id": "book 2_p27_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 27, "page_end": 27, "topic_guess": "number_theory", "text": "GMAT number properties note: TD: “Okay, keep looking at this. Are the 2’s and 3’s stacked somehow?”.", "topic": "number_properties"}
{"id": "book 2_p28_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 28, "page_end": 28, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. There’s 2−3−2 twice as you go across, but so what?", "topic": "algebra"}
{"id": "book 2_p28_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 28, "page_end": 28, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: And the 1 is weird by itself. Original and try taking differences.” BU notices no pattern.", "topic": "algebra"}
{"id": "book 2_p28_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 28, "page_end": 28, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The numbers look even uglier. Go back to original numbers again.", "topic": "algebra"}
{"id": "book 2_p28_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 28, "page_end": 28, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: What’s going on there?” BU notices that the numbers are growing quickly, like squares or exponentials. 12 2 32 82 __ TD: “Must have something to do with those squares.", "topic": "algebra"}
{"id": "book 2_p28_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 28, "page_end": 28, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: I should look at those again.” BU notices a gap on the le, among the powers. 21 32 82 __ TD: “How about looking at 2.", "topic": "algebra"}
{"id": "book 2_p28_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 28, "page_end": 28, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Write it with exponents: 2 = 21. Actually, 1 doesn’t have to be 12.", "topic": "algebra"}
{"id": "book 2_p28_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 28, "page_end": 28, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: One can be to any power and still be 1. The power is a question mark.” BU notices 21 then 32.", "topic": "algebra"}
{"id": "book 2_p28_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 28, "page_end": 28, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: BU really wants 1, 2, 3, 4 somehow. __ TD: “Try 4 in that last position.", "topic": "algebra"}
{"id": "book 2_p28_c1_r9", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 28, "page_end": 28, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Could the last term be 4 somehow?” BU likes the look of this. 21 32 43 __ TD: “64 is 4 to the what?", "topic": "algebra"}
{"id": "book 2_p28_c1_r10", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 28, "page_end": 28, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: 42 = 16, times another 4 equals 64, so.", "topic": "algebra"}
{"id": "book 2_p29_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 29, "page_end": 29, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. That fits.” BU is thrilled: 1, 2, 3, 4 below and 1, 2, 3 up top.", "topic": "number_properties"}
{"id": "book 2_p29_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 29, "page_end": 29, "topic_guess": "number_theory", "text": "GMAT number properties note: 10 21 32 43 __ TD: “Extend le. The bases are 1, 2, 3, 4, etc., and the powers are 0, 1, 2, 3, etc.” BU is content.", "topic": "number_properties"}
{"id": "book 2_p29_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 29, "page_end": 29, "topic_guess": "number_theory", "text": "GMAT number properties note: 10 21 32 43 54 TD: “So the answer is 54, which is 252, or 625.” Your own process was almost certainly different in the details. Also, your internal dialog was very rapid—parts of it probably only took fractions of a second to transpire.", "topic": "number_properties"}
{"id": "book 2_p29_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 29, "page_end": 29, "topic_guess": "number_theory", "text": "GMAT number properties note: A er all, you think at the speed of thought. The important thing is to recognize how the bottom-up bloodhound and the top-down detective worked together in the case above.", "topic": "number_properties"}
{"id": "book 2_p29_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 29, "page_end": 29, "topic_guess": "number_theory", "text": "GMAT number properties note: The TD detective set the overall agenda and then pointed the BU bloodhound at the clues. The bloodhound did practically all the “noticing,” which in some sense is where all the magic happened.", "topic": "number_properties"}
{"id": "book 2_p29_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 29, "page_end": 29, "topic_guess": "number_theory", "text": "GMAT number properties note: But sometimes the bloodhound got stuck, so the detective had to intervene, consciously trying a new path. For instance, 64 reads so strongly as 82 that the detective had to actively give up on that reading.", "topic": "number_properties"}
{"id": "book 2_p29_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 29, "page_end": 29, "topic_guess": "number_theory", "text": "GMAT number properties note: There are so many possible meaningful sequences that it wouldn’t have made sense to apply a strict recipe from the outset: “Try X first, then Y, then Z...” Such an algorithm would require hundreds of possibilities. Should you always look for 1, 2, 3, 4?", "topic": "number_properties"}
{"id": "book 2_p30_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 30, "page_end": 30, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Factors because they weren’t that useful here?", "topic": "number_properties"}
{"id": "book 2_p30_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 30, "page_end": 30, "topic_guess": "number_theory", "text": "GMAT number properties note: A computer can rapidly and easily apply a complicated algorithm with hundreds of steps, but humans can’t. (If you are an engineer or programmer, maybe you wish you could program your own brain, but so far, that’s not possible!) What humans are good at, though, is noticing patterns.", "topic": "number_properties"}
{"id": "book 2_p30_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 30, "page_end": 30, "topic_guess": "number_theory", "text": "GMAT number properties note: The bottom-up brain is extremely powerful—far more powerful than any computer yet built. As you gather problem-solving tools, the task becomes knowing when to apply which tool.", "topic": "number_properties"}
{"id": "book 2_p30_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 30, "page_end": 30, "topic_guess": "number_theory", "text": "GMAT number properties note: This task becomes harder as problem structures become more complex. But if you deploy your bottom-up bloodhound according to a general problem-solving process such as Understand, Plan, Solve, then you can count on the bloodhound to notice the relevant aspects of the problem—the aspects that tell you which tool to use.", "topic": "number_properties"}
{"id": "book 2_p30_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 30, "page_end": 30, "topic_guess": "number_theory", "text": "GMAT number properties note: You can break down Understand, Plan, Solve into several discrete steps:. Understand Glance at the problem briefly: does anything stand out?", "topic": "number_properties"}
{"id": "book 2_p30_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 30, "page_end": 30, "topic_guess": "number_theory", "text": "GMAT number properties note: Jot down any obvious formulas or numbers. Plan Reflect on what you were given: what clues might help tell you how to approach this problem?", "topic": "number_properties"}
{"id": "book 2_p31_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 31, "page_end": 31, "topic_guess": "general", "text": "GMAT quant note: Focus on equations, expressions, variables, and step-by-step manipulation. Organize your approach: choose a solution path.", "topic": "algebra"}
{"id": "book 2_p31_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 31, "page_end": 31, "topic_guess": "general", "text": "GMAT quant note: You’ll get lots of practice using the UPS process throughout this guide.", "topic": "algebra"}
{"id": "book 2_p32_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 32, "page_end": 32, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Learning How to Think This book is intended to make you smarter.", "topic": "word_problems"}
{"id": "book 2_p32_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 32, "page_end": 32, "topic_guess": "word_problems", "text": "GMAT word problem note: It is also intended to make you scrappier. That description encompasses two main ideas: employing GMAT strategies as well as textbook solution methods and knowing when to let go.", "topic": "word_problems"}
{"id": "book 2_p32_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 32, "page_end": 32, "topic_guess": "word_problems", "text": "GMAT word problem note: If you have traditionally been good at paper-based standardized tests, then you may be used to solving practically every problem the “textbook” way. Problems that forced you to get down and dirty—to work backwards from the choices, to estimate and eliminate—may have annoyed you.", "topic": "word_problems"}
{"id": "book 2_p32_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 32, "page_end": 32, "topic_guess": "word_problems", "text": "GMAT word problem note: A major purpose of this book is to help you learn to choose the best GMAT approach. On the hardest Quant problems, the textbook approach is o en not the best GMAT approach.", "topic": "word_problems"}
{"id": "book 2_p32_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 32, "page_end": 32, "topic_guess": "word_problems", "text": "GMAT word problem note: Unfortunately, advanced test-takers are sometimes very stubborn. Sometimes they feel they should solve a problem according to some theoretical approach.", "topic": "word_problems"}
{"id": "book 2_p32_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 32, "page_end": 32, "topic_guess": "word_problems", "text": "GMAT word problem note: Or they fail to move to Plan B or C rapidly enough, so they don’t have enough time le to execute that plan. In the end, they might wind up guessing purely at random—and that’s a shame.", "topic": "word_problems"}
{"id": "book 2_p33_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 33, "page_end": 33, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. GMAT problems o en have back doors—ways to solve that don’t involve crazy computation or genius-level insights.", "topic": "word_problems"}
{"id": "book 2_p33_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 33, "page_end": 33, "topic_guess": "word_problems", "text": "GMAT word problem note: in theory, GMAT problems can all be solved in two minutes. By searching for the back door, you might avoid all the bear traps that the problem writer set out by the front door!", "topic": "word_problems"}
{"id": "book 2_p33_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 33, "page_end": 33, "topic_guess": "word_problems", "text": "GMAT word problem note: In addition to learning alternative solution methods, you also need to learn when to let go. As you know, the GMAT is an adaptive test.", "topic": "word_problems"}
{"id": "book 2_p33_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 33, "page_end": 33, "topic_guess": "word_problems", "text": "GMAT word problem note: If you keep getting questions correct, the test will keep getting harder...and harder... At some point, there will appear a monster problem, one that announces “I must break you.” In your battle with this problem, you could actually lose the bigger war—even if you ultimately conquer this particular problem.", "topic": "word_problems"}
{"id": "book 2_p33_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 33, "page_end": 33, "topic_guess": "word_problems", "text": "GMAT word problem note: Maybe it takes you eight minutes, or it beats you up so badly that your head starts pounding. This will take its toll on your score.", "topic": "word_problems"}
{"id": "book 2_p33_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 33, "page_end": 33, "topic_guess": "word_problems", "text": "GMAT word problem note: This will happen to everyone, no matter how good you are at the GMAT. The GMAT is not an academic test, though it certainly appears to be.", "topic": "word_problems"}
{"id": "book 2_p33_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 33, "page_end": 33, "topic_guess": "word_problems", "text": "GMAT word problem note: Business schools are primarily interested in whether you’re going to be an effective businessperson. Good businesspeople are able to assess a situation rapidly, manage scarce resources, distinguish between good opportunities and bad ones, and make de.", "topic": "word_problems"}
{"id": "book 2_p34_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 34, "page_end": 34, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Whether you have the presence of mind to recognize a bad opportunity and the discipline to let it go.", "topic": "word_problems"}
{"id": "book 2_p34_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 34, "page_end": 34, "topic_guess": "word_problems", "text": "GMAT word problem note: Show the GMAT that you know how to manage your scarce resources (time and mental energy) and that you can recognize and cut off a bad opportunity.", "topic": "word_problems"}
{"id": "book 2_p35_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 35, "page_end": 35, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Plan of This Book The rest of this book has three parts:.", "topic": "word_problems"}
{"id": "book 2_p35_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 35, "page_end": 35, "topic_guess": "word_problems", "text": "GMAT word problem note: Strategies & Tactics Part Two:. Sixteen sets of 10 problems each The four chapters in Part I focus on principles, strategies, and tactics related to the two types of GMAT math problems:.", "topic": "word_problems"}
{"id": "book 2_p35_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 35, "page_end": 35, "topic_guess": "word_problems", "text": "GMAT word problem note: The next four chapters, in Part II, focus on techniques.", "topic": "word_problems"}
{"id": "book 2_p36_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 36, "page_end": 36, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. That apply across several topics but are more specific than the approaches in Part I.", "topic": "statistics"}
{"id": "book 2_p36_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 36, "page_end": 36, "topic_guess": "statistics", "text": "GMAT statistics note: Each of the eight chapters in Part I and Part II contains the following:. Many of these problems will be GMAT-like in format, but many will not.", "topic": "statistics"}
{"id": "book 2_p36_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 36, "page_end": 36, "topic_guess": "statistics", "text": "GMAT statistics note: Part III contains sets of GMAT-like Workout problems, designed to exercise your skills as if you were taking the GMAT and seeing its hardest problems. Several of these sets contain clusters of problems relating to the chapters in Parts I and II, although the problems within each set do not all resemble each other in obvious ways.", "topic": "statistics"}
{"id": "book 2_p36_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 36, "page_end": 36, "topic_guess": "statistics", "text": "GMAT statistics note: Other Workout problem sets are mixed by both approach and topic. Note that these problems are not arranged in order of difficulty.", "topic": "statistics"}
{"id": "book 2_p36_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 36, "page_end": 36, "topic_guess": "statistics", "text": "GMAT statistics note: Also, you should know that some of these problems draw on advanced content covered in the Manhattan Prep All the Quant guide. Try-It Problems embedded throughout the text Problem Sets at the end of the chapter.", "topic": "statistics"}
{"id": "book 2_p37_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 37, "page_end": 37, "topic_guess": "general", "text": "GMAT quant note: Solutions to Try-It Problems If you haven’t tried to solve the first three Try-It problems in the Try It Yourself section at the beginning of this chapter, then go back and try them now. Think about how to get your top-down brain and your bottom-up brain to work together like a detective and a bloodhound.", "topic": "geometry"}
{"id": "book 2_p37_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 37, "page_end": 37, "topic_guess": "general", "text": "GMAT quant note: Come back when you’ve tackled the problems, even if you don’t get to an answer (in this case, do make a guess). In these solutions, we’ll outline sample dialogs between the top-down detective and the bottom-up bloodhound.", "topic": "geometry"}
{"id": "book 2_p37_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 37, "page_end": 37, "topic_guess": "general", "text": "GMAT quant note: Try-It #0-1 A jar is filled with red, white, and blue tokens that are equivalent except for their color. The chance of randomly selecting a red token, replacing it, then randomly selecting a white token is the same as the chance of randomly selecting a blue token.", "topic": "geometry"}
{"id": "book 2_p37_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 37, "page_end": 37, "topic_guess": "general", "text": "GMAT quant note: If the number of tokens of every color is a multiple of 3, what is the smallest possible total number of tokens in the jar?", "topic": "geometry"}
{"id": "book 2_p38_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 38, "page_end": 38, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Jar is filled with red, white, and blue tokens...", "topic": "probability"}
{"id": "book 2_p38_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 38, "page_end": 38, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Chance of randomly selecting... TD: “I need to Understand this problem first.", "topic": "probability"}
{"id": "book 2_p38_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 38, "page_end": 38, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: There’s a jar, and it’s got red, white, and blue tokens in it.” BU notices “chance” and “randomly.” That’s probability. TD: “All right, this is a probability problem.", "topic": "probability"}
{"id": "book 2_p38_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 38, "page_end": 38, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Now, what’s the situation?” BU notices that there are two situations. Chance of randomly selecting a red token, replacing it, then randomly selecting a white token is the same as the chance of randomly selecting a blue token.", "topic": "probability"}
{"id": "book 2_p38_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 38, "page_end": 38, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: In simpler words, if I pick a red, then a white, that’s the same chance as if I pick a blue. Okay, what else?” “multiple of 3” thing yet.", "topic": "probability"}
{"id": "book 2_p38_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 38, "page_end": 38, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Smallest possible total number of tokens in the jar? They’re small, but not tiny.", "topic": "probability"}
{"id": "book 2_p39_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 39, "page_end": 39, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. TD: “Let’s Reflect for a moment to figure out a Plan.", "topic": "algebra"}
{"id": "book 2_p39_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 39, "page_end": 39, "topic_guess": "algebra", "text": "GMAT algebra note: How about algebra— if I name the number of each color, then I can represent each fact and also what I’m looking for. Multiply red and white fractions.", "topic": "algebra"}
{"id": "book 2_p39_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 39, "page_end": 39, "topic_guess": "algebra", "text": "GMAT algebra note: Simplify algebraically.” BU is now unsure. The chance of randomly selecting a red token, replacing it, then randomly selecting a white token is the same as the chance of randomly selecting a blue token...", "topic": "algebra"}
{"id": "book 2_p39_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 39, "page_end": 39, "topic_guess": "algebra", "text": "GMAT algebra note: TD: “Let’s start over conceptually. Can I learn anything interesting?” BU notices that blues are different.", "topic": "algebra"}
{"id": "book 2_p39_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 39, "page_end": 39, "topic_guess": "algebra", "text": "GMAT algebra note: TD: “How are blues different? Picking a red, then a white is as likely as picking a blue.", "topic": "algebra"}
{"id": "book 2_p39_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 39, "page_end": 39, "topic_guess": "algebra", "text": "GMAT algebra note: What does that mean?” BU notices that it’s unlikely to pick a blue. So there aren’t many blues compared to reds or whites.", "topic": "algebra"}
{"id": "book 2_p39_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 39, "page_end": 39, "topic_guess": "algebra", "text": "GMAT algebra note: Fewer blues than reds or whites B < R and B < W TD: “Are there fewer blues? Focus on the algebraic setup.”.", "topic": "algebra"}
{"id": "book 2_p40_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 40, "page_end": 40, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. In the very first equation above, each fraction on the le is less than 1, so their product is even smaller.", "topic": "algebra"}
{"id": "book 2_p40_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 40, "page_end": 40, "topic_guess": "algebra", "text": "GMAT algebra note: The denominators of the three fractions are all the same. BU notices fractions less than 1.", "topic": "algebra"}
{"id": "book 2_p40_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 40, "page_end": 40, "topic_guess": "algebra", "text": "GMAT algebra note: TD: “Two positive fractions less than 1 multiplied together give an even smaller number.” TD: “Yes, there are fewer blues.” BU is quiet. If the number of tokens of every color is a multiple of 3, what is the smallest possible total number of tokens in the jar?", "topic": "algebra"}
{"id": "book 2_p40_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 40, "page_end": 40, "topic_guess": "algebra", "text": "GMAT algebra note: Neither R nor W can equal 3 (since B is smaller than either). The smallest possible total is now 15.", "topic": "algebra"}
{"id": "book 2_p40_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 40, "page_end": 40, "topic_guess": "algebra", "text": "GMAT algebra note: Algebra by itself isn’t getting me there. Try the most constrained variable:.", "topic": "algebra"}
{"id": "book 2_p40_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 40, "page_end": 40, "topic_guess": "algebra", "text": "GMAT algebra note: Since it’s the smallest quantity, but still positive, pretend B is 3. Divide by RW.” BU likes having only two variables.", "topic": "algebra"}
{"id": "book 2_p40_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 40, "page_end": 40, "topic_guess": "algebra", "text": "GMAT algebra note: TD: “Need to test other numbers. Apply constraints I know—B is the smallest number.", "topic": "algebra"}
{"id": "book 2_p41_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 41, "page_end": 41, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Try the next possibility.” BU doesn’t like breaking the symmetry between R and W.", "topic": "probability"}
{"id": "book 2_p41_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 41, "page_end": 41, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: TD: “Does it matter whether R = 6 and W = 9 or the other way around? Plug in and go.” TD: “This works.", "topic": "probability"}
{"id": "book 2_p41_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 41, "page_end": 41, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Let’s look at another pathway—one that moves more quickly to the back door. ALTERNATIVE SOLUTION TO TRY-IT #0-1 ...", "topic": "probability"}
{"id": "book 2_p41_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 41, "page_end": 41, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Chance of randomly selecting... BU notices “chance.” BU doesn’t like probability.", "topic": "probability"}
{"id": "book 2_p41_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 41, "page_end": 41, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Okay, let’s make sense of this and see whether there are any back doors. The number of tokens of every color is BU notices that there are only limited possibilities for each number.", "topic": "probability"}
{"id": "book 2_p41_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 41, "page_end": 41, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: TD: “Okay, every quantity is a multiple of 3. There are 3, 6, 9, etc., of each color.”.", "topic": "probability"}
{"id": "book 2_p42_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 42, "page_end": 42, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. A jar is filled with red, white, and blue tokens...", "topic": "probability"}
{"id": "book 2_p42_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 42, "page_end": 42, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Beginning assumes that there actually are tokens of each color. Let’s start at the easy end—in this case, the smallest number.” BU notices 9 = 3 + 3 + 3.", "topic": "probability"}
{"id": "book 2_p42_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 42, "page_end": 42, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Select a white: , which is not \"select a blue\" TD: “The only possible way to have 9 total tokens that work? TD: “Only way to have 12 total is 3, 3, and 6.", "topic": "probability"}
{"id": "book 2_p42_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 42, "page_end": 42, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Picking a red and then a white is the same as picking a blue, so the blue should be one of the 3’s. Let’s say red is 3 and white is 6.”.", "topic": "probability"}
{"id": "book 2_p43_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 43, "page_end": 43, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating. Keep going.” BU notices 15 has a few options.", "topic": "word_problems"}
{"id": "book 2_p43_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 43, "page_end": 43, "topic_guess": "general", "text": "GMAT quant note: Make blue the 3.” TD: “Nope. Make blue the 3.” TD: “That’s it!", "topic": "word_problems"}
{"id": "book 2_p44_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 44, "page_end": 44, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Many people find this second approach less stressful and more efficient than the textbook approach. In fact, there is no way to find the correct answer by pure algebra.", "topic": "geometry"}
{"id": "book 2_p44_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 44, "page_end": 44, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Ultimately, you have to test suitable numbers. Try-It #0-2 Arrow , which is a line segment exactly 5 units long with an arrowhead at A, is constructed in the xy-plane.", "topic": "geometry"}
{"id": "book 2_p44_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 44, "page_end": 44, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: The x- and y- coordinates of A and B are integers that satisfy the inequalities 0 ≤ x ≤ 9 and 0 ≤ y ≤ 9. How many different arrows with these properties can be constructed?", "topic": "geometry"}
{"id": "book 2_p44_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 44, "page_end": 44, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: SOLUTION TO TRY-IT #0-2 BU notices “xy-plane.” TD: “Let’s Understand first. This is a coordinate- plane problem.", "topic": "geometry"}
{"id": "book 2_p44_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 44, "page_end": 44, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Put in boundaries as necessary.” BU wonders where this is going. How many different arrows with these TD: “What is the question asking for again?", "topic": "geometry"}
{"id": "book 2_p45_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 45, "page_end": 45, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Properties can be constructed?", "topic": "number_properties"}
{"id": "book 2_p45_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 45, "page_end": 45, "topic_guess": "number_theory", "text": "GMAT number properties note: Reread the question.” BU wonders which properties. The x- and y-coordinates of A and B are to be integers that satisfy the inequalities 0 ≤ x ≤ 9 and 0 ≤ y ≤ 9.", "topic": "number_properties"}
{"id": "book 2_p45_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 45, "page_end": 45, "topic_guess": "number_theory", "text": "GMAT number properties note: TD: “What are the properties of the arrows supposed to be again? Each arrow is 5 units long.” BU notices “integers” and “coordinates” and pictures a pegboard.", "topic": "number_properties"}
{"id": "book 2_p45_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 45, "page_end": 45, "topic_guess": "number_theory", "text": "GMAT number properties note: The tip and the end of the arrow have to touch holes in the pegboard exactly. The Plan is to start counting.", "topic": "number_properties"}
{"id": "book 2_p45_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 45, "page_end": 45, "topic_guess": "number_theory", "text": "GMAT number properties note: How to Organize?” BU imagines many possible arrows. Brute force can’t be the right way forward.", "topic": "number_properties"}
{"id": "book 2_p45_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 45, "page_end": 45, "topic_guess": "number_theory", "text": "GMAT number properties note: The arrows can point in all sorts of different ways. TD: “Let’s simplify the Plan.", "topic": "number_properties"}
{"id": "book 2_p45_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 45, "page_end": 45, "topic_guess": "number_theory", "text": "GMAT number properties note: Let’s focus on just one orientation of arrows—pointing straight up. How many places can the arrow be?” BU wants to go up & down, then right & le.", "topic": "number_properties"}
{"id": "book 2_p45_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 45, "page_end": 45, "topic_guess": "number_theory", "text": "GMAT number properties note: In one column, there are 5 positions for the arrowhead: y = 5, 6, 7, 8, or 9. That’s the same as 9 − 5 + 1, by the way.", "topic": "number_properties"}
{"id": "book 2_p45_c1_r9", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 45, "page_end": 45, "topic_guess": "number_theory", "text": "GMAT number properties note: TD: “Count the positions in one column, then multiply by the number of columns. Be careful to count endpoints.”.", "topic": "number_properties"}
{"id": "book 2_p46_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 46, "page_end": 46, "topic_guess": "geometry", "text": "GMAT geometry note: There are 10 identical columns: x = 0 through x = 9. 5 × 10 = 50 possible positions for the arrow pointing straight up.", "topic": "geometry"}
{"id": "book 2_p46_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 46, "page_end": 46, "topic_guess": "geometry", "text": "GMAT geometry note: 50 × 2 = 100 possible positions for the arrow if it points straight up or to the right. Other possibilities?” BU notices the square is the same vertically as horizontally.", "topic": "geometry"}
{"id": "book 2_p46_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 46, "page_end": 46, "topic_guess": "geometry", "text": "GMAT geometry note: TD: “I get the same result for arrows pointing right. Am I done?” 50 × 4 = 200 possible positions BU wonders about “down” and “le.” TD: “These arrows can point straight down or straight le, too.", "topic": "geometry"}
{"id": "book 2_p46_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 46, "page_end": 46, "topic_guess": "geometry", "text": "GMAT geometry note: Those would have the same result. So there are 50 positions in each of the four directions.", "topic": "geometry"}
{"id": "book 2_p46_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 46, "page_end": 46, "topic_guess": "geometry", "text": "GMAT geometry note: Calculate at this point and evaluate answers. BU is suspicious: somehow too easy.", "topic": "geometry"}
{"id": "book 2_p46_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 46, "page_end": 46, "topic_guess": "geometry", "text": "GMAT geometry note: TD: “Could the arrows be at an angle?” BU notices that the arrow is 5 units long, associated with 3−4−5 triangles.", "topic": "geometry"}
{"id": "book 2_p47_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 47, "page_end": 47, "topic_guess": "geometry", "text": "GMAT geometry note: Put the arrow as the hypotenuse of a 3−4−5 triangle. Looks like there are four ways if I go 3 up and 4 across: up right, up le, down right, down le.” BU is happy.", "topic": "geometry"}
{"id": "book 2_p47_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 47, "page_end": 47, "topic_guess": "geometry", "text": "GMAT geometry note: TD: “Likewise, there must be four ways if I go 4 up and 3 across: again, up right, up le, down right, and down le. There are 7 positions vertically for the arrowhead (9 − 3 + 1) and 6 positions horizontally (9 − 4 + 1), for a total of 7 × 6 = 42 positions.", "topic": "geometry"}
{"id": "book 2_p47_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 47, "page_end": 47, "topic_guess": "geometry", "text": "GMAT geometry note: 8 × 42 = 336 possible positions at an angle. In total, there are 200 + 336 = 536 positions.", "topic": "geometry"}
{"id": "book 2_p47_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 47, "page_end": 47, "topic_guess": "geometry", "text": "GMAT geometry note: TD: “Now count just one of these ways. Be sure to include endpoints.” BU notices the symmetry.", "topic": "geometry"}
{"id": "book 2_p47_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 47, "page_end": 47, "topic_guess": "geometry", "text": "GMAT geometry note: The 3 up, 4 across is the same as the 4 up, 3 across, if you turn the square. TD: “Each of these angled ways will be the same.", "topic": "geometry"}
{"id": "book 2_p47_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 47, "page_end": 47, "topic_guess": "geometry", "text": "GMAT geometry note: There are 8 ways to point the arrow at an angle. There isn’t much of an alternative to the approach above.", "topic": "geometry"}
{"id": "book 2_p48_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 48, "page_end": 48, "topic_guess": "geometry", "text": "GMAT geometry note: SOLUTION TO TRY-IT #0-3 TD: “Okay, let’s Understand this. The problem wants the value of x.", "topic": "geometry"}
{"id": "book 2_p48_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 48, "page_end": 48, "topic_guess": "geometry", "text": "GMAT geometry note: How about a Plan?” BU notices this is an isosceles triangle, because there are two sides labeled x. How about the two equal angles?", "topic": "geometry"}
{"id": "book 2_p48_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 48, "page_end": 48, "topic_guess": "geometry", "text": "GMAT geometry note: Divide 135° equally across the two missing angles. TD: “Figure out the two missing angles.", "topic": "geometry"}
{"id": "book 2_p48_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 48, "page_end": 48, "topic_guess": "geometry", "text": "GMAT geometry note: Use the 180° rule.” BU doesn’t recognize this triangle.", "topic": "geometry"}
{"id": "book 2_p49_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 49, "page_end": 49, "topic_guess": "geometry", "text": "GMAT geometry note: TD: “Hmm...here’s a Plan: add a perpendicular line to make right triangles. Drop the line from the top point.", "topic": "geometry"}
{"id": "book 2_p49_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 49, "page_end": 49, "topic_guess": "geometry", "text": "GMAT geometry note: I'll label corners while I'm at it. Now fill in angles.” BU notices 45−45−90 and is happy.", "topic": "geometry"}
{"id": "book 2_p49_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 49, "page_end": 49, "topic_guess": "geometry", "text": "GMAT geometry note: TD: “Use the 45−45−90 to write expressions for its sides. Then can be split up into two pieces, and I can set up the Pythagorean theorem.” BU feels that this process is kind of ugly.", "topic": "geometry"}
{"id": "book 2_p49_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 49, "page_end": 49, "topic_guess": "geometry", "text": "GMAT geometry note: Write the Pythagorean theorem for the small triangle on the le, using the as the hypotenuse.” BU thinks this equation is really ugly. Expand the quadratic and simplify.” BU doesn’t like the square root on the bottom.", "topic": "geometry"}
{"id": "book 2_p49_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 49, "page_end": 49, "topic_guess": "geometry", "text": "GMAT geometry note: TD: “Multiply by to get rid of it on the - bottom of the fraction.”.", "topic": "geometry"}
{"id": "book 2_p50_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 50, "page_end": 50, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. BU has no idea how to take the square root of this.", "topic": "algebra"}
{"id": "book 2_p50_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 50, "page_end": 50, "topic_guess": "algebra", "text": "GMAT algebra note: If x2 is about 3.5, then the square root must be a bit less than 2 (since the square root of 4 is 2). That matches the squared value, not the square root.", "topic": "algebra"}
{"id": "book 2_p50_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 50, "page_end": 50, "topic_guess": "algebra", "text": "GMAT algebra note: The method you just saw is algebraically intensive, and so your bottom-up bloodhound might have kicked up a fuss along the way. Sometimes, your top-down brain needs to ignore the bottom-up brain.", "topic": "algebra"}
{"id": "book 2_p50_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 50, "page_end": 50, "topic_guess": "algebra", "text": "GMAT algebra note: Remember, when you’re actually taking the GMAT, you have to solve problems quickly—and you don’t need to publish your solutions in a mathematics journal. What you want is to get the correct answer as quickly and as easily as possible.", "topic": "algebra"}
{"id": "book 2_p50_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 50, "page_end": 50, "topic_guess": "algebra", "text": "GMAT algebra note: In this regard, the solution above works perfectly well.", "topic": "algebra"}
{"id": "book 2_p51_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 51, "page_end": 51, "topic_guess": "geometry", "text": "GMAT geometry note: Alternatively, the question stem asks for an approximate answer, so you can also try estimating from the start. Draw the triangle carefully and start with the same perpendicular line as before.", "topic": "geometry"}
{"id": "book 2_p51_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 51, "page_end": 51, "topic_guess": "geometry", "text": "GMAT geometry note: This line is a little shorter than the side of length (which is about 1.4). Call the two shorter legs 1.2 and calculate the hypotenuse.", "topic": "geometry"}
{"id": "book 2_p51_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 51, "page_end": 51, "topic_guess": "geometry", "text": "GMAT geometry note: It equals 1.2 multiplied by 1.4, or approximately 1.7. How can you estimate that math quickly?", "topic": "geometry"}
{"id": "book 2_p51_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 51, "page_end": 51, "topic_guess": "geometry", "text": "GMAT geometry note: Unfortunately, you might guess wrong at this point. But the odds are much better than they were at the outset.", "topic": "geometry"}
{"id": "book 2_p51_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 51, "page_end": 51, "topic_guess": "geometry", "text": "GMAT geometry note: It is worthwhile to look for multiple solution paths as you practice. Your top-down brain will become faster, more organized, and more flexible, enabling your bottom-up brain to have more flashes of insight.", "topic": "geometry"}
{"id": "book 2_p51_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 51, "page_end": 51, "topic_guess": "geometry", "text": "GMAT geometry note: That was a substantial introduction.", "topic": "geometry"}
{"id": "book 2_p52_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 52, "page_end": 52, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "book 2_p54_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 54, "page_end": 54, "topic_guess": "strategy", "text": "GMAT strategy note: Focus on equations, expressions, variables, and step-by-step manipulation. Understand the Basics Principle #2:.", "topic": "algebra"}
{"id": "book 2_p54_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 54, "page_end": 54, "topic_guess": "strategy", "text": "GMAT strategy note: Solve—and Put Pen to Paper Principle #4:.", "topic": "algebra"}
{"id": "book 2_p55_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 55, "page_end": 55, "topic_guess": "strategy", "text": "GMAT strategy note: Track mean, median, range, distribution, and how operations change spread. Advanced Principles Chapters 1 and 2 of this book focus on the more fundamental of the two types of GMAT math questions:.", "topic": "statistics"}
{"id": "book 2_p55_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 55, "page_end": 55, "topic_guess": "strategy", "text": "GMAT strategy note: However, Chapters 3 and 4 deal specifically with DS issues. You’ve already seen very basic versions of the first three principles in the introduction, in the dialogs between the top-down and the bottom-up brain.", "topic": "statistics"}
{"id": "book 2_p55_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 55, "page_end": 55, "topic_guess": "strategy", "text": "GMAT strategy note: As mentioned earlier, these principles draw on the work of George Pólya, who was a brilliant mathematician and teacher of mathematics. Pólya was teaching future mathematicians, not GMAT test-takers, but what he said still applies.", "topic": "statistics"}
{"id": "book 2_p55_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 55, "page_end": 55, "topic_guess": "strategy", "text": "GMAT strategy note: His little book How to Solve It has been in print since 1945—it’s worth getting a copy. In the meantime, keep reading.", "topic": "statistics"}
{"id": "book 2_p56_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 56, "page_end": 56, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Understand the Basics Take time to think and plan before you start solving a difficult problem.", "topic": "word_problems"}
{"id": "book 2_p56_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 56, "page_end": 56, "topic_guess": "word_problems", "text": "GMAT word problem note: If Quant is your strength, you may want to dive straight into every problem as soon as you see it, without pausing to consider all of the angles. There are two good reasons to slow down:.", "topic": "word_problems"}
{"id": "book 2_p56_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 56, "page_end": 56, "topic_guess": "word_problems", "text": "GMAT word problem note: To remind yourself to slow down and plan, understand each problem by taking three steps:. Glance at the entire problem: is it PS or DS?", "topic": "word_problems"}
{"id": "book 2_p56_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 56, "page_end": 56, "topic_guess": "word_problems", "text": "GMAT word problem note: If it’s DS, glance at the statements. Knowing what type of problem you’re dealing with will help you read more effectively.", "topic": "word_problems"}
{"id": "book 2_p56_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 56, "page_end": 56, "topic_guess": "word_problems", "text": "GMAT word problem note: Pólya recommended that you ask yourself simple questions as you read a problem. Here are some great Pólya-style questions that can help you understand:.", "topic": "word_problems"}
{"id": "book 2_p56_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 56, "page_end": 56, "topic_guess": "word_problems", "text": "GMAT word problem note: You need to manage your time and mental energy across the entire GMAT. If you pause briefly to find a more efficient solution, you’ll save time and energy for other problems.", "topic": "word_problems"}
{"id": "book 2_p56_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 56, "page_end": 56, "topic_guess": "word_problems", "text": "GMAT word problem note: If you start doing math without thinking first, you might have to change your approach later in the problem, which takes time that you don’t have. You also risk falling for traps.", "topic": "word_problems"}
{"id": "book 2_p57_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 57, "page_end": 57, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. You don’t need to meticulously go through every one of these questions whenever you solve a problem.", "topic": "word_problems"}
{"id": "book 2_p57_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 57, "page_end": 57, "topic_guess": "word_problems", "text": "GMAT word problem note: (However, that’s a good thing to do when you review a problem!) They’re here to help you consider how you might read more productively. As you read the problem, jot down any given numbers or formulas on your scrap paper.", "topic": "word_problems"}
{"id": "book 2_p57_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 57, "page_end": 57, "topic_guess": "word_problems", "text": "GMAT word problem note: That doesn’t mean you should start doing math while you’re trying to read. If you start trying to solve the problem when you haven’t even finished reading it, you’re getting ahead of yourself.", "topic": "word_problems"}
{"id": "book 2_p57_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 57, "page_end": 57, "topic_guess": "word_problems", "text": "GMAT word problem note: Start the following problem by taking these three steps:. What exactly is the problem asking for?", "topic": "word_problems"}
{"id": "book 2_p57_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 57, "page_end": 57, "topic_guess": "word_problems", "text": "GMAT word problem note: What information would I need in order to find the answer? What information do I already have?", "topic": "word_problems"}
{"id": "book 2_p57_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 57, "page_end": 57, "topic_guess": "word_problems", "text": "GMAT word problem note: What information don’t I have? Sometimes you care about something you don’t know.", "topic": "word_problems"}
{"id": "book 2_p57_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 57, "page_end": 57, "topic_guess": "word_problems", "text": "GMAT word problem note: This could be an intermediate unknown quantity that you didn’t think of earlier. Other times, you don’t know something, and you don’t care.", "topic": "word_problems"}
{"id": "book 2_p57_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 57, "page_end": 57, "topic_guess": "word_problems", "text": "GMAT word problem note: For instance, if a problem includes the quantity 11! (11 factorial), you will practically never need to know the exact value of that quantity.", "topic": "word_problems"}
{"id": "book 2_p57_c1_r9", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 57, "page_end": 57, "topic_guess": "word_problems", "text": "GMAT word problem note: What is this problem testing? In other words, why is this problem on the GMAT?", "topic": "word_problems"}
{"id": "book 2_p57_c1_r10", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 57, "page_end": 57, "topic_guess": "word_problems", "text": "GMAT word problem note: What aspect of math are they testing? What kind of reasoning do they want me to demonstrate?", "topic": "word_problems"}
{"id": "book 2_p58_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 58, "page_end": 58, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Try-It #1-1 x = 910 − 317 and is an integer.", "topic": "number_properties"}
{"id": "book 2_p58_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 58, "page_end": 58, "topic_guess": "number_theory", "text": "GMAT number properties note: If n is a positive integer that has exactly two factors, how many different values for n are possible? The answers are numbers but in written form.", "topic": "number_properties"}
{"id": "book 2_p58_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 58, "page_end": 58, "topic_guess": "number_theory", "text": "GMAT number properties note: This format is reserved for problems that ask for the number of numbers or number of possibilities for something. Here are some possible answers to the Pólya questions:.", "topic": "number_properties"}
{"id": "book 2_p58_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 58, "page_end": 58, "topic_guess": "number_theory", "text": "GMAT number properties note: What exactly is the problem asking for? The number of possible values for n.", "topic": "number_properties"}
{"id": "book 2_p58_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 58, "page_end": 58, "topic_guess": "number_theory", "text": "GMAT number properties note: This means that n might have multiple possible values. In fact, it probably can take on more than one value.", "topic": "number_properties"}
{"id": "book 2_p58_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 58, "page_end": 58, "topic_guess": "number_theory", "text": "GMAT number properties note: I may not need these actual values. What are the quantities I care about?", "topic": "number_properties"}
{"id": "book 2_p58_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 58, "page_end": 58, "topic_guess": "number_theory", "text": "GMAT number properties note: I’m given x and n as variables. These are the quantities I care about.", "topic": "number_properties"}
{"id": "book 2_p59_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 59, "page_end": 59, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. X = 910 − 317 That is, x = a specific large integer, expressed in terms of powers of 9 and 3.", "topic": "number_properties"}
{"id": "book 2_p59_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 59, "page_end": 59, "topic_guess": "number_theory", "text": "GMAT number properties note: That is, x is divisible by n, or n is a factor of x. Finally, n is a positive integer that has exactly two factors.", "topic": "number_properties"}
{"id": "book 2_p59_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 59, "page_end": 59, "topic_guess": "number_theory", "text": "GMAT number properties note: Prime numbers have exactly two factors. So I can rephrase the information: n is prime.", "topic": "number_properties"}
{"id": "book 2_p59_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 59, "page_end": 59, "topic_guess": "number_theory", "text": "GMAT number properties note: (Primes are always positive.) What don’t I know? Here’s something I don’t know:.", "topic": "number_properties"}
{"id": "book 2_p59_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 59, "page_end": 59, "topic_guess": "number_theory", "text": "GMAT number properties note: I don’t know the value of x as a series of digits. Using a calculator or Excel, I could find out that x equals 3,357,644,238.", "topic": "number_properties"}
{"id": "book 2_p59_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 59, "page_end": 59, "topic_guess": "number_theory", "text": "GMAT number properties note: But I don’t know this number at the outset. Moreover, because this calculation is far too cumbersome, it must be the case that I don’t need to find this number.", "topic": "number_properties"}
{"id": "book 2_p59_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 59, "page_end": 59, "topic_guess": "number_theory", "text": "GMAT number properties note: What is this problem testing? From the foregoing, I can infer that this problem is testing Divisibility & Primes.", "topic": "number_properties"}
{"id": "book 2_p59_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 59, "page_end": 59, "topic_guess": "number_theory", "text": "GMAT number properties note: I will probably also need to manipulate exponents, since I see them in the expression for x. You can ask these questions in whatever order is most helpful for the problem.", "topic": "number_properties"}
{"id": "book 2_p59_c1_r9", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 59, "page_end": 59, "topic_guess": "number_theory", "text": "GMAT number properties note: For instance, you might not look at what the problem is asking for until you’ve understood the given information. As you decide that a piece of information is important, jot it down on your scrap paper.", "topic": "number_properties"}
{"id": "book 2_p59_c1_r10", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 59, "page_end": 59, "topic_guess": "number_theory", "text": "GMAT number properties note: At this point, your scrap paper might look something like this.", "topic": "number_properties"}
{"id": "book 2_p61_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 61, "page_end": 61, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Build a Plan Next, think about how you will solve the problem:.", "topic": "statistics"}
{"id": "book 2_p61_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 61, "page_end": 61, "topic_guess": "statistics", "text": "GMAT statistics note: Here are some Pólya questions that help you think about what you know and come up with a plan:. Is a good approach already obvious?", "topic": "statistics"}
{"id": "book 2_p61_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 61, "page_end": 61, "topic_guess": "statistics", "text": "GMAT statistics note: From your answers above, you may already see a way to reach the answer. If you can envision the rough outlines of the correct path, then go ahead and get started.", "topic": "statistics"}
{"id": "book 2_p61_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 61, "page_end": 61, "topic_guess": "statistics", "text": "GMAT statistics note: If not, what in the problem can help me figure out a good approach? If you are stuck, look for particular clues to tell you what to do next.", "topic": "statistics"}
{"id": "book 2_p61_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 61, "page_end": 61, "topic_guess": "statistics", "text": "GMAT statistics note: Revisit your answers to the basic questions. Can you rephrase or reword them?", "topic": "statistics"}
{"id": "book 2_p61_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 61, "page_end": 61, "topic_guess": "statistics", "text": "GMAT statistics note: Can you combine two pieces of information in any way, or can you rephrase the question, given everything you know? Can I remember a similar problem?", "topic": "statistics"}
{"id": "book 2_p61_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 61, "page_end": 61, "topic_guess": "statistics", "text": "GMAT statistics note: Try relating the problem to other problems you’ve faced. This can help you categorize the problem or recall a solution process.", "topic": "statistics"}
{"id": "book 2_p62_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 62, "page_end": 62, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. For the Try-It #1-1 problem, some of the information is already rephrased (reorganized).", "topic": "number_properties"}
{"id": "book 2_p62_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 62, "page_end": 62, "topic_guess": "number_theory", "text": "GMAT number properties note: Go further now, combining information and simplifying the question:. Given: n is a prime number AND n is a factor of x.", "topic": "number_properties"}
{"id": "book 2_p62_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 62, "page_end": 62, "topic_guess": "number_theory", "text": "GMAT number properties note: Combined: n is a prime factor of x. How many different values for n are possible?", "topic": "number_properties"}
{"id": "book 2_p62_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 62, "page_end": 62, "topic_guess": "number_theory", "text": "GMAT number properties note: How many different values for n, a prime factor of x, are possible? How many distinct prime factors does x have?", "topic": "number_properties"}
{"id": "book 2_p62_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 62, "page_end": 62, "topic_guess": "number_theory", "text": "GMAT number properties note: You need the prime factorization of x. Notice that n is not even in the question anymore.", "topic": "number_properties"}
{"id": "book 2_p62_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 62, "page_end": 62, "topic_guess": "number_theory", "text": "GMAT number properties note: The variable n just gave you a way to ask this underlying question. It can be helpful initially to put certain complicated facts to the side.", "topic": "number_properties"}
{"id": "book 2_p62_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 62, "page_end": 62, "topic_guess": "number_theory", "text": "GMAT number properties note: At this stage, however, you know that you need the prime factors of x. So now you have the beginning of a plan: factor this expression into its primes.", "topic": "number_properties"}
{"id": "book 2_p63_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 63, "page_end": 63, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Solve—and Put Pen to Paper The third step is to do the work: solve.", "topic": "number_properties"}
{"id": "book 2_p63_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 63, "page_end": 63, "topic_guess": "number_theory", "text": "GMAT number properties note: You’ll want to execute that solution in an error-free way—it would be terrible to get all the thinking correct, then make a careless computational mistake. That’s why we say you should put pen to paper.", "topic": "number_properties"}
{"id": "book 2_p63_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 63, "page_end": 63, "topic_guess": "number_theory", "text": "GMAT number properties note: In the expression 910 − 317, the 3 is prime but the 9 is not. Since the problem is asking about prime factors, rewrite the equation in terms of prime numbers:.", "topic": "number_properties"}
{"id": "book 2_p63_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 63, "page_end": 63, "topic_guess": "number_theory", "text": "GMAT number properties note: Next, pull out a common factor from both terms. The largest common factor is Now, you have what you need: the prime factorization of x.", "topic": "number_properties"}
{"id": "book 2_p63_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 63, "page_end": 63, "topic_guess": "number_theory", "text": "GMAT number properties note: The number x has three distinct prime factors: 2, 3, and 13. The idea of putting pen to paper also applies when you get stuck anywhere along the way on a monster problem.", "topic": "number_properties"}
{"id": "book 2_p64_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 64, "page_end": 64, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Think back to those killer Try-It problems in the introduction.", "topic": "algebra"}
{"id": "book 2_p64_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 64, "page_end": 64, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Those are not the kinds of problems you can figure out just by looking at them. When you get stuck on a tough problem, take action.", "topic": "algebra"}
{"id": "book 2_p64_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 64, "page_end": 64, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Do not just stare, hoping that you suddenly get it. Instead, ask yourself the Pólya questions again and write down whatever you can:.", "topic": "algebra"}
{"id": "book 2_p64_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 64, "page_end": 64, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: This way, your top-down brain can help your bottom-up brain find the correct leads—or help it let go. In particular, it’s almost impossible to abandon an unpromising line of thinking without writing something down.", "topic": "algebra"}
{"id": "book 2_p64_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 64, "page_end": 64, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Think back to the sequence problem in the introduction. You’ll keep seeing 64 as 82 unless you try writing it in another way.", "topic": "algebra"}
{"id": "book 2_p64_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 64, "page_end": 64, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Do not try to juggle everything in your head. Your working memory has limited capacity, and your bottom-up brain needs that space to work.", "topic": "algebra"}
{"id": "book 2_p64_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 64, "page_end": 64, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: A multistep problem cannot be solved in your brain as quickly, easily, and accurately as it can be on paper. As you put pen to paper, there are three themes you'll want to keep in mind.", "topic": "algebra"}
{"id": "book 2_p64_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 64, "page_end": 64, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: LOOK FOR PATTERNS Reinterpretations of given information or of the question Intermediate results, whether specific or general Avenues or approaches that didn’t work.", "topic": "algebra"}
{"id": "book 2_p65_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 65, "page_end": 65, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Every GMAT Quant problem has a two-minutes-or-faster solution path, which may depend upon a pattern that you’ll need to extrapolate.", "topic": "algebra"}
{"id": "book 2_p65_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 65, "page_end": 65, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: You’ll know a pattern is needed when a problem asks something that would be impossible to calculate (without a calculator) in two minutes. When this happens, write out the first five to eight items in the sequence or list in order to try to spot the pattern.", "topic": "algebra"}
{"id": "book 2_p65_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 65, "page_end": 65, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Try-It #1-2 for all integer values of n greater than 1. If S1 = 1, what is the sum of the first 61 terms in the sequence?", "topic": "algebra"}
{"id": "book 2_p65_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 65, "page_end": 65, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Nobody is going to write out all 61 terms and then add them up in two minutes. The recursive definition of Sn doesn’t yield any secrets upon first glance.", "topic": "algebra"}
{"id": "book 2_p66_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 66, "page_end": 66, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. The terms of the sequence are ....", "topic": "algebra"}
{"id": "book 2_p66_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 66, "page_end": 66, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Three terms repeat in this cyclical pattern forever. Every third term is the same.", "topic": "algebra"}
{"id": "book 2_p66_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 66, "page_end": 66, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: If you don’t spot a pattern within the first five to eight terms, stop using this approach and see whether there’s another way (including guessing!). The problem asks for the sum, so find the sum of each group of three consecutive terms: .", "topic": "algebra"}
{"id": "book 2_p66_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 66, "page_end": 66, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: There are 20 groups in the first 61 terms, and one additional term that hasn’t been counted yet. So the sum of the first 61 terms is as follows.", "topic": "algebra"}
{"id": "book 2_p67_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 67, "page_end": 67, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. It is almost impossible to stare at the recursive definition of this sequence and discern the resulting pattern.", "topic": "algebra"}
{"id": "book 2_p67_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 67, "page_end": 67, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The best way to identify the pattern is to calculate a few values of the sequence and look for the pattern. DRAW IT OUT Some problems are much easier to solve if you draw out what’s happening in the problem.", "topic": "algebra"}
{"id": "book 2_p67_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 67, "page_end": 67, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Whenever a story problem describes something that could actually happen in the real world, you could try to draw out the solution. For instance, if a problem involves motion, you can draw snapshots representing the problem at different points in time.", "topic": "algebra"}
{"id": "book 2_p67_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 67, "page_end": 67, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Try-It #1-3 Truck A is on a straight highway heading due south at the same time Truck B is on a different straight highway heading due east. At 1:00 p.m., Truck A is exactly 14 miles north of Truck B.", "topic": "algebra"}
{"id": "book 2_p67_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 67, "page_end": 67, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: If both trucks are traveling at a constant speed of 30 miles per hour, at which of the following times will they be exactly 10 miles apart?", "topic": "algebra"}
{"id": "book 2_p68_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 68, "page_end": 68, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Represent Truck A and Truck B as of 1:00 p.m.", "topic": "word_problems"}
{"id": "book 2_p68_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 68, "page_end": 68, "topic_guess": "word_problems", "text": "GMAT word problem note: How does the distance between Truck A and Truck B change as time goes by? Since the answers are all a matter of minutes a er 1:00 p.m., try a convenient increment of a few minutes.", "topic": "word_problems"}
{"id": "book 2_p68_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 68, "page_end": 68, "topic_guess": "word_problems", "text": "GMAT word problem note: A er 10 minutes, each truck will have traveled 5 miles (30 miles per 60 minutes = 5 miles in 10 minutes). How far apart will the trucks be then?", "topic": "word_problems"}
{"id": "book 2_p68_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 68, "page_end": 68, "topic_guess": "word_problems", "text": "GMAT word problem note: Because Truck A is traveling due south and Truck B is traveling due east, the triangle must be a right triangle. At this point, you could solve the problem in one of two ways.", "topic": "word_problems"}
{"id": "book 2_p68_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 68, "page_end": 68, "topic_guess": "word_problems", "text": "GMAT word problem note: Therefore, of an hour later, at 1:12 p.m., the trucks will be exactly 10 miles apart. Alternatively, you could set up an algebraic equation and solve for the unknown number of miles traveled, such that the distance between the trucks is 10.", "topic": "word_problems"}
{"id": "book 2_p69_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 69, "page_end": 69, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Therefore, y could equal 6 or 8 miles.", "topic": "algebra"}
{"id": "book 2_p69_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 69, "page_end": 69, "topic_guess": "algebra", "text": "GMAT algebra note: In other words, the trucks will be exactly 10 miles apart at 1:12 p.m. However, many other kinds of problems can benefit from visual thinking.", "topic": "algebra"}
{"id": "book 2_p69_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 69, "page_end": 69, "topic_guess": "algebra", "text": "GMAT algebra note: SOLVE AN EASIER PROBLEM A problem may contain large numbers or complicated expressions that actually distract you from the task at hand: finding a solution path. When this happens, one tactic is to simplify part of the problem and solve that.", "topic": "algebra"}
{"id": "book 2_p69_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 69, "page_end": 69, "topic_guess": "algebra", "text": "GMAT algebra note: Once you understand how the math works, return to the more complex problem and apply the same solution path.", "topic": "algebra"}
{"id": "book 2_p70_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 70, "page_end": 70, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Try-It #1-4 If x and y are positive integers and is the square of an odd integer, what is the smallest possible value of xy?", "topic": "number_properties"}
{"id": "book 2_p70_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 70, "page_end": 70, "topic_guess": "number_theory", "text": "GMAT number properties note: As you read, jot down the given information. Note that you might not immediately write down the square of an odd integer info if you still have to puzzle out what it means:.", "topic": "number_properties"}
{"id": "book 2_p70_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 70, "page_end": 70, "topic_guess": "number_theory", "text": "GMAT number properties note: What does the square of an odd integer look like?", "topic": "number_properties"}
{"id": "book 2_p71_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 71, "page_end": 71, "topic_guess": "algebra", "text": "GMAT algebra note: Watch divisibility, factors, remainders, parity, and prime structure. Are there any patterns or commonalities?", "topic": "number_properties"}
{"id": "book 2_p71_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 71, "page_end": 71, "topic_guess": "algebra", "text": "GMAT algebra note: All of the numbers are perfect squares. Therefore, is an odd perfect square.", "topic": "number_properties"}
{"id": "book 2_p71_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 71, "page_end": 71, "topic_guess": "algebra", "text": "GMAT algebra note: The question asks for the smallest possible value of xy. What do you need to figure out in order to find that?", "topic": "number_properties"}
{"id": "book 2_p71_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 71, "page_end": 71, "topic_guess": "algebra", "text": "GMAT algebra note: If the expression is distracting you, try figuring out what this would mean for a simpler version of the expression. What if is an odd perfect square?", "topic": "number_properties"}
{"id": "book 2_p71_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 71, "page_end": 71, "topic_guess": "algebra", "text": "GMAT algebra note: In order for the number to be odd, you have to get rid of the even number 20 (because an even number times any number equals an even number). The only way to get rid of the even part is to divide it out by y.", "topic": "number_properties"}
{"id": "book 2_p71_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 71, "page_end": 71, "topic_guess": "algebra", "text": "GMAT algebra note: If y is 4, then the expression would become . As long as x is an odd number, 5x will be odd, too.", "topic": "number_properties"}
{"id": "book 2_p71_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 71, "page_end": 71, "topic_guess": "algebra", "text": "GMAT algebra note: How can you apply that thinking to the real problem? It’s still true that, in order for to be odd, you have to get rid of the even factors in the numerator.", "topic": "number_properties"}
{"id": "book 2_p72_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 72, "page_end": 72, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. In other words, y2 must cancel out all the even factors in 1,620.", "topic": "number_properties"}
{"id": "book 2_p72_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 72, "page_end": 72, "topic_guess": "number_theory", "text": "GMAT number properties note: The y2 must contain at least two 2’s, so y itself has to contain at least one 2. Okay, that takes care of y: at minimum, y must be 2.", "topic": "number_properties"}
{"id": "book 2_p72_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 72, "page_end": 72, "topic_guess": "number_theory", "text": "GMAT number properties note: If so, then the expression becomes Now, what about x? If you’re not sure, return to your simpler problem thinking.", "topic": "number_properties"}
{"id": "book 2_p72_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 72, "page_end": 72, "topic_guess": "number_theory", "text": "GMAT number properties note: In the last step, in order to make 5x odd, x has to be odd. 5x also has to be a perfect square.", "topic": "number_properties"}
{"id": "book 2_p72_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 72, "page_end": 72, "topic_guess": "number_theory", "text": "GMAT number properties note: If you make x = 5, then 5x = 25, an odd perfect square. The expression 5x, therefore, needs a second 5 to make this a perfect square.", "topic": "number_properties"}
{"id": "book 2_p72_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 72, "page_end": 72, "topic_guess": "number_theory", "text": "GMAT number properties note: Make sure that 405x has two pairs of every factor. 405 contains only one 5, so x must contain another 5.", "topic": "number_properties"}
{"id": "book 2_p72_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 72, "page_end": 72, "topic_guess": "number_theory", "text": "GMAT number properties note: Also contained in 405 is 81, which is 92. That set of factors already represents a perfect square, so the minimum requirement is that x equals 5.", "topic": "number_properties"}
{"id": "book 2_p72_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 72, "page_end": 72, "topic_guess": "number_theory", "text": "GMAT number properties note: If y must be 2, at minimum, and x must be 5, at minimum, then the smallest - possible value of xy is 10. You can generalize this approach.", "topic": "number_properties"}
{"id": "book 2_p72_c1_r9", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 72, "page_end": 72, "topic_guess": "number_theory", "text": "GMAT number properties note: If a problem has many complexities, you can attack it by ignoring some of the complexities at first.", "topic": "number_properties"}
{"id": "book 2_p73_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 73, "page_end": 73, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Then, see whether you can adjust the solution to the simpler problem in order to solve the original.", "topic": "algebra"}
{"id": "book 2_p73_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 73, "page_end": 73, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: To recap, put your work on paper. Don’t try to solve hard problems in your head.", "topic": "algebra"}
{"id": "book 2_p73_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 73, "page_end": 73, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: In general, jot down intermediate results as you go. You may see them in a new light and consider how they fit into the solution.", "topic": "algebra"}
{"id": "book 2_p73_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 73, "page_end": 73, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: For instance, make tables to keep track of cases. The more organized you are, the more insights you will have into difficult problems.", "topic": "algebra"}
{"id": "book 2_p73_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 73, "page_end": 73, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Write out the first few cases. Solve an easier problem, then apply your method to the harder problem.", "topic": "algebra"}
{"id": "book 2_p74_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 74, "page_end": 74, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Review Your Work When you are done with a test or practice set, you are not really done.", "topic": "word_problems"}
{"id": "book 2_p74_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 74, "page_end": 74, "topic_guess": "word_problems", "text": "GMAT word problem note: When you first do a problem under timed conditions, your brain is too busy solving the problem to effectively learn and remember. What you learn from a new problem comes a er you’ve finished it and picked your answer, when you look at it with a clear head and no timer.", "topic": "word_problems"}
{"id": "book 2_p74_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 74, "page_end": 74, "topic_guess": "word_problems", "text": "GMAT word problem note: Give yourself twice as much time to review each problem as you spent doing the problem in the first place. Here are some things you might consider as you review a problem.", "topic": "word_problems"}
{"id": "book 2_p74_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 74, "page_end": 74, "topic_guess": "word_problems", "text": "GMAT word problem note: Most of these questions are useful even if you got the problem correct. Don’t restrict yourself to reviewing problems you got wrong.", "topic": "word_problems"}
{"id": "book 2_p74_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 74, "page_end": 74, "topic_guess": "word_problems", "text": "GMAT word problem note: Review any problem you might learn something from and ask yourself:. What are all of the pathways to the answer?", "topic": "word_problems"}
{"id": "book 2_p74_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 74, "page_end": 74, "topic_guess": "word_problems", "text": "GMAT word problem note: What is the easiest and fastest way to implement it? What clues in the problem told me to use a certain approach or take a certain step?", "topic": "word_problems"}
{"id": "book 2_p74_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 74, "page_end": 74, "topic_guess": "word_problems", "text": "GMAT word problem note: If I see one of those clues in a different problem, what should I do? What traps or tricks are built into this problem?", "topic": "word_problems"}
{"id": "book 2_p74_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 74, "page_end": 74, "topic_guess": "word_problems", "text": "GMAT word problem note: Where could I have made a mistake? If I did make a mistake, what went wrong in my problem-solving process?", "topic": "word_problems"}
{"id": "book 2_p74_c1_r9", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 74, "page_end": 74, "topic_guess": "word_problems", "text": "GMAT word problem note: Do I need to change how I approach similar problems?", "topic": "word_problems"}
{"id": "book 2_p75_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 75, "page_end": 75, "topic_guess": "general", "text": "GMAT quant note: Focus on equations, expressions, variables, and step-by-step manipulation. What could I take from this problem to help me solve other problems in the future?", "topic": "algebra"}
{"id": "book 2_p76_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 76, "page_end": 76, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Understand, Plan, and Solve.", "topic": "probability"}
{"id": "book 2_p76_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 76, "page_end": 76, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Then, review each problem in depth. As you review, do two things:.", "topic": "probability"}
{"id": "book 2_p76_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 76, "page_end": 76, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: The solutions include our own responses to these two tasks. Yours might look different, and that's fine.", "topic": "probability"}
{"id": "book 2_p76_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 76, "page_end": 76, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Problem Set Identify exactly what the problem is asking for and what that means in the simplest possible terms. Note at least one general takeaway that might be useful on other problems in the future.", "topic": "probability"}
{"id": "book 2_p76_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 76, "page_end": 76, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Each factor of 210 is inscribed on its own plastic ball and all of the balls are placed in a jar. If a ball is randomly selected from the jar, what is the probability that the ball is inscribed with a multiple of.", "topic": "probability"}
{"id": "book 2_p77_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 77, "page_end": 77, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. If x is a positive integer, what is the units digit of (24)5 + 2x(36)6(17)3?", "topic": "algebra"}
{"id": "book 2_p77_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 77, "page_end": 77, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: A baker makes a combination of chocolate chip cookies and peanut butter cookies for a school bake sale. His recipes only allow him to make chocolate chip cookies in batches of 7 and peanut butter cookies in batches of 6.", "topic": "algebra"}
{"id": "book 2_p77_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 77, "page_end": 77, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: If he makes exactly 95 cookies for the bake sale, what is the minimum number of chocolate chip cookies that he could make?", "topic": "algebra"}
{"id": "book 2_p78_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 78, "page_end": 78, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. A rectangular solid is changed such that the width and length are each increased by 1 inch and the height is decreased by 9 inches.", "topic": "algebra"}
{"id": "book 2_p78_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 78, "page_end": 78, "topic_guess": "algebra", "text": "GMAT algebra note: Despite these changes, the new rectangular solid has the same volume as the original rectangular solid. If the width and length of the original rectangular solid are equal and the height of the new rectangular solid is 4 times the width of the original rectangular solid, what is the volume of the rectangular solid?", "topic": "algebra"}
{"id": "book 2_p78_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 78, "page_end": 78, "topic_guess": "algebra", "text": "GMAT algebra note: The sum of all distinct solutions for x in the equation x2 − 8x + 21 = |x − 4| + 5 is equal to which of the following?", "topic": "algebra"}
{"id": "book 2_p79_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 79, "page_end": 79, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "book 2_p80_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 80, "page_end": 80, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Solutions Each solution addresses the two steps from the instructions:.", "topic": "number_properties"}
{"id": "book 2_p80_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 80, "page_end": 80, "topic_guess": "number_theory", "text": "GMAT number properties note: Identify exactly what the problem is asking for, and what that means in the simplest possible terms. Note at least one general takeaway that might useful on other problems in the future.", "topic": "number_properties"}
{"id": "book 2_p80_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 80, "page_end": 80, "topic_guess": "number_theory", "text": "GMAT number properties note: The problem is asking for the probability that the selected ball is a multiple of 42. The quantities you care about are the factors of 210.", "topic": "number_properties"}
{"id": "book 2_p80_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 80, "page_end": 80, "topic_guess": "number_theory", "text": "GMAT number properties note: There are many balls, each with a different factor of 210. Each factor of 210 is represented.", "topic": "number_properties"}
{"id": "book 2_p80_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 80, "page_end": 80, "topic_guess": "number_theory", "text": "GMAT number properties note: One ball is selected randomly. Some balls have a multiple of 42 (e.g., 42 itself).", "topic": "number_properties"}
{"id": "book 2_p80_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 80, "page_end": 80, "topic_guess": "number_theory", "text": "GMAT number properties note: How many factors of 210 there are How many of these factors are multiples of 42 What the problem is testing:.", "topic": "number_properties"}
{"id": "book 2_p81_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 81, "page_end": 81, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Plan: 210 to primes → build full list of factors from prime components → distinguish between multiples of 42 and non-multiples → count factors → compute probability.", "topic": "number_properties"}
{"id": "book 2_p81_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 81, "page_end": 81, "topic_guess": "number_theory", "text": "GMAT number properties note: Alternatively, you could list all the factors of 210 using factor pairs.", "topic": "number_properties"}
{"id": "book 2_p82_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 82, "page_end": 82, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. There are 16 factors of 210, and two of them (42 and 210) are multiples of 42.", "topic": "number_properties"}
{"id": "book 2_p82_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 82, "page_end": 82, "topic_guess": "number_theory", "text": "GMAT number properties note: You can also count the factors using 210’s prime factorization: (2)(3)(5) Here’s a shortcut to determine the number of distinct factors of 210. Add 1 to the power of each prime factor and multiply:.", "topic": "number_properties"}
{"id": "book 2_p82_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 82, "page_end": 82, "topic_guess": "number_theory", "text": "GMAT number properties note: There are 16 different factors of 210: 2 x 2 x 2 x 2 = 16. How many of these 16 factors are multiples of 42?", "topic": "number_properties"}
{"id": "book 2_p82_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 82, "page_end": 82, "topic_guess": "number_theory", "text": "GMAT number properties note: 42 itself is a multiple of 42, of course. To find any others, divide 210 by 42 to get 5.", "topic": "number_properties"}
{"id": "book 2_p82_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 82, "page_end": 82, "topic_guess": "number_theory", "text": "GMAT number properties note: This number is a prime, so the only other possible factor is 42 × 5, or 210. The problem is straightforward in one sense: it says the word factor explicitly.", "topic": "number_properties"}
{"id": "book 2_p82_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 82, "page_end": 82, "topic_guess": "number_theory", "text": "GMAT number properties note: Listing all the factors is feasible in two minutes, but you do need to be going down that solution path fairly quickly because it will take some time. It may be slightly faster to use.", "topic": "number_properties"}
{"id": "book 2_p83_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 83, "page_end": 83, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. The factor-counting shortcut, but only if you do know how to deal with the multiples of 42.", "topic": "number_properties"}
{"id": "book 2_p83_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 83, "page_end": 83, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The problem asks for the units digit. Because the problem talks about a product, you care only about the units digits, not the overall values.", "topic": "number_properties"}
{"id": "book 2_p83_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 83, "page_end": 83, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Furthermore, the problem provides crazy numbers. You are absolutely not going to multiply these out.", "topic": "number_properties"}
{"id": "book 2_p83_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 83, "page_end": 83, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: There must be some kind of pattern at work. Use the Last Digit Shortcut (discussed in the All the Quant guide).", "topic": "number_properties"}
{"id": "book 2_p83_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 83, "page_end": 83, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: If x is a positive integer, then 2x must be even and 5 + 2x must be odd. Units digit of (24)5 + 2x = units digit of (4)odd.", "topic": "number_properties"}
{"id": "book 2_p83_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 83, "page_end": 83, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The pattern for the units digit of 4integer = [4, 6]. Units digit of (36)6 must be 6, as every power of 6 ends in 6.", "topic": "number_properties"}
{"id": "book 2_p83_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 83, "page_end": 83, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Units digit of (17)3 = units digit of (7)3. The pattern for the units digit of 7integer = [7, 9, 3, 1].", "topic": "number_properties"}
{"id": "book 2_p83_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 83, "page_end": 83, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The product of the units digits is (4)(6)(3) = 72, which has a units digit of 2. Patterns were very important on this one!", "topic": "number_properties"}
{"id": "book 2_p83_c1_r9", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 83, "page_end": 83, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: If you forget any of the units digit patterns, start listing out the early cases. At most, you’ll need to list four cases to find the pattern.", "topic": "number_properties"}
{"id": "book 2_p84_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 84, "page_end": 84, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The problem asks for the minimum number of chocolate chip cookies.", "topic": "number_properties"}
{"id": "book 2_p84_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 84, "page_end": 84, "topic_guess": "number_theory", "text": "GMAT number properties note: He can only make chocolate chip cookies in batches of 7 and peanut butter cookies in batches of 6. He makes exactly 95 cookies total.", "topic": "number_properties"}
{"id": "book 2_p84_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 84, "page_end": 84, "topic_guess": "number_theory", "text": "GMAT number properties note: Make a chart: 7C 6P = 95 − 7C Is 6P a multiple of 6? (i.e., Is P an integer?) 7 88 N 14 81 N 21 74 N.", "topic": "number_properties"}
{"id": "book 2_p85_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 85, "page_end": 85, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. 7C 6P = 95 − 7C Is 6P a multiple of 6?", "topic": "algebra"}
{"id": "book 2_p85_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 85, "page_end": 85, "topic_guess": "algebra", "text": "GMAT algebra note: The two competing constraints made testing choices the most efficient method. There are three equations and three variables.", "topic": "algebra"}
{"id": "book 2_p85_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 85, "page_end": 85, "topic_guess": "algebra", "text": "GMAT algebra note: What is the easiest way to solve for the volume? The question asks for the volume of the box.", "topic": "algebra"}
{"id": "book 2_p86_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 86, "page_end": 86, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. The width, w, appears in all three constraint equations, so solve for the other variables in terms of w and substitute into the longest constraint:.", "topic": "algebra"}
{"id": "book 2_p86_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 86, "page_end": 86, "topic_guess": "algebra", "text": "GMAT algebra note: Since w can’t be zero, you can divide it out safely. The question is complex enough that you could check your work at the end by also calculating the volume of the new solid (w + 1)(l + 1)(h − 9).", "topic": "algebra"}
{"id": "book 2_p86_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 86, "page_end": 86, "topic_guess": "algebra", "text": "GMAT algebra note: As always, you have to decide whether to spend that time here versus elsewhere.", "topic": "algebra"}
{"id": "book 2_p87_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 87, "page_end": 87, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Notice that, though the initial volume formula seemed long and annoying, the calculations canceled out nicely in the end.", "topic": "algebra"}
{"id": "book 2_p87_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 87, "page_end": 87, "topic_guess": "algebra", "text": "GMAT algebra note: This is common on the GMAT—common enough, in fact, to suspect that you may be doing something wrong if the algebra becomes very messy. The question implies that there may be multiple solutions, as does the nonlinear given equation.", "topic": "algebra"}
{"id": "book 2_p87_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 87, "page_end": 87, "topic_guess": "algebra", "text": "GMAT algebra note: What is the most efficient way to find those solutions? There are actually two good approaches.", "topic": "algebra"}
{"id": "book 2_p87_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 87, "page_end": 87, "topic_guess": "algebra", "text": "GMAT algebra note: Choose the one that is easier for you. Split it into two equations, the “positive” version and the “negative” version:.", "topic": "algebra"}
{"id": "book 2_p87_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 87, "page_end": 87, "topic_guess": "algebra", "text": "GMAT algebra note: The question asks for the sum of the distinct solutions. In other words, if the number 2 were to show up twice as a solution, you would count it only once.", "topic": "algebra"}
{"id": "book 2_p88_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 88, "page_end": 88, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Sum of the different solutions: 5 + 4 + 3 = 12.", "topic": "algebra"}
{"id": "book 2_p88_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 88, "page_end": 88, "topic_guess": "algebra", "text": "GMAT algebra note: Approach #2: simplify the equation and use theory to finish it off. You square a number and get the absolute value of that same number (not squared!).", "topic": "algebra"}
{"id": "book 2_p88_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 88, "page_end": 88, "topic_guess": "algebra", "text": "GMAT algebra note: Only a few numbers can make that true: 1 squared equals |1|, 0 squared equals |0|, −1 squared equals |−1|. When a problem asks for distinct solutions, count each unique solution.", "topic": "algebra"}
{"id": "book 2_p88_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 88, "page_end": 88, "topic_guess": "algebra", "text": "GMAT algebra note: Ignore multiple instances of the same value. Both are valid solution methods.", "topic": "algebra"}
{"id": "book 2_p88_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 88, "page_end": 88, "topic_guess": "algebra", "text": "GMAT algebra note: Which one do you think you will be able to remember and use more easily?", "topic": "algebra"}
{"id": "book 2_p89_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 89, "page_end": 89, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "book 2_p90_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 90, "page_end": 90, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Advanced Strategies Advanced Guessing Tactics.", "topic": "word_problems"}
{"id": "book 2_p91_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 91, "page_end": 91, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Strategies & Tactics Sometimes you will encounter a Problem Solving (PS) problem that you can’t answer—either because its content is difficult or obscure or because you don’t have enough time to solve completely in two minutes.", "topic": "word_problems"}
{"id": "book 2_p91_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 91, "page_end": 91, "topic_guess": "word_problems", "text": "GMAT word problem note: Here, we make the distinction between solution strategies and guessing tactics. Solution strategies are broad: they apply to a wide variety of problems, they provide a complete approach, and they can be used safely in most circumstances.", "topic": "word_problems"}
{"id": "book 2_p91_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 91, "page_end": 91, "topic_guess": "word_problems", "text": "GMAT word problem note: Moreover, a particular tactic may only be useful in special situations or for parts of a problem.", "topic": "word_problems"}
{"id": "book 2_p92_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 92, "page_end": 92, "topic_guess": "strategy", "text": "GMAT strategy note: Translate the wording into variables, relationships, and equations before calculating. One of the most productive strategies on the GMAT is to pick good numbers and plug them into unknowns.", "topic": "word_problems"}
{"id": "book 2_p92_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 92, "page_end": 92, "topic_guess": "strategy", "text": "GMAT strategy note: Try this when the concepts are especially complex or when conditions are placed on key inputs that are otherwise unspecified (e.g., n is a prime number). Doing so can o en help you avoid demanding calculations or the need to set up and solve complicated algebraic expressions.", "topic": "word_problems"}
{"id": "book 2_p92_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 92, "page_end": 92, "topic_guess": "strategy", "text": "GMAT strategy note: Test Cases In certain circumstances, a problem allows multiple possible scenarios, or cases. On PS questions, the problem usually asks you to find something that must be true or could be true.", "topic": "word_problems"}
{"id": "book 2_p92_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 92, "page_end": 92, "topic_guess": "strategy", "text": "GMAT strategy note: On these problems, you can test different numbers to eliminate answers until only one remains. Avoid Needless Computation The GMAT rarely requires you to carry out intensive calculations to arrive at an answer.", "topic": "word_problems"}
{"id": "book 2_p92_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 92, "page_end": 92, "topic_guess": "strategy", "text": "GMAT strategy note: Look for opportunities to avoid tedious computation by factoring, simplifying, or estimating.", "topic": "word_problems"}
{"id": "book 2_p93_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 93, "page_end": 93, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. The correct answer may be part of one of these pairs.", "topic": "word_problems"}
{"id": "book 2_p93_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 93, "page_end": 93, "topic_guess": "word_problems", "text": "GMAT word problem note: In such cases, look for clues as to the correct sign of the correct answer.", "topic": "word_problems"}
{"id": "book 2_p94_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 94, "page_end": 94, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. CHOOSE SMART NUMBERS Some types of problems allow you to pick real numbers and solve the problem arithmetically rather than algebraically.", "topic": "algebra"}
{"id": "book 2_p94_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 94, "page_end": 94, "topic_guess": "algebra", "text": "GMAT algebra note: Likewise, you can o en pick a smart number for a fraction or percent problem without specified absolute value amounts. Other problem types allow this strategy as well.", "topic": "algebra"}
{"id": "book 2_p94_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 94, "page_end": 94, "topic_guess": "algebra", "text": "GMAT algebra note: For instance, a problem may put specific conditions on the inputs but not give you exact numbers. In this case, you can go ahead and just pick inputs that fit the conditions.", "topic": "algebra"}
{"id": "book 2_p94_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 94, "page_end": 94, "topic_guess": "algebra", "text": "GMAT algebra note: If a problem specifies that “x must be a positive even integer” but does not specify the value of x, picking 2 for x will probably get you to a solution quickly and easily. The GMAT Official Guide contains many problems that are difficult to solve algebraically but much easier to solve with real numbers.", "topic": "algebra"}
{"id": "book 2_p94_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 94, "page_end": 94, "topic_guess": "algebra", "text": "GMAT algebra note: However, as an advanced test-taker, you might consider it a point of pride not to plug in a number. You might want to prove a “theoretically correct” answer.", "topic": "algebra"}
{"id": "book 2_p94_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 94, "page_end": 94, "topic_guess": "algebra", "text": "GMAT algebra note: The GMAT tests you on your flexibility of thinking and your ability to manage a very limited amount of time. Use the easiest and most efficient solution path, not the te.", "topic": "algebra"}
{"id": "book 2_p95_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 95, "page_end": 95, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Andra, Elif, and Grady each invested in a certain stock.", "topic": "algebra"}
{"id": "book 2_p95_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 95, "page_end": 95, "topic_guess": "algebra", "text": "GMAT algebra note: Andra invested q dollars, which was 40% more than Elif invested. If Elif invested 25% less than Grady invested, what was the total amount invested by all three, in terms of q?", "topic": "algebra"}
{"id": "book 2_p95_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 95, "page_end": 95, "topic_guess": "algebra", "text": "GMAT algebra note: This problem can be solved algebraically: write a couple of equations and solve for all three variables, then add them up. A glance at the answers, though, indicates that the algebra is likely to get messy.", "topic": "algebra"}
{"id": "book 2_p95_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 95, "page_end": 95, "topic_guess": "algebra", "text": "GMAT algebra note: Instead, choose a real number and solve the problem arithmetically. If you’ve made it to the GMAT Advanced Quant book, then you have likely used this strategy before (or at least learned about it).", "topic": "algebra"}
{"id": "book 2_p95_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 95, "page_end": 95, "topic_guess": "algebra", "text": "GMAT algebra note: At times, you may have been frustrated because this technique didn’t actually seem easier than doing the math algebraically. If so, here’s the missing piece: you need to learn how to choose smart numbers in the best possible way.", "topic": "algebra"}
{"id": "book 2_p95_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 95, "page_end": 95, "topic_guess": "algebra", "text": "GMAT algebra note: Most of the time, you’re going to choose for the variable given (in this case, q). In some cases, though, starting with the given variable doesn’t make your life any easier.", "topic": "algebra"}
{"id": "book 2_p95_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 95, "page_end": 95, "topic_guess": "algebra", "text": "GMAT algebra note: The problem above actually has three unknowns: one for Andra, one for Elif, and one for Grady. Take a look at the relationship between those unknowns before you decide which one is the best starting point.", "topic": "algebra"}
{"id": "book 2_p96_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 96, "page_end": 96, "topic_guess": "strategy", "text": "GMAT strategy note: Percent problems usually relate part, whole, rate, and percent change. Andra invests 40% more than Elif.", "topic": "percent"}
{"id": "book 2_p96_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 96, "page_end": 96, "topic_guess": "strategy", "text": "GMAT strategy note: For just these two, it would be easier to pick a number for Elif and then calculate Andra’s amount. Elif invests 25% less than Grady.", "topic": "percent"}
{"id": "book 2_p96_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 96, "page_end": 96, "topic_guess": "strategy", "text": "GMAT strategy note: For these two, it is easier to start with Grady and then calculate Elif. As a result, start with Grady, then find Elif, then find Andra.", "topic": "percent"}
{"id": "book 2_p96_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 96, "page_end": 96, "topic_guess": "strategy", "text": "GMAT strategy note: If Grady invests $100, then Elif invests 25% less, or $75. Andra invests 40% more than Elif, or $75 + $30 = $105.", "topic": "percent"}
{"id": "book 2_p96_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 96, "page_end": 96, "topic_guess": "strategy", "text": "GMAT strategy note: Make sure to note on your scrap paper that q = 105. Collectively, the three invest $100 + $75 + $105 = $280.", "topic": "percent"}
{"id": "book 2_p96_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 96, "page_end": 96, "topic_guess": "strategy", "text": "GMAT strategy note: Note a few important aspects that will help you to choose smart numbers efficiently and effectively. First, if there are multiple unknowns and you have to choose where to start, pause to think about how to make the math as easy as possible.", "topic": "percent"}
{"id": "book 2_p96_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 96, "page_end": 96, "topic_guess": "strategy", "text": "GMAT strategy note: In the case of the problem above, if you had picked q = 100 for Andra, your next step would 2q = 2(105) = $210.", "topic": "percent"}
{"id": "book 2_p97_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 97, "page_end": 97, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Have been to figure out Elif’s amount.", "topic": "word_problems"}
{"id": "book 2_p97_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 97, "page_end": 97, "topic_guess": "word_problems", "text": "GMAT word problem note: It is not the case that Elif would be $60, or 40% less than Andra. Rather, Andra is 40% more than Elif: 1.4e = $100, so .", "topic": "word_problems"}
{"id": "book 2_p97_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 97, "page_end": 97, "topic_guess": "word_problems", "text": "GMAT word problem note: Elif would actually equal approximately $71.42857. Nobody’s going to want to go down that path!", "topic": "word_problems"}
{"id": "book 2_p97_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 97, "page_end": 97, "topic_guess": "word_problems", "text": "GMAT word problem note: At this stage, you have two choices: you can go back and pick for someone else or you can think about what numbers would make this particular path easier. (In this case, that would still result in a messy number for the next calculation.", "topic": "word_problems"}
{"id": "book 2_p97_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 97, "page_end": 97, "topic_guess": "word_problems", "text": "GMAT word problem note: You can stop and eliminate a choice whenever you can tell that it will not equal $280. Third, practice this strategy extensively in order to expose yourself to these little variations and possible sticking points.", "topic": "word_problems"}
{"id": "book 2_p97_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 97, "page_end": 97, "topic_guess": "word_problems", "text": "GMAT word problem note: As you become proficient with the strategy, you’ll be amazed at how much time and mental effort it can save.", "topic": "word_problems"}
{"id": "book 2_p98_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 98, "page_end": 98, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Rewards finding the correct answer as efficiently as possible or guessing when needed.", "topic": "number_properties"}
{"id": "book 2_p98_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 98, "page_end": 98, "topic_guess": "number_theory", "text": "GMAT number properties note: Try-It #2-2 If , which of these integers could be the value of z? Look at all of those fractions!", "topic": "number_properties"}
{"id": "book 2_p98_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 98, "page_end": 98, "topic_guess": "number_theory", "text": "GMAT number properties note: Solving for z algebraically in this problem would not be easy. The numerators are always 2, so what needs to happen to bring the two halves of the equation together?", "topic": "number_properties"}
{"id": "book 2_p98_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 98, "page_end": 98, "topic_guess": "number_theory", "text": "GMAT number properties note: The le side needs to be made smaller, so the denominator needs to be bigger.", "topic": "number_properties"}
{"id": "book 2_p99_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 99, "page_end": 99, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Try-It #2-3 A certain college party is attended by both male and female students.", "topic": "algebra"}
{"id": "book 2_p99_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 99, "page_end": 99, "topic_guess": "algebra", "text": "GMAT algebra note: The ratio of male-to-female students is 3 to 5. If 5 of the male students were to leave the party, the ratio would change to 1 to 2.", "topic": "algebra"}
{"id": "book 2_p99_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 99, "page_end": 99, "topic_guess": "algebra", "text": "GMAT algebra note: How many total students are at the party? Of course, you could set up equations for the unknowns in the problem and solve them algebraically.", "topic": "algebra"}
{"id": "book 2_p99_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 99, "page_end": 99, "topic_guess": "algebra", "text": "GMAT algebra note: However, the numbers in the answers are pretty straightforward integers.", "topic": "algebra"}
{"id": "book 2_p100_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 100, "page_end": 100, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Translate the wording into variables, relationships, and equations before calculating. What characteristic must be true of the correct answer?", "topic": "word_problems"}
{"id": "book 2_p100_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 100, "page_end": 100, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Once you have found the answer using this technique, you can stop. You don’t need to test any remaining answers.", "topic": "word_problems"}
{"id": "book 2_p100_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 100, "page_end": 100, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Also, you don’t have to translate equations or figure out how to eliminate variables and solve. You just work each number through the problem until you’re done.", "topic": "word_problems"}
{"id": "book 2_p100_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 100, "page_end": 100, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: As you work through an answer, think about how the math is playing out. Again, you will usually be able to eliminate some answers without actually having to try them.", "topic": "word_problems"}
{"id": "book 2_p100_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 100, "page_end": 100, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: If there are 30 students total, then the unknown multiplier is = 3.75, leaving you with a non-integer number of students. This is impossible, so this answer must be incorrect.", "topic": "word_problems"}
{"id": "book 2_p100_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 100, "page_end": 100, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: If 5 males leave, there will be 4 le, leaving a ratio of 4: 15. This ratio is also pretty far away from the correct ratio of 1: 2, so consider trying the larger remaining answer, 80, next.", "topic": "word_problems"}
{"id": "book 2_p101_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 101, "page_end": 101, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Some problems allow you to choose real numbers to solve, but you can’t choose just one set of numbers as you do when you choose smart numbers.", "topic": "number_properties"}
{"id": "book 2_p101_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 101, "page_end": 101, "topic_guess": "number_theory", "text": "GMAT number properties note: Rather, you have to test multiple scenarios to get yourself to the one correct answer. On Problem Solving problems, this tends to occur with must be true or could be true problems.", "topic": "number_properties"}
{"id": "book 2_p101_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 101, "page_end": 101, "topic_guess": "number_theory", "text": "GMAT number properties note: Try It #2-4 If n is a positive integer, what must be true of n3 − n? This problem is asking about a theoretical concept: what mathematical characteristic must be true of this expression?", "topic": "number_properties"}
{"id": "book 2_p101_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 101, "page_end": 101, "topic_guess": "number_theory", "text": "GMAT number properties note: In order to solve, you can test allowable cases to narrow down the answers until only one remains. Note that in this PS problem, one of the answers must be correct.", "topic": "number_properties"}
{"id": "book 2_p101_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 101, "page_end": 101, "topic_guess": "number_theory", "text": "GMAT number properties note: In this case, any positive integer n will help you get to the answer, so start with the simplest possible positive integer: 1. In general, when Testing Cases on PS or DS, start with the simplest number that fits the problem’s parameters.", "topic": "number_properties"}
{"id": "book 2_p102_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 102, "page_end": 102, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Because one of the statements talks about having more than a certain number of prime factors, try a larger number next: n n3 − n Yes, 0 is divisible by 4.", "topic": "number_properties"}
{"id": "book 2_p103_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 103, "page_end": 103, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. You could save yourself some time on this one by recognizing that the expression n3 − n can be rewritten as:.", "topic": "number_properties"}
{"id": "book 2_p103_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 103, "page_end": 103, "topic_guess": "number_theory", "text": "GMAT number properties note: In other words, the expression represents three consecutive integers. Test some cases to discover what must be true about the product of three consecutive integers.", "topic": "number_properties"}
{"id": "book 2_p103_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 103, "page_end": 103, "topic_guess": "number_theory", "text": "GMAT number properties note: If n = 1, then the three consecutive integers are 0, 1, 2. If n = 2, then the three consecutive integers are 1, 2, and 3.", "topic": "number_properties"}
{"id": "book 2_p103_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 103, "page_end": 103, "topic_guess": "number_theory", "text": "GMAT number properties note: This product does not contain two 4’s, so it is not a multiple of 4. If n = 3, then the three consecutive integers are 3, 4, and 5.", "topic": "number_properties"}
{"id": "book 2_p103_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 103, "page_end": 103, "topic_guess": "number_theory", "text": "GMAT number properties note: (The prime factors are 2, 3, and 5.) Eliminate.", "topic": "number_properties"}
{"id": "book 2_p104_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 104, "page_end": 104, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. In this case, you’re trying to find factors, and the three consecutive integers tell you that directly.", "topic": "number_properties"}
{"id": "book 2_p104_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 104, "page_end": 104, "topic_guess": "number_theory", "text": "GMAT number properties note: You don’t need to multiply them out. (You’ll learn more about avoiding unnecessary computation in the next section.) Instead, if possible, try to notice a pattern.", "topic": "number_properties"}
{"id": "book 2_p104_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 104, "page_end": 104, "topic_guess": "number_theory", "text": "GMAT number properties note: In every set of three consecutive numbers, you will always have at least one even number. You will also always have a multiple of 3.", "topic": "number_properties"}
{"id": "book 2_p104_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 104, "page_end": 104, "topic_guess": "number_theory", "text": "GMAT number properties note: (In Case 1, 0 is a multiple of 3.) As a result, the product will always be a multiple of 6. You may also notice that the problem contains no upper limit.", "topic": "number_properties"}
{"id": "book 2_p104_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 104, "page_end": 104, "topic_guess": "number_theory", "text": "GMAT number properties note: By choosing a large enough number, you’re going to be able to create a number that contains more than two distinct prime numbers. You can test cases directly to eliminate the four incorrect answers, or you can use a few cases to help you figure out the theory underlying the problem, which will also get you to the correct answer.", "topic": "number_properties"}
{"id": "book 2_p104_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 104, "page_end": 104, "topic_guess": "number_theory", "text": "GMAT number properties note: AVOID NEEDLESS COMPUTATION You won’t see many GMAT problems that require substantial calculation to arrive at a precise answer. Rather, correct answers on difficult problems will generally be relatively easy to compute once the difficult concept or trick in the problem has been correctly identified and addressed.", "topic": "number_properties"}
{"id": "book 2_p104_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 104, "page_end": 104, "topic_guess": "number_theory", "text": "GMAT number properties note: On several types of GMAT problems, a significant amount of computation can be a.", "topic": "number_properties"}
{"id": "book 2_p105_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 105, "page_end": 105, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Estimation Intelligent estimation can save you time and effort on many problems.", "topic": "percent"}
{"id": "book 2_p105_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 105, "page_end": 105, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Try-It #2-5 The percent change from 29 to 43 is approximately what percent of the percent change from 43 to 57? In this case, the question stem straight up tells you that you can estimate.", "topic": "percent"}
{"id": "book 2_p105_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 105, "page_end": 105, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Any time you see the word approximately (or a synonym), definitely do not try to solve for the exact answer. The question stem is pretty complex.", "topic": "percent"}
{"id": "book 2_p105_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 105, "page_end": 105, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Use the percent change formula:.", "topic": "percent"}
{"id": "book 2_p106_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 106, "page_end": 106, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. You can simplify things by solving the question in parts.", "topic": "percent"}
{"id": "book 2_p106_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 106, "page_end": 106, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: First, it talks about the percent change from 29 to 43. The percent change for the top half of the fraction, is , but that number is cumbersome.", "topic": "percent"}
{"id": "book 2_p106_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 106, "page_end": 106, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: This is approximately The percent change from 43 to 57 is . The question is really asking:.", "topic": "percent"}
{"id": "book 2_p106_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 106, "page_end": 106, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: The fraction is equivalent to 150%. The wording of the question can sometimes provide a strong clue that estimation should be used.", "topic": "percent"}
{"id": "book 2_p106_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 106, "page_end": 106, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Look for phrases such as these:. The test doesn’t have to tell you that you can estimate, though.", "topic": "percent"}
{"id": "book 2_p106_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 106, "page_end": 106, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Heavy Long Division is what percent of ... The number is approximately equal to which of the following?", "topic": "percent"}
{"id": "book 2_p106_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 106, "page_end": 106, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: This result is closest to which of the following? Which of the following is most nearly equal to...?", "topic": "percent"}
{"id": "book 2_p107_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 107, "page_end": 107, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Very few problems on the GMAT truly require long division, even though it might appear otherwise.", "topic": "number_properties"}
{"id": "book 2_p107_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 107, "page_end": 107, "topic_guess": "number_theory", "text": "GMAT number properties note: You can almost always approximate the answer or reduce the division by taking out common factors. Try-It #2-6 is equivalent to which of the following?", "topic": "number_properties"}
{"id": "book 2_p107_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 107, "page_end": 107, "topic_guess": "number_theory", "text": "GMAT number properties note: At first glance, it appears that precise long division is necessary. However, with some manipulation and factoring, the solution is much more straightforward.", "topic": "number_properties"}
{"id": "book 2_p107_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 107, "page_end": 107, "topic_guess": "number_theory", "text": "GMAT number properties note: The key to factoring this fraction is to move the decimals of both the numerator and denominator three places to the right so that you’re dealing with integers. Then you might notice that 3,507 is divisible by 7 (3,500 and 7 are both divisible by 7, providing a clue that you may be able to factor out 7).", "topic": "number_properties"}
{"id": "book 2_p107_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 107, "page_end": 107, "topic_guess": "number_theory", "text": "GMAT number properties note: Moreover, 10,020 is divisible by 10 and by 2 (10,020 ends in a 0, and 1,002 is even):.", "topic": "number_properties"}
{"id": "book 2_p108_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 108, "page_end": 108, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Alternatively, you might observe that 10.02 is very slightly larger than 10.", "topic": "algebra"}
{"id": "book 2_p108_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 108, "page_end": 108, "topic_guess": "algebra", "text": "GMAT algebra note: Try-It #2-7 What is the value of Since 102 is extremely small compared to 105, and the choices are somewhat spread out, estimate:. Notice that by ignoring the 102 term, you made the denominator slightly larger than it originally was.", "topic": "algebra"}
{"id": "book 2_p108_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 108, "page_end": 108, "topic_guess": "algebra", "text": "GMAT algebra note: Therefore, 0.81918 is slightly smaller than the correct answer, 0.82. Quadratic Expressions in Word Problems Some Word Problems result in a quadratic equation.", "topic": "algebra"}
{"id": "book 2_p108_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 108, "page_end": 108, "topic_guess": "algebra", "text": "GMAT algebra note: You are probably pretty good at solving quadratic equations, so your natural bias would be to set up and solve the equation.", "topic": "algebra"}
{"id": "book 2_p109_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 109, "page_end": 109, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. May be very difficult to solve.", "topic": "algebra"}
{"id": "book 2_p109_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 109, "page_end": 109, "topic_guess": "algebra", "text": "GMAT algebra note: Try-It #2-8 A shoe cobbler charges n dollars to repair a single pair of loafers. Tomorrow, he intends to earn $240 repairing loafers.", "topic": "algebra"}
{"id": "book 2_p109_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 109, "page_end": 109, "topic_guess": "algebra", "text": "GMAT algebra note: If he were to reduce his fee per pair by $20, he would have to repair an additional pair of the loafers to earn the same amount of revenue. How many pairs of loafers does he intend to repair tomorrow?", "topic": "algebra"}
{"id": "book 2_p109_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 109, "page_end": 109, "topic_guess": "algebra", "text": "GMAT algebra note: The problem may not seem too bad... Until you try to set it up algebraically.", "topic": "algebra"}
{"id": "book 2_p109_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 109, "page_end": 109, "topic_guess": "algebra", "text": "GMAT algebra note: Assign x to represent the number of pairs of loafers the cobbler intends to repair tomorrow. Using the equation for revenue gives you the following:.", "topic": "algebra"}
{"id": "book 2_p110_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 110, "page_end": 110, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. From here, the algebra gets complicated very quickly.", "topic": "algebra"}
{"id": "book 2_p110_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 110, "page_end": 110, "topic_guess": "algebra", "text": "GMAT algebra note: Do you need to make x larger or smaller?", "topic": "algebra"}
{"id": "book 2_p112_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 112, "page_end": 112, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Advanced Guessing Tactics To repeat, the tactics below are less universally useful than the strategies we just covered.", "topic": "word_problems"}
{"id": "book 2_p112_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 112, "page_end": 112, "topic_guess": "word_problems", "text": "GMAT word problem note: However, when all else fails, “break the glass” and try one or more of these tactics. They’re almost always better than guessing completely randomly.", "topic": "word_problems"}
{"id": "book 2_p112_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 112, "page_end": 112, "topic_guess": "word_problems", "text": "GMAT word problem note: If you’re way behind on time and you have to sacrifice a couple of problems, though, guess immediately and move on. Of course, if you can solve the problem directly, do so.", "topic": "word_problems"}
{"id": "book 2_p112_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 112, "page_end": 112, "topic_guess": "word_problems", "text": "GMAT word problem note: But also study the tactic, so you’re ready to use it on a harder problem of the same type. Look for Answer Pairs Certainty:.", "topic": "word_problems"}
{"id": "book 2_p112_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 112, "page_end": 112, "topic_guess": "word_problems", "text": "GMAT word problem note: The GMAT likes to put in a final obstacle. Say you do everything correct, except you solve Add up to 1 on a probability or fraction question Add up to 100% on questions involving percents Add up to 0 (be opposites of each other) Multiply to 1 (be reciprocals of each other).", "topic": "word_problems"}
{"id": "book 2_p113_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 113, "page_end": 113, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. For the wrong unknown or forget to subtract from 1.", "topic": "statistics"}
{"id": "book 2_p113_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 113, "page_end": 113, "topic_guess": "statistics", "text": "GMAT statistics note: Under the pressure of the exam, people make this sort of penultimate error all the time (penultimate means next to last). Also, the way in which the answers are paired may provide clues about the correct solution method and/or traps in the problem.", "topic": "statistics"}
{"id": "book 2_p113_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 113, "page_end": 113, "topic_guess": "statistics", "text": "GMAT statistics note: Try-It #2-9 At a certain high school, the junior class is twice the size of the senior class. If of the seniors and of the juniors study Japanese, what fraction of the students in both classes do not study Japanese?", "topic": "statistics"}
{"id": "book 2_p113_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 113, "page_end": 113, "topic_guess": "statistics", "text": "GMAT statistics note: Note that two pairs of answers each add up to 1: , and .", "topic": "statistics"}
{"id": "book 2_p114_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 114, "page_end": 114, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. The fact that these pairs sum to 1 also provides a clue to double-check the wording of the question: do vs.", "topic": "percent"}
{"id": "book 2_p114_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 114, "page_end": 114, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Do not study Japanese (the sum of the fractions of the students that do study Japanese and those who do not will equal 1). To solve, you could use a double-set matrix that shows juniors versus seniors and Japanese studiers versus non-Japanese studiers.", "topic": "percent"}
{"id": "book 2_p114_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 114, "page_end": 114, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Pick a smart number that is a multiple of the denominators in the problem:. If the junior class has 12 people, then the senior class thus has 6.", "topic": "percent"}
{"id": "book 2_p114_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 114, "page_end": 114, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: If of the seniors study Japanese, then seniors study Japanese. If of the juniors study Japanese, then juniors study Japanese.", "topic": "percent"}
{"id": "book 2_p114_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 114, "page_end": 114, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: There are 12 + 6 = 18 students total, and 2 + 3 = 5 of them study Japanese. Thus, of the students do study Japanese, so the fraction of the students that do not study Japanese is .", "topic": "percent"}
{"id": "book 2_p114_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 114, "page_end": 114, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: High You may be able to eliminate answers above or below some easily calculated threshold value.", "topic": "percent"}
{"id": "book 2_p115_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 115, "page_end": 115, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. This strategy can sometimes be used in combination with an answer pairs strategy, as pairs of answers are o en composed of a high and a low value.", "topic": "word_problems"}
{"id": "book 2_p115_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 115, "page_end": 115, "topic_guess": "word_problems", "text": "GMAT word problem note: Therefore, somewhere between and of the students overall, or less than half, must study Japanese. This implies that the fraction of students who do not study Japanese must be more than half.", "topic": "word_problems"}
{"id": "book 2_p115_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 115, "page_end": 115, "topic_guess": "word_problems", "text": "GMAT word problem note: In 2013, 60% of the eSoroban devices sold were purchased by women, of whom purchased the orange device. If an equal number of orange and green eSoroban devices were sold in 2013, what fraction of men who purchased an eSoroban in 2013 purchased the green device?", "topic": "word_problems"}
{"id": "book 2_p116_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 116, "page_end": 116, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. More than half of the people purchasing this device are women, and more than half of them buy the green version.", "topic": "statistics"}
{"id": "book 2_p116_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 116, "page_end": 116, "topic_guess": "statistics", "text": "GMAT statistics note: If an equal number of green and orange devices are sold, then more than half of the men must buy the orange version. Furthermore, women represent 60% of purchases, while men represent just 40%.", "topic": "statistics"}
{"id": "book 2_p116_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 116, "page_end": 116, "topic_guess": "statistics", "text": "GMAT statistics note: Pretend for a moment that there were equal numbers of men and women. If that were the case, then of women would buy orange and would buy green.", "topic": "statistics"}
{"id": "book 2_p116_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 116, "page_end": 116, "topic_guess": "statistics", "text": "GMAT statistics note: On the flip side, of men would buy green and would buy orange. In fact, there are more women than men, so the proportion of men buying orange devices has to be higher than .", "topic": "statistics"}
{"id": "book 2_p116_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 116, "page_end": 116, "topic_guess": "statistics", "text": "GMAT statistics note: Fewer than must buy the green version. The actual split is , so the correct value is not that much lower than .", "topic": "statistics"}
{"id": "book 2_p116_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 116, "page_end": 116, "topic_guess": "statistics", "text": "GMAT statistics note: But here’s the actual solution, using a double-set matrix and a smart number of 100 total people:.", "topic": "statistics"}
{"id": "book 2_p117_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 117, "page_end": 117, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Men Women Total Green 50 − 35 = 15 60 − 25 = 35 50 Total 40 60 100 Because 15 men purchased a green device out of 40 men total, the proportion is .", "topic": "word_problems"}
{"id": "book 2_p117_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 117, "page_end": 117, "topic_guess": "word_problems", "text": "GMAT word problem note: Look at Positive/Negative Certainty:. (Again, this is something to do when you have run out of direct approaches or you are short on time!) Try-It #2-11 If x □ y is defined to equal for all x and y, then (−1 □ 2) □ 3 is equivalent to which of the following?", "topic": "word_problems"}
{"id": "book 2_p118_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 118, "page_end": 118, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. The negative sign in the term −1 will be extinguished, because the term before the □ symbol is squared when this function is calculated.", "topic": "word_problems"}
{"id": "book 2_p118_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 118, "page_end": 118, "topic_guess": "word_problems", "text": "GMAT word problem note: Therefore, the correct answer will be positive. Answer Properties Certainty:.", "topic": "word_problems"}
{"id": "book 2_p118_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 118, "page_end": 118, "topic_guess": "word_problems", "text": "GMAT word problem note: Moderate You may not have time, while doing a Problem Solving problem, to figure out the correct answer. However, it might take less time to figure out something about the correct answer.", "topic": "word_problems"}
{"id": "book 2_p118_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 118, "page_end": 118, "topic_guess": "word_problems", "text": "GMAT word problem note: Here are some other things you may be able to determine: 5. Moderate The estimation technique can be extended to some Geometry problems.", "topic": "word_problems"}
{"id": "book 2_p118_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 118, "page_end": 118, "topic_guess": "word_problems", "text": "GMAT word problem note: You can approximate the length of a line segment, size of an angle, or area of an object by drawing it as accurately as possible on your scrap paper. Is the correct answer divisible by a certain value?", "topic": "word_problems"}
{"id": "book 2_p118_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 118, "page_end": 118, "topic_guess": "word_problems", "text": "GMAT word problem note: What is the units digit of the correct answer? Does the correct answer include square roots?", "topic": "word_problems"}
{"id": "book 2_p118_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 118, "page_end": 118, "topic_guess": "word_problems", "text": "GMAT word problem note: Does the correct answer include decimals?", "topic": "word_problems"}
{"id": "book 2_p119_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 119, "page_end": 119, "topic_guess": "geometry", "text": "GMAT geometry note: When you take the GMAT, the laminated scratch booklet is printed with a light grid. This grid can help you draw very accurate scale pictures, and o en the picture alone is enough to answer the question.", "topic": "geometry"}
{"id": "book 2_p119_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 119, "page_end": 119, "topic_guess": "geometry", "text": "GMAT geometry note: A circle is then inscribed inside triangle ADE. What is the ratio of the area of the circle to the area of the square?", "topic": "geometry"}
{"id": "book 2_p119_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 119, "page_end": 119, "topic_guess": "geometry", "text": "GMAT geometry note: The circle is about the height of the square and about the width. Therefore, the area of the circle should be approximately of the area of the square.", "topic": "geometry"}
{"id": "book 2_p120_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 120, "page_end": 120, "topic_guess": "geometry", "text": "GMAT geometry note: The most efficient way to solve this problem fully is to assign a radius of 1 to the circle. If the center of the circle is O, then OF = 1.", "topic": "geometry"}
{"id": "book 2_p120_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 120, "page_end": 120, "topic_guess": "geometry", "text": "GMAT geometry note: Because ADE is an equilateral triangle, angle ADE = 60°. OD bisects angle ADE, so angle ODE = 30°.", "topic": "geometry"}
{"id": "book 2_p120_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 120, "page_end": 120, "topic_guess": "geometry", "text": "GMAT geometry note: Therefore, triangle OFD is a 30: 60: 90 triangle, and . DE must be twice the length of DF, meaning that , and the area of square BCDE = So the ratio of the area of the circle to the area of the square is Notice how much easier the draw to scale tactic was!", "topic": "geometry"}
{"id": "book 2_p120_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 120, "page_end": 120, "topic_guess": "geometry", "text": "GMAT geometry note: But it works o en enough that you want to think about using it.", "topic": "geometry"}
{"id": "book 2_p121_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 121, "page_end": 121, "topic_guess": "strategy", "text": "GMAT strategy note: Translate the wording into variables, relationships, and equations before calculating. Problem Set Solve problems 1–12.", "topic": "word_problems"}
{"id": "book 2_p121_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 121, "page_end": 121, "topic_guess": "strategy", "text": "GMAT strategy note: In each case, identify whether you could use advanced strategies (Choose Smart Numbers, Work Backwards, Test Cases, or Avoid Needless Computation) and guessing tactics (Look for Answer Pairs, Apply Cutoffs, Look at Positive-Negative, or Draw to Scale) in any beneficial way. Of course, there are textbook ways to solve these problems directly.", "topic": "word_problems"}
{"id": "book 2_p121_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 121, "page_end": 121, "topic_guess": "strategy", "text": "GMAT strategy note: Instead of trying the textbook method first, focus on applying the strategies and tactics described in this chapter. A popular yoga studio that is always filled to capacity moves to a new location that is able to serve 45% more students.", "topic": "word_problems"}
{"id": "book 2_p121_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 121, "page_end": 121, "topic_guess": "strategy", "text": "GMAT strategy note: Unfortunately, 20% of the current students will no longer attend classes at the new location. If classes at the new location are also filled to capacity, what fraction of the students at the new location will be new students?", "topic": "word_problems"}
{"id": "book 2_p122_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 122, "page_end": 122, "topic_guess": "general", "text": "GMAT quant note: If x < 10, y < 8, and y < x, what must be true? If , which of the following could be the value of x?", "topic": "general"}
{"id": "book 2_p123_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 123, "page_end": 123, "topic_guess": "general", "text": "GMAT quant note: If , which of the following could be the value of x? If 154 is of x, approximately what is the value of 2x?", "topic": "general"}
{"id": "book 2_p124_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 124, "page_end": 124, "topic_guess": "general", "text": "GMAT quant note: In a certain clothing store, the most expensive pair of socks sells for $1 less than twice the price of the cheapest pair of socks. A customer notices that for exactly $18, she can buy three fewer pairs of the most expensive socks than the cheapest socks.", "topic": "general"}
{"id": "book 2_p124_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 124, "page_end": 124, "topic_guess": "general", "text": "GMAT quant note: What could be the number of pairs of the cheapest socks she could have purchased? If , then m must equal which of the following?", "topic": "general"}
{"id": "book 2_p125_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 125, "page_end": 125, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If 3x + 3x + 3x = 1, what is x?", "topic": "number_properties"}
{"id": "book 2_p125_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 125, "page_end": 125, "topic_guess": "number_theory", "text": "GMAT number properties note: If a and b are integers and a is a factor of b, what must be true? The distinct prime factors of a2 are also factors of b.II.", "topic": "number_properties"}
{"id": "book 2_p126_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 126, "page_end": 126, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Without solving problems 13–16, which answers could you confidently eliminate and why?", "topic": "number_properties"}
{"id": "book 2_p126_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 126, "page_end": 126, "topic_guess": "number_theory", "text": "GMAT number properties note: The integer k is positive but less than 400. If 21k is a multiple of 180, how many unique prime factors does k have?", "topic": "number_properties"}
{"id": "book 2_p127_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 127, "page_end": 127, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Is equivalent to which of the following?", "topic": "percent"}
{"id": "book 2_p127_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 127, "page_end": 127, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: In the 7-inch square to the right, another square is inscribed. What fraction of the larger square is shaded?", "topic": "percent"}
{"id": "book 2_p127_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 127, "page_end": 127, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: If Mason is now twice as old as Gunther was 10 years ago, and G is Gunther’s current age in years, which of the following represents the sum of Mason’s and Gunther’s ages 4 years from now?", "topic": "percent"}
{"id": "book 2_p128_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 128, "page_end": 128, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "book 2_p129_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 129, "page_end": 129, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Choose a smart number of 100 for this Percents problem.", "topic": "percent"}
{"id": "book 2_p129_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 129, "page_end": 129, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: If the new studio is filled to capacity, at 145 students, but only 80 old students continue to attend, then the studio will have 65 new students. Old capacity: 100 students New capacity: 145 students Old students who will stay with the studio: 80 2.", "topic": "percent"}
{"id": "book 2_p129_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 129, "page_end": 129, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Make sure to choose only values that are allowed by the problem: x < 10, y < 8, and y < x. If x = 9 and y = 7, then xy = 63, which is less than 80.", "topic": "percent"}
{"id": "book 2_p129_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 129, "page_end": 129, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: If, on the other hand, x = −10 and y = −20, then xy = 200, which is greater than 80. Statement I does not have to be true.", "topic": "percent"}
{"id": "book 2_p129_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 129, "page_end": 129, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Positive values for both x and y will make , but if x = 9 and y = −1, then , which is not greater than 1.", "topic": "percent"}
{"id": "book 2_p130_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 130, "page_end": 130, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. If x = 3 and y = 2, then x2 + y2 is 13, which is greater than 1.", "topic": "algebra"}
{"id": "book 2_p130_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 130, "page_end": 130, "topic_guess": "algebra", "text": "GMAT algebra note: If x = −2 and y = −3, then x2 + y2 is still 13, which is still greater than 1. Don’t forget about fractions!", "topic": "algebra"}
{"id": "book 2_p130_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 130, "page_end": 130, "topic_guess": "algebra", "text": "GMAT algebra note: If x = 0.3 and y = 0.2, then x2 + y2 = 0.09 + 0.04, which is not greater than 1. Fractions between 0 and 1 get smaller when squared.", "topic": "algebra"}
{"id": "book 2_p130_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 130, "page_end": 130, "topic_guess": "algebra", "text": "GMAT algebra note: None of the statements must be true. You’d actually need to use the quadratic formula, and the equation would turn out to be really nasty.", "topic": "algebra"}
{"id": "book 2_p130_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 130, "page_end": 130, "topic_guess": "algebra", "text": "GMAT algebra note: Much easier: 122 = 144, and that goes very nicely with 288.", "topic": "algebra"}
{"id": "book 2_p131_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 131, "page_end": 131, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. It would be much easier if 154 were 150.", "topic": "percent"}
{"id": "book 2_p131_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 131, "page_end": 131, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: First of all, put it over 100:. This is almost , so use that fraction instead.", "topic": "percent"}
{"id": "book 2_p132_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 132, "page_end": 132, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Translate the equation with the estimated values:.", "topic": "algebra"}
{"id": "book 2_p132_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 132, "page_end": 132, "topic_guess": "algebra", "text": "GMAT algebra note: Finally, make sure to answer the right question that was asked! Call the number of cheap pairs c and the number of expensive pairs e.", "topic": "algebra"}
{"id": "book 2_p132_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 132, "page_end": 132, "topic_guess": "algebra", "text": "GMAT algebra note: Call the cost of a cheap pair $c and the cost of an expensive pair $e.", "topic": "algebra"}
{"id": "book 2_p133_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 133, "page_end": 133, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Note that you don’t actually have to figure out the value of the right-hand side of the equation, as long as you know that it will not equal 9.", "topic": "algebra"}
{"id": "book 2_p133_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 133, "page_end": 133, "topic_guess": "algebra", "text": "GMAT algebra note: Here’s how the algebra would have to be set up.", "topic": "algebra"}
{"id": "book 2_p134_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 134, "page_end": 134, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. From sentence 2: 18 = cx and 18 = ey = (c − 3)y From here, you would solve the equations from sentence 2 for x and y, respectively, and plug them into the equation from sentence 1:.", "topic": "algebra"}
{"id": "book 2_p134_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 134, "page_end": 134, "topic_guess": "algebra", "text": "GMAT algebra note: With the correct manipulation (over multiple, complicated steps!), that equation would eventually become c2 − 21 + 108 = 0 and you could solve for the two solutions, c = 9 and c = 12. In this circumstance, start with the middle number of the remaining choices.", "topic": "algebra"}
{"id": "book 2_p134_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 134, "page_end": 134, "topic_guess": "algebra", "text": "GMAT algebra note: (Avoid Needless Computation) 10.", "topic": "algebra"}
{"id": "book 2_p135_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 135, "page_end": 135, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Testing choices is fast, too.", "topic": "number_properties"}
{"id": "book 2_p135_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 135, "page_end": 135, "topic_guess": "number_theory", "text": "GMAT number properties note: Stop when you find the correct answer. Which number should you try next, −1 or 1?", "topic": "number_properties"}
{"id": "book 2_p135_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 135, "page_end": 135, "topic_guess": "number_theory", "text": "GMAT number properties note: Since 0 leads to a number that’s too big, try −1 next. A factor can be smaller than the main number, but a factor can also equal the main number: 6 is a factor of 6.", "topic": "number_properties"}
{"id": "book 2_p135_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 135, "page_end": 135, "topic_guess": "number_theory", "text": "GMAT number properties note: Statement I 2 is a factor of 6 6 is not a factor of 2 1, 2, 3, and 6 are all factors of 6.", "topic": "number_properties"}
{"id": "book 2_p136_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 136, "page_end": 136, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The distinct prime factors of a2 are also factors of b.", "topic": "number_properties"}
{"id": "book 2_p136_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 136, "page_end": 136, "topic_guess": "number_theory", "text": "GMAT number properties note: There is just one distinct prime factor of a2: 2. In this case, a is a factor of b, yes.", "topic": "number_properties"}
{"id": "book 2_p136_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 136, "page_end": 136, "topic_guess": "number_theory", "text": "GMAT number properties note: There are two distinct prime factors of a2: 2 and 3. In this case, a is still a factor of b.", "topic": "number_properties"}
{"id": "book 2_p136_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 136, "page_end": 136, "topic_guess": "number_theory", "text": "GMAT number properties note: The distinct prime factors of a2 are the same as the distinct prime factors of a, since a2 is made up of a multiplied by itself. So, first, this statement is really saying that the distinct prime factors of a are also factors of b.", "topic": "number_properties"}
{"id": "book 2_p136_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 136, "page_end": 136, "topic_guess": "number_theory", "text": "GMAT number properties note: If a is a factor of b, then by definition all of a’s prime factors also have to be factors of b. No matter what numbers you try for statement II, a will be a factor of b.", "topic": "number_properties"}
{"id": "book 2_p136_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 136, "page_end": 136, "topic_guess": "number_theory", "text": "GMAT number properties note: If b is 6, then a could be 6, in which case . Alternatively, if b is 6, then a could be 2, in which case So far, this statement looks good.", "topic": "number_properties"}
{"id": "book 2_p136_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 136, "page_end": 136, "topic_guess": "number_theory", "text": "GMAT number properties note: Don’t forget about negative numbers! The problem doesn’t specify positive integers.", "topic": "number_properties"}
{"id": "book 2_p136_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 136, "page_end": 136, "topic_guess": "number_theory", "text": "GMAT number properties note: In this case, a could still be 2, so . Statement III does not have to be true.", "topic": "number_properties"}
{"id": "book 2_p136_c1_r9", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 136, "page_end": 136, "topic_guess": "number_theory", "text": "GMAT number properties note: The GMAT doesn’t o en test the factors of negative numbers, but the definiti.", "topic": "number_properties"}
{"id": "book 2_p137_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 137, "page_end": 137, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. An easier technique is to pick a smart number: one value of k that satisfies the constraints.", "topic": "number_properties"}
{"id": "book 2_p137_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 137, "page_end": 137, "topic_guess": "number_theory", "text": "GMAT number properties note: Any value of k that fits the constraints must have the same number of different prime factors as any other legal value of k. Otherwise, the problem could not exist as written.", "topic": "number_properties"}
{"id": "book 2_p137_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 137, "page_end": 137, "topic_guess": "number_theory", "text": "GMAT number properties note: There would be more than one correct answer. The problem states that 21k is a multiple of 180, so must be an integer.", "topic": "number_properties"}
{"id": "book 2_p137_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 137, "page_end": 137, "topic_guess": "number_theory", "text": "GMAT number properties note: In other words, k must be divisible by 60. The easiest number to choose is k = 60.", "topic": "number_properties"}
{"id": "book 2_p137_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 137, "page_end": 137, "topic_guess": "number_theory", "text": "GMAT number properties note: The prime factorization of 60 is 2 × 2 × 3 × 5, so 60 has the unique prime factors 2, 3, and 5. Thus, k has three unique prime factors.", "topic": "number_properties"}
{"id": "book 2_p137_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 137, "page_end": 137, "topic_guess": "number_theory", "text": "GMAT number properties note: If you ignore the 103 in the denominator, the division is . You have slightly overstated the denominator, thus slightly understated the result.", "topic": "number_properties"}
{"id": "book 2_p138_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 138, "page_end": 138, "topic_guess": "geometry", "text": "GMAT geometry note: Ignoring the dimensions 3 and 4 for a moment, think about the types of squares that might be inscribed in the larger square:. The shaded area can be at most of the larger square, which occurs when the smallest possible square is inscribed in the larger square.", "topic": "geometry"}
{"id": "book 2_p138_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 138, "page_end": 138, "topic_guess": "geometry", "text": "GMAT geometry note: This gives you a great cutoff. The larger the inscribed square, the smaller the shaded area and the more the inscribed square must be rotated from the vertical orientation of the minimum inscribed square.", "topic": "geometry"}
{"id": "book 2_p139_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 139, "page_end": 139, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. There is a way to arrive at the exact answer: compute the relevant areas and take the ratio.", "topic": "percent"}
{"id": "book 2_p139_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 139, "page_end": 139, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: The point of this exercise, though, is to practice guessing tactics. Gunther must be at least 10 years old for him to have had a non-negative age “10 years ago” and for Mason to have a non-negative age now.", "topic": "percent"}
{"id": "book 2_p140_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 140, "page_end": 140, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "book 2_p141_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 141, "page_end": 141, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating. Follow a Consistent Process Principle #2:.", "topic": "word_problems"}
{"id": "book 2_p141_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 141, "page_end": 141, "topic_guess": "general", "text": "GMAT quant note: Never Rephrase Yes/No as Value Principle #3:. Work from Facts to Question Principle #4:.", "topic": "word_problems"}
{"id": "book 2_p141_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 141, "page_end": 141, "topic_guess": "general", "text": "GMAT quant note: Be a Contrarian Principle #5:.", "topic": "word_problems"}
{"id": "book 2_p142_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 142, "page_end": 142, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. This gives you a significant advantage.", "topic": "word_problems"}
{"id": "book 2_p142_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 142, "page_end": 142, "topic_guess": "word_problems", "text": "GMAT word problem note: Once you know whether a piece of information lets you answer the given question, you can stop calculating. You do not have to waste time finishing that calculation.", "topic": "word_problems"}
{"id": "book 2_p142_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 142, "page_end": 142, "topic_guess": "word_problems", "text": "GMAT word problem note: However, this type of problem presents its own challenges. That is, unlike Problem Solving (PS), DS may contain math problems that cannot be solved!", "topic": "word_problems"}
{"id": "book 2_p142_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 142, "page_end": 142, "topic_guess": "word_problems", "text": "GMAT word problem note: This aspect of DS is unsettling. Practice with other material and then return here when you feel fully comfortable with how DS works.) On DS problems, the issue being tested is “answer-ability” itself—can the given question be answered, and if so, with what information?", "topic": "word_problems"}
{"id": "book 2_p142_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 142, "page_end": 142, "topic_guess": "word_problems", "text": "GMAT word problem note: So the GMAT disguises “answer-ability” as best it can. The given facts and the question.", "topic": "word_problems"}
{"id": "book 2_p143_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 143, "page_end": 143, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Itself are generally presented in ways that make this determination difficult.", "topic": "word_problems"}
{"id": "book 2_p143_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 143, "page_end": 143, "topic_guess": "word_problems", "text": "GMAT word problem note: For instance, information that seems to be sufficient may actually be insufficient if it permits an alternative scenario that leads to a different answer. Likewise, information that seems to be insufficient may actually be sufficient if all the possible scenarios lead to the same answer for the question.", "topic": "word_problems"}
{"id": "book 2_p143_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 143, "page_end": 143, "topic_guess": "word_problems", "text": "GMAT word problem note: Advanced DS problems require you to step up your game. You will have to get really good at Testing Cases, which is even more important for advanced DS than for advanced PS problems.", "topic": "word_problems"}
{"id": "book 2_p143_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 143, "page_end": 143, "topic_guess": "word_problems", "text": "GMAT word problem note: You will also have to get really good at simplifying the given facts and the given question. The GMAT increases the trickiness of the phrasing of the question and/or statements even more than the complexity of the underlying concepts.", "topic": "word_problems"}
{"id": "book 2_p143_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 143, "page_end": 143, "topic_guess": "word_problems", "text": "GMAT word problem note: So how do you approach advanced DS problems? Unfortunately, there is no “one-size-fits-all” approach:.", "topic": "word_problems"}
{"id": "book 2_p143_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 143, "page_end": 143, "topic_guess": "word_problems", "text": "GMAT word problem note: All that said, there are a few guiding principles you can follow. The best approach may involve precise application of theory, or it may involve a “quick and dirty” approach.", "topic": "word_problems"}
{"id": "book 2_p143_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 143, "page_end": 143, "topic_guess": "word_problems", "text": "GMAT word problem note: The statements may be easy or difficult to interpret. The question may require no rephrasing or elaborate rephrasing.", "topic": "word_problems"}
{"id": "book 2_p143_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 143, "page_end": 143, "topic_guess": "word_problems", "text": "GMAT word problem note: In fact, the crux of the problem may rest entirely on a careful rephrasing of the question.", "topic": "word_problems"}
{"id": "book 2_p144_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 144, "page_end": 144, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Follow a Consistent Process This is the most important principle.", "topic": "word_problems"}
{"id": "book 2_p144_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 144, "page_end": 144, "topic_guess": "word_problems", "text": "GMAT word problem note: A consistent process will prevent the most common errors. It will focus your efforts at any stage of the problem.", "topic": "word_problems"}
{"id": "book 2_p144_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 144, "page_end": 144, "topic_guess": "word_problems", "text": "GMAT word problem note: Perhaps most importantly, it will reduce your stress level, because you will have confidence in the approach that you’ve practiced. Follow these rules of the road:.", "topic": "word_problems"}
{"id": "book 2_p144_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 144, "page_end": 144, "topic_guess": "word_problems", "text": "GMAT word problem note: Do your work on paper, not in your head. Writing down each thought as it occurs helps you keep track of the work you’ve done.", "topic": "word_problems"}
{"id": "book 2_p144_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 144, "page_end": 144, "topic_guess": "word_problems", "text": "GMAT word problem note: Your mind is also freed up to think ahead. Many DS questions are explicitly designed to confuse you if you do all the work in your head.", "topic": "word_problems"}
{"id": "book 2_p144_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 144, "page_end": 144, "topic_guess": "word_problems", "text": "GMAT word problem note: Label everything and separate everything physically on your paper. If you mix up the elements of the problem, you will o en mess up the problem itself.", "topic": "word_problems"}
{"id": "book 2_p144_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 144, "page_end": 144, "topic_guess": "word_problems", "text": "GMAT word problem note: Keep these four elements straight:. Facts given in the question stem.", "topic": "word_problems"}
{"id": "book 2_p144_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 144, "page_end": 144, "topic_guess": "word_problems", "text": "GMAT word problem note: You can leave them unlabeled or you can put them with each statement. Label the question with a question mark.", "topic": "word_problems"}
{"id": "book 2_p144_c1_r9", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 144, "page_end": 144, "topic_guess": "word_problems", "text": "GMAT word problem note: Amazingly, many people fail to take this simple step. Without a question mark, you might think the question is a fact—and you will.", "topic": "word_problems"}
{"id": "book 2_p145_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 145, "page_end": 145, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. It may be worth rewriting the facts from the question stem alongside each statement.", "topic": "word_problems"}
{"id": "book 2_p145_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 145, "page_end": 145, "topic_guess": "word_problems", "text": "GMAT word problem note: Although this may seem redundant, the time is well spent if it prevents you from forgetting to use those facts. Keep the question mark as you rephrase.", "topic": "word_problems"}
{"id": "book 2_p145_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 145, "page_end": 145, "topic_guess": "word_problems", "text": "GMAT word problem note: It’s also helpful to keep the helping verb in a Yes/No question:. Rephrase the question and statements whenever possible.", "topic": "word_problems"}
{"id": "book 2_p145_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 145, "page_end": 145, "topic_guess": "word_problems", "text": "GMAT word problem note: Question stems and statements are o en more complex than they need to be. If you can simplify the information up front, you will save yourself time and effort later in the problem.", "topic": "word_problems"}
{"id": "book 2_p145_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 145, "page_end": 145, "topic_guess": "word_problems", "text": "GMAT word problem note: In particular, try to rephrase the question before you dive into the statements. Evaluate the easier statement first.", "topic": "word_problems"}
{"id": "book 2_p145_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 145, "page_end": 145, "topic_guess": "word_problems", "text": "GMAT word problem note: If the second statement looks much easier to you than the first, then start with the second statement. Physically separate the work that you do on the individual statements.", "topic": "word_problems"}
{"id": "book 2_p145_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 145, "page_end": 145, "topic_guess": "word_problems", "text": "GMAT word problem note: Doing so can help reduce the risk of statement carryover— unintentionally letting one statement influence you as you evaluate the other.", "topic": "word_problems"}
{"id": "book 2_p146_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 146, "page_end": 146, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Translate the wording into variables, relationships, and equations before calculating. Scrapwork for Rephrasing: ~~~~ Constraints: ~~~~ AD BCE FACT from question stem: ~~~~ FACT from (1): ~~~~ ...", "topic": "word_problems"}
{"id": "book 2_p146_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 146, "page_end": 146, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: And any work you do to combine these facts FACT from question stem: ~~~~ FACT from (2): ~~~~ ... And any work you do to combine these facts QUESTION:.", "topic": "word_problems"}
{"id": "book 2_p146_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 146, "page_end": 146, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Also observe that this schema explicitly parses out the facts given in the question stem and evaluates those facts alongside each statement.", "topic": "word_problems"}
{"id": "book 2_p147_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 147, "page_end": 147, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Try-It #3-1 If x and y are integers and 4xy = x2y + 4y, what is the value of xy?", "topic": "number_properties"}
{"id": "book 2_p147_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 147, "page_end": 147, "topic_guess": "number_theory", "text": "GMAT number properties note: Because the question stem contains an equation, simplify it before considering the statements:. Therefore, either x = 2 or y = 0, or both.", "topic": "number_properties"}
{"id": "book 2_p147_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 147, "page_end": 147, "topic_guess": "number_theory", "text": "GMAT number properties note: One of the following scenarios must be true: x y xy = ? 2 Not 0 2y Not 2 0 0 y − x = 0(1) x3 < 0(2).", "topic": "number_properties"}
{"id": "book 2_p148_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 148, "page_end": 148, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. It turns out that you only need to know the value of y.", "topic": "algebra"}
{"id": "book 2_p148_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 148, "page_end": 148, "topic_guess": "algebra", "text": "GMAT algebra note: Otherwise, x must equal 2, in which case the value of xy is still determined by the value of y. Rather than trying to combine this algebraically with the equation in the question stem, try a couple of the possible scenarios that fit the statement y − x = 2.", "topic": "algebra"}
{"id": "book 2_p148_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 148, "page_end": 148, "topic_guess": "algebra", "text": "GMAT algebra note: Construct scenarios using the earlier table as inspiration:. Since there are two possible answers, this statement is not sufficient.", "topic": "algebra"}
{"id": "book 2_p148_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 148, "page_end": 148, "topic_guess": "algebra", "text": "GMAT algebra note: If x does not equal 2, then y must equal 0, according to the fact from the question stem. Notice how valuable it was to evaluate the fact in the question stem first and to use it to rephrase the question.", "topic": "algebra"}
{"id": "book 2_p148_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 148, "page_end": 148, "topic_guess": "algebra", "text": "GMAT algebra note: Then, when you reach the statements, your work is made much easier. Here is approximately how your paper could look:.", "topic": "algebra"}
{"id": "book 2_p148_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 148, "page_end": 148, "topic_guess": "algebra", "text": "GMAT algebra note: X = 2, y = 0 or both If y = 0, value of xy = 0. If x = 2, value of xy depends on y.", "topic": "algebra"}
{"id": "book 2_p148_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 148, "page_end": 148, "topic_guess": "algebra", "text": "GMAT algebra note: If x = 2, then y = 2 + x = 4, so xy = (2)(4) = 8. If y = 0, then x = y − 2 = −2, so xy = (−2)(0) = 0.", "topic": "algebra"}
{"id": "book 2_p149_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 149, "page_end": 149, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Constraints: x and y integers Case 1:.", "topic": "number_properties"}
{"id": "book 2_p149_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 149, "page_end": 149, "topic_guess": "number_theory", "text": "GMAT number properties note: If y = 0, x = −2, so xy = 0. If x is negative, y must equal 0.", "topic": "number_properties"}
{"id": "book 2_p149_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 149, "page_end": 149, "topic_guess": "number_theory", "text": "GMAT number properties note: You can lay out your paper in many other ways. For instance, you might go with this version:.", "topic": "number_properties"}
{"id": "book 2_p150_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 150, "page_end": 150, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating. In this second layout, facts go on the le, while the question and any rephrasing go on the right.", "topic": "word_problems"}
{"id": "book 2_p150_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 150, "page_end": 150, "topic_guess": "general", "text": "GMAT quant note: Then the process is always to see whether you can bridge the gap, going from le to right. The important thing is that you develop a consistent layout that you always use.", "topic": "word_problems"}
{"id": "book 2_p151_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 151, "page_end": 151, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. Never Rephrase Yes/No as Value All DS questions can be divided into two types:.", "topic": "number_properties"}
{"id": "book 2_p151_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 151, "page_end": 151, "topic_guess": "general", "text": "GMAT quant note: Value questions (such as “What is a?”) and Yes/No questions (such as “Is x an integer?”). Value questions and Yes/No questions are fundamentally different: they require different levels of information to answer the question.", "topic": "number_properties"}
{"id": "book 2_p151_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 151, "page_end": 151, "topic_guess": "general", "text": "GMAT quant note: Therefore, never rephrase a Yes/No question as a Value question. Value questions usually require more information than Yes/No questions.", "topic": "number_properties"}
{"id": "book 2_p151_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 151, "page_end": 151, "topic_guess": "general", "text": "GMAT quant note: Try-It #3-2 Is the integer n odd? You don’t need to know which value n might be, just whether n is odd.", "topic": "number_properties"}
{"id": "book 2_p151_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 151, "page_end": 151, "topic_guess": "general", "text": "GMAT quant note: Therefore, do not rephrase this question to “What is integer n?” Doing so unnecessarily increases the amount of information you need to answer the question. Of course, if you happen to know what n is, then great, you can answer any Yes/No question about n.", "topic": "number_properties"}
{"id": "book 2_p151_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 151, "page_end": 151, "topic_guess": "general", "text": "GMAT quant note: But you generally don’t need to know the value of n to answer Yes/No questions about n, and the GMAT loves to exploit that truth at your expense. N2 − 2n is not a multiple of 4.(1) n is a multiple of 3.(2).", "topic": "number_properties"}
{"id": "book 2_p152_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 152, "page_end": 152, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. (1) SUFFICIENT: n2 − 2n = n(n − 2).", "topic": "number_properties"}
{"id": "book 2_p152_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 152, "page_end": 152, "topic_guess": "general", "text": "GMAT quant note: If n is even, both terms in this product will be even and the product will be divisible by 4. Since n2 − 2n is not a multiple of 4, integer n cannot be even—it must be odd.", "topic": "number_properties"}
{"id": "book 2_p152_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 152, "page_end": 152, "topic_guess": "general", "text": "GMAT quant note: Multiples of 3 can be either odd or even. Rephrasing a Yes/No question into a Value question makes the question unnecessarily picky.", "topic": "number_properties"}
{"id": "book 2_p152_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 152, "page_end": 152, "topic_guess": "general", "text": "GMAT quant note: Yes/No questions can o en be sufficiently answered despite having multiple possible values for the answer. Note that the converse of this principle is not always true.", "topic": "number_properties"}
{"id": "book 2_p152_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 152, "page_end": 152, "topic_guess": "general", "text": "GMAT quant note: Occasionally, it’s okay to rephrase a Value question as a Yes/No question—specifically, when it turns out that there are only two possible values. Try-It #3-3 If x is a positive integer, what is the remainder of Some quick analysis will show that x2 − 1 can be factored into (x + 1)(x − 1).", "topic": "number_properties"}
{"id": "book 2_p152_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 152, "page_end": 152, "topic_guess": "general", "text": "GMAT quant note: If x is odd, then both of these terms will be even and the product will be divisible by 4, yielding a remainder of 0 when divided by 4. If x is even, then x2 will be divisible by 4, so the remainder of x2 − 1 will be 3.", "topic": "number_properties"}
{"id": "book 2_p152_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 152, "page_end": 152, "topic_guess": "general", "text": "GMAT quant note: There are only two possible values of the remainder: 0 and 3.", "topic": "number_properties"}
{"id": "book 2_p153_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 153, "page_end": 153, "topic_guess": "general", "text": "GMAT quant note: Separate total outcomes from desired outcomes and count carefully. “Is x even?” Although this Value question seemed at first to have several different potential outcomes, only two are possible, so you are able to change the question to a Yes/No format by suitable rephrasing.", "topic": "probability"}
{"id": "book 2_p154_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 154, "page_end": 154, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Work from Facts to Question Especially for simple Yes/No questions, people o en assume the answer to the question is Yes before looking at the statements.", "topic": "word_problems"}
{"id": "book 2_p154_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 154, "page_end": 154, "topic_guess": "word_problems", "text": "GMAT word problem note: They only test cases for which the answer is Yes, rather than testing all of the possible cases allowed by the statements. This line of thinking is backwards—and tempting, because of the order in which things are presented.", "topic": "word_problems"}
{"id": "book 2_p154_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 154, "page_end": 154, "topic_guess": "word_problems", "text": "GMAT word problem note: Instead ask, “If I start with the applicable facts and consider all possibilities, do I get a definitive answer to the question?” Always work from the given facts to the question—never the reverse! This is why you have to keep the facts separated from the question and why you should always clearly mark the question on your paper.", "topic": "word_problems"}
{"id": "book 2_p154_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 154, "page_end": 154, "topic_guess": "word_problems", "text": "GMAT word problem note: Try-It #3-4 If x ≠ 0, is x = 1?", "topic": "word_problems"}
{"id": "book 2_p155_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 155, "page_end": 155, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. If you work from the question to the facts, you would assume a Yes for the question, then plug this information into the statements.", "topic": "algebra"}
{"id": "book 2_p155_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 155, "page_end": 155, "topic_guess": "algebra", "text": "GMAT algebra note: For instance, you would plug x = 1 into each statement. That’s especially easy to do in this case because this particular question is much simpler to think about than the statements (which are nasty little equations).", "topic": "algebra"}
{"id": "book 2_p155_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 155, "page_end": 155, "topic_guess": "algebra", "text": "GMAT algebra note: Don’t start from the question! No matter what, when you are judging sufficiency, always proceed from the facts to the question.", "topic": "algebra"}
{"id": "book 2_p155_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 155, "page_end": 155, "topic_guess": "algebra", "text": "GMAT algebra note: It doesn’t matter how easy or hard the question is at that point. A er you’ve rephrased, put the question on hold and work from the statements and any other given facts to the question:.", "topic": "algebra"}
{"id": "book 2_p155_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 155, "page_end": 155, "topic_guess": "algebra", "text": "GMAT algebra note: In a Yes/No question, when evaluating the statements, always try to determine whether the question can be answered the same way under any possibility that is consistent with.", "topic": "algebra"}
{"id": "book 2_p156_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 156, "page_end": 156, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. A Yes answer means Always Yes, for all allowed scenarios.", "topic": "statistics"}
{"id": "book 2_p156_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 156, "page_end": 156, "topic_guess": "statistics", "text": "GMAT statistics note: Likewise, a No answer means Always No.", "topic": "statistics"}
{"id": "book 2_p157_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 157, "page_end": 157, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Be a Contrarian To avoid statement carryover and to gain insight into the nature of a problem, deliberately try to violate one statement as you evaluate the other statement.", "topic": "word_problems"}
{"id": "book 2_p157_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 157, "page_end": 157, "topic_guess": "word_problems", "text": "GMAT word problem note: This will make it much harder for you to make a faulty assumption that leads to an incorrect answer. Think outside the first statement’s box.", "topic": "word_problems"}
{"id": "book 2_p157_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 157, "page_end": 157, "topic_guess": "word_problems", "text": "GMAT word problem note: Try-It #3-5 If x ≠ 0, is xy > 0? The question is actually asking whether x and y have the same sign.", "topic": "word_problems"}
{"id": "book 2_p157_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 157, "page_end": 157, "topic_guess": "word_problems", "text": "GMAT word problem note: This indicates nothing about the sign of y. Besides, it is natural to assume that a given variable will have a positive value, because positive numbers are much more intuitive than negative numbers.", "topic": "word_problems"}
{"id": "book 2_p158_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 158, "page_end": 158, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. In this case, it is necessary to flip the sign of the inequality when you cross-multiply.", "topic": "algebra"}
{"id": "book 2_p158_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 158, "page_end": 158, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: That is, if x < 0, then means that 1 > xy, and the answer to the question is MAYBE. When evaluating individual statements, deliberately trying to violate the other statement can help you see the full pattern or trick in the problem.", "topic": "algebra"}
{"id": "book 2_p158_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 158, "page_end": 158, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: You will be less likely to fall victim to statement carryover.", "topic": "algebra"}
{"id": "book 2_p159_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 159, "page_end": 159, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Assume Nothing This principle is a corollary of the previous principle: avoid assuming constraints that aren’t actually given in the problem—particularly assumptions that seem natural to make.", "topic": "number_properties"}
{"id": "book 2_p159_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 159, "page_end": 159, "topic_guess": "number_theory", "text": "GMAT number properties note: Try-It #3-6 Is z an even integer? The wording of this question has a tendency to bias people toward integers.", "topic": "number_properties"}
{"id": "book 2_p159_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 159, "page_end": 159, "topic_guess": "number_theory", "text": "GMAT number properties note: A er all, the “opposite” of even is odd, and odd numbers are integers, too. However, the question does not state that z must be an integer in the first place, so do not assume that it is.", "topic": "number_properties"}
{"id": "book 2_p159_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 159, "page_end": 159, "topic_guess": "number_theory", "text": "GMAT number properties note: The fact that is an even integer implies that z = 2 × (an even integer), so z must be an even integer. The fact that 3z is an even integer implies that z = , so z might not be an integer at all.", "topic": "number_properties"}
{"id": "book 2_p160_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 160, "page_end": 160, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Equal One way to avoid assuming is to invoke Principle #3: work from facts to question.", "topic": "number_properties"}
{"id": "book 2_p160_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 160, "page_end": 160, "topic_guess": "number_theory", "text": "GMAT number properties note: No even integers have been skipped over, nor have you allowed the question to suggest z values. That is how assumptions sneak in.", "topic": "number_properties"}
{"id": "book 2_p160_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 160, "page_end": 160, "topic_guess": "number_theory", "text": "GMAT number properties note: Next, divide the numbers in your list by 3: , etc. Note that you could have stopped computing a er the second value, since you already achieved a Yes and a No.", "topic": "number_properties"}
{"id": "book 2_p160_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 160, "page_end": 160, "topic_guess": "number_theory", "text": "GMAT number properties note: Another common assumption is that a variable must be positive. Do not assume that any unknown is positive unless it is stated as such in the information given (or if the unknown counts physical things or measures some other positive-only quantity).", "topic": "number_properties"}
{"id": "book 2_p160_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 160, "page_end": 160, "topic_guess": "number_theory", "text": "GMAT number properties note: 3 × even = even, so z could be even. 3 × odd = odd, so z is definitely not odd.", "topic": "number_properties"}
{"id": "book 2_p161_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 161, "page_end": 161, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. Your process may be different.", "topic": "number_properties"}
{"id": "book 2_p161_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 161, "page_end": 161, "topic_guess": "general", "text": "GMAT quant note: There are 19 batters on a baseball team. Every batter bats either right-handed only, le -handed only, or both right-handed and le - handed.", "topic": "number_properties"}
{"id": "book 2_p161_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 161, "page_end": 161, "topic_guess": "general", "text": "GMAT quant note: How many of the 19 batters bat le -handed? Seven of the batters bat right-handed but do not bat le - handed.", "topic": "number_properties"}
{"id": "book 2_p161_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 161, "page_end": 161, "topic_guess": "general", "text": "GMAT quant note: Four of the batters bat both right-handed and le -handed.(2) 2. If a is a positive integer and 81 divided by a results in a remainder of 1, what is the value of a?", "topic": "number_properties"}
{"id": "book 2_p161_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 161, "page_end": 161, "topic_guess": "general", "text": "GMAT quant note: The remainder when a is divided by 40 is 0.(1) The remainder when 40 is divided by a is 40.(2).", "topic": "number_properties"}
{"id": "book 2_p162_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 162, "page_end": 162, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Apply Principle #2 (Never Rephrase Yes/No as Value).", "topic": "number_properties"}
{"id": "book 2_p162_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 162, "page_end": 162, "topic_guess": "number_theory", "text": "GMAT number properties note: Describe why these Yes/No questions cannot be rephrased as Value questions. If a, b, c, d, and e are positive integers such that is an integer?", "topic": "number_properties"}
{"id": "book 2_p162_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 162, "page_end": 162, "topic_guess": "number_theory", "text": "GMAT number properties note: If a, b, and c are each integers greater than 1, is the product abc divisible by 6? The product ab is even.(1) The product bc is divisible by 3.(2) 5.", "topic": "number_properties"}
{"id": "book 2_p162_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 162, "page_end": 162, "topic_guess": "number_theory", "text": "GMAT number properties note: If n is a positive integer, is n − 1 divisible by 3? N2 + n is not divisible by 6.(1) 3n = k + 3, where k is a positive multiple of 3(2) 6.", "topic": "number_properties"}
{"id": "book 2_p162_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 162, "page_end": 162, "topic_guess": "number_theory", "text": "GMAT number properties note: Revisit problems #4 and #5 above, this time deliberately violating Principle #3 (Work from Facts to Question). Determine the incorrect answer you might have selected if you had reversed the process and worked from the question to the facts.", "topic": "number_properties"}
{"id": "book 2_p163_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 163, "page_end": 163, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Then, re-solve the problem by applying Principle #4 (be a contrarian).", "topic": "algebra"}
{"id": "book 2_p163_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 163, "page_end": 163, "topic_guess": "algebra", "text": "GMAT algebra note: Verify that applying Principle #4 leads to the correct answer, whereas not following the principle could lead to an incorrect answer. For problems 8 and 9, apply Principle #5 (assume nothing) by identifying the explicit constraints given in the problem.", "topic": "algebra"}
{"id": "book 2_p163_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 163, "page_end": 163, "topic_guess": "algebra", "text": "GMAT algebra note: What values are still permissible? Next, solve using these constraints.", "topic": "algebra"}
{"id": "book 2_p164_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 164, "page_end": 164, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "book 2_p165_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 165, "page_end": 165, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Your notes may be different.) 1.", "topic": "number_properties"}
{"id": "book 2_p165_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 165, "page_end": 165, "topic_guess": "number_theory", "text": "GMAT number properties note: Rephrasing: 19 batters total → integers only! Some R only Some L only Some R & L 0 “neither” How many le -handed batters?", "topic": "number_properties"}
{"id": "book 2_p165_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 165, "page_end": 165, "topic_guess": "number_theory", "text": "GMAT number properties note: L Not L Total R Not R 0 Total x 19 What is the integer x?", "topic": "number_properties"}
{"id": "book 2_p166_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 166, "page_end": 166, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. L Not L Total R 7 Not R 0 Total x 7 19 Question:.", "topic": "number_properties"}
{"id": "book 2_p166_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 166, "page_end": 166, "topic_guess": "number_theory", "text": "GMAT number properties note: Rephrasing: 81 divided by a → remainder of 1. A goes evenly into 80, and a ≠ 1.", "topic": "number_properties"}
{"id": "book 2_p166_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 166, "page_end": 166, "topic_guess": "number_theory", "text": "GMAT number properties note: A is one of the factors of 80 other than 1. A = 2, 4, 5, 8, 10, 16, 20, 40, or 80.", "topic": "number_properties"}
{"id": "book 2_p166_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 166, "page_end": 166, "topic_guess": "number_theory", "text": "GMAT number properties note: Which of the numbers listed above is the value of a?", "topic": "number_properties"}
{"id": "book 2_p167_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 167, "page_end": 167, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. Which of the listed numbers is a?", "topic": "number_properties"}
{"id": "book 2_p167_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 167, "page_end": 167, "topic_guess": "general", "text": "GMAT quant note: (INSUFFICIENT) a must be 80. Yes (SUFFICIENT) Yes (SUFFICIENT).", "topic": "number_properties"}
{"id": "book 2_p168_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 168, "page_end": 168, "topic_guess": "word_problems", "text": "GMAT word problem note: Watch divisibility, factors, remainders, parity, and prime structure. Each of the below is an accurate Yes/No rephrase:.", "topic": "number_properties"}
{"id": "book 2_p168_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 168, "page_end": 168, "topic_guess": "word_problems", "text": "GMAT word problem note: Alternatively, you could ask “Is there an even integer and a multiple of 3 among a, b, and c?” (1) INSUFFICIENT: ab is divisible by 2, but it’s unclear whether it is divisible by 3 (or whether c is divisible by 3). (2) INSUFFICIENT: bc is divisible by 3, but it’s unclear whether it is divisible by 2 (or whether a is divisible by 2).", "topic": "number_properties"}
{"id": "book 2_p168_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 168, "page_end": 168, "topic_guess": "word_problems", "text": "GMAT word problem note: Therefore abc is divisible by both 2 and 3. Is Is abc = 6, 12, 18, 24, 30, 36, 42, 48, etc.?", "topic": "number_properties"}
{"id": "book 2_p168_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 168, "page_end": 168, "topic_guess": "word_problems", "text": "GMAT word problem note: Is abc divisible by 2 and by 3? An accurate Yes/No rephrase is the following.", "topic": "number_properties"}
{"id": "book 2_p169_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 169, "page_end": 169, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. This narrows down the values of interest to a certain type of number, which follows a pattern: 1, 4, 7, 10, etc.", "topic": "algebra"}
{"id": "book 2_p169_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 169, "page_end": 169, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: If n2 + n = n (n + 1) is not divisible by 6, you can rule out certain values for n. N n + 1 n (n + 1) not divisible by 6 The pattern from the rephrasing is apparent here: n can only be 1, 4, 7, 10, etc., all integers that are one greater than a multiple of 3.", "topic": "algebra"}
{"id": "book 2_p170_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 170, "page_end": 170, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The integers n − 1, n, and n + 1 must be consecutive.", "topic": "number_properties"}
{"id": "book 2_p170_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 170, "page_end": 170, "topic_guess": "number_theory", "text": "GMAT number properties note: If n(n + 1) is not divisible by 3, then n − 1 must be divisible by 3, since in any set of three consecutive integers, one of the integers must be divisible by 3. If 3n = 3 × pos integer + 3, then n = pos integer + 1.", "topic": "number_properties"}
{"id": "book 2_p170_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 170, "page_end": 170, "topic_guess": "number_theory", "text": "GMAT number properties note: This does not resolve whether n is definitely one greater than a multiple of 3. Revisiting #4, working incorrectly from the Question to the Facts:.", "topic": "number_properties"}
{"id": "book 2_p170_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 170, "page_end": 170, "topic_guess": "number_theory", "text": "GMAT number properties note: Manipulating the question to abc = 6 × integer (and losing track of the question mark), someone might be tempted to check whether it is possible for abc to be a multiple of 6, instead of whether abc is definitely a multiple of 6:. Be sure to revisit how to do this problem correctly!", "topic": "number_properties"}
{"id": "book 2_p170_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 170, "page_end": 170, "topic_guess": "number_theory", "text": "GMAT number properties note: Revisiting #5, working incorrectly from the Question to the Facts:. Someone working incorrectly might try multiples of 3 for n − 1 to see whether they “work” with the statements: (1) abc = 6 × integer, so ab is even.", "topic": "number_properties"}
{"id": "book 2_p170_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 170, "page_end": 170, "topic_guess": "number_theory", "text": "GMAT number properties note: ✓ (2) abc = 6 × integer, so bc is divisible by 3.", "topic": "number_properties"}
{"id": "book 2_p171_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 171, "page_end": 171, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating. (1) SUFFICIENT: n − 1 n n + 1 n2 + n = n(n+ 1) not divis by 6 9 10 11 110 ✓ (2) SEEMS SUFFICIENT (incorrectly):.", "topic": "word_problems"}
{"id": "book 2_p171_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 171, "page_end": 171, "topic_guess": "general", "text": "GMAT quant note: Be sure to revisit how to do this problem correctly, so you are certain how to do so for the future!", "topic": "word_problems"}
{"id": "book 2_p172_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 172, "page_end": 172, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Non-Contrarian Approach Contrarian Approach (1) INSUFFICIENT: m > 0 or m = pos, so “Is pos3 > pos2?” (1) INSUFFICIENT: m > 0 or m = pos, so “Is pos3 > pos2?” (2) SEEMS SUFFICIENT: m2 > m implies that m is not a fraction or 1, therefore m3 > m2.", "topic": "percent"}
{"id": "book 2_p172_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 172, "page_end": 172, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Or, if you assume that m > 0, carrying over from (1), you might do the following: m2 > m m > 1 (dividing by m) Therefore, m3 > m2. (2) INSUFFICIENT: m2 > m implies that m is not a fraction or 1, so if m > 1 then the answer is Yes.", "topic": "percent"}
{"id": "book 2_p172_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 172, "page_end": 172, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: BUT, contradicting (1), what if m is negative? That is possible according to (2), since neg2 > neg.", "topic": "percent"}
{"id": "book 2_p172_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 172, "page_end": 172, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: No, if m is a proper fraction or 1.", "topic": "percent"}
{"id": "book 2_p173_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 173, "page_end": 173, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Non-Contrarian Approach Contrarian Approach (1) AND (2) SUFFICIENT:.", "topic": "percent"}
{"id": "book 2_p173_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 173, "page_end": 173, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Combined, the statements eliminate negative, zero, positive proper fractions, and 1 for the value of m. Note that both solutions were hampered by inadequate rephrasing.", "topic": "percent"}
{"id": "book 2_p173_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 173, "page_end": 173, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Ideally, you would first rephrase as follows:. With proper rephrasing, other errors are less likely.", "topic": "percent"}
{"id": "book 2_p173_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 173, "page_end": 173, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: But even with inadequate rephrasing, taking a contrarian approach can save you from a wrong answer. (It’s okay to divide by m2, which must be positive: a square is never negative and m is also not 0, according to the question stem.) 8.", "topic": "percent"}
{"id": "book 2_p173_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 173, "page_end": 173, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: The explicit constraint is yz ≠ 0, which indicates that y ≠ 0 and z ≠ 0. Both y and z could be any nonzero value, including positive integers, negative integers, positive fractions, negative fractions, etc.", "topic": "percent"}
{"id": "book 2_p174_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 174, "page_end": 174, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. For (1), a faulty assumption could be made by those who plug in values for z.", "topic": "percent"}
{"id": "book 2_p174_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 174, "page_end": 174, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: That would yield a definite No answer to the question, as all the y values are at least as great as 1. The (unverbalized) assumption is that z is an integer, but that’s not necessarily so.", "topic": "percent"}
{"id": "book 2_p174_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 174, "page_end": 174, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Correct solution: (1) INSUFFICIENT: y must be positive, but is it a fraction or an integer? If z = 2, then y = 4, and the answer is No.", "topic": "percent"}
{"id": "book 2_p175_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 175, "page_end": 175, "topic_guess": "general", "text": "GMAT quant note: Since you’re assuming y < 0, the rephrasing is y < −1. YES NO (1) AND (2) SUFFICIENT:.", "topic": "general"}
{"id": "book 2_p175_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 175, "page_end": 175, "topic_guess": "general", "text": "GMAT quant note: If z = , then y = , and the answer is Yes.", "topic": "general"}
{"id": "book 2_p176_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 176, "page_end": 176, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Avoid limiting yourself to cases in which x is greater than y.", "topic": "percent"}
{"id": "book 2_p176_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 176, "page_end": 176, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Since “Is x > y?” is a question and not a statement, x might be greater than or less than y. Actively seek cases that give a No answer.", "topic": "percent"}
{"id": "book 2_p176_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 176, "page_end": 176, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: 3 4 |–1| < |3| (valid) No 4 3 |1| < |3| (valid) Yes Since x could be greater than y or less than y, given the information in the statement, this statement is insufficient. The second case tested above also fits this statement and yields an answer of Yes.", "topic": "percent"}
{"id": "book 2_p176_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 176, "page_end": 176, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Find another case in which |x| > |y|, but x is not greater than y. 4 3 |4| > |3| (valid) Yes –4 3 |–4| > |3| (valid) No (1) AND (2) INSUFFICIENT:.", "topic": "percent"}
{"id": "book 2_p176_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 176, "page_end": 176, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: A case that fits both statements and gives an answer of Yes has already been found. Try to find another case for which both statements are true, but which gives an answer of No.", "topic": "percent"}
{"id": "book 2_p176_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 176, "page_end": 176, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: As you test different values for x and y, you may find certain cases for which one or both statements are not true. Cross these off on your paper, and do not take them into consideration when choosing your answer.", "topic": "percent"}
{"id": "book 2_p177_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 177, "page_end": 177, "topic_guess": "general", "text": "GMAT quant note: X y (1): |x – y| < |x| (2): |x| > |y| Is x > y?x y (1): |x – y| < |x| (2): |x| > |y| Is x > y? 4 3 |1| < |3| (valid) |4| > |3| (valid) Yes –4 3 |–7| < |–4|: (not valid) n/a n/a The answer to the question can be either Yes (if x = 4 and y = 3) or No (if x = –4 and y = –3), so the statements together are insufficient.", "topic": "general"}
{"id": "book 2_p178_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 178, "page_end": 178, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "book 2_p179_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 179, "page_end": 179, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Advanced Strategies Advanced Guessing Tactics Summary Common Wrong Answers.", "topic": "word_problems"}
{"id": "book 2_p180_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 180, "page_end": 180, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Strategies are broad: they apply to a wide variety of problems, they provide a complete approach, and they can be used safely in most circumstances.", "topic": "word_problems"}
{"id": "book 2_p180_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 180, "page_end": 180, "topic_guess": "word_problems", "text": "GMAT word problem note: Moreover, a particular tactic may only be useful in special situations or for parts of a problem. Compute to Completion For some problems, you won’t necessarily be able to tell whether the answer can be calculated until you follow through on the calculations all.", "topic": "word_problems"}
{"id": "book 2_p181_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 181, "page_end": 181, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Extract the Equation For many Word Problems, you need to represent the problem with algebraic equations to avoid embedded tricks that can be difficult to spot otherwise.", "topic": "word_problems"}
{"id": "book 2_p181_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 181, "page_end": 181, "topic_guess": "word_problems", "text": "GMAT word problem note: Know the Code The most challenging part of some DS problems is figuring out what the question and statements are actually saying. If the problem is written in “GMAT code,” translate it into simple language before you do anything else.", "topic": "word_problems"}
{"id": "book 2_p181_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 181, "page_end": 181, "topic_guess": "word_problems", "text": "GMAT word problem note: Use the Constraints Many DS problems provide explicit constraints on the variables. In other problems, these constraints will be implicit (e.g., a variable that refers to a number of people, houses, or airplanes must be both positive and an integer).", "topic": "word_problems"}
{"id": "book 2_p181_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 181, "page_end": 181, "topic_guess": "word_problems", "text": "GMAT word problem note: In either case, these constraints frequently determine the correct answer, so you must identify and use them. Beware of Inequalities Whenever a DS problem involves inequality symbols, be especially careful—the GMAT loves to trick people with inequalities.", "topic": "word_problems"}
{"id": "book 2_p181_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 181, "page_end": 181, "topic_guess": "word_problems", "text": "GMAT word problem note: Test Cases One of the best ways to show that a statement is insufficient is to test different scenarios, or cases, in which that statement is true.", "topic": "word_problems"}
{"id": "book 2_p182_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 182, "page_end": 182, "topic_guess": "strategy", "text": "GMAT strategy note: Watch divisibility, factors, remainders, parity, and prime structure. These tactics are listed later in the chapter.", "topic": "number_properties"}
{"id": "book 2_p182_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 182, "page_end": 182, "topic_guess": "strategy", "text": "GMAT strategy note: As with Problem Solving (PS) tactics, some of these D S tactics work wonders when used correctly. Others only slightly improve your guessing odds.", "topic": "number_properties"}
{"id": "book 2_p183_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 183, "page_end": 183, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. For some problems, however, you cannot determine whether a single answer can be obtained until you’ve calculated the problem all the way through.", "topic": "algebra"}
{"id": "book 2_p183_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 183, "page_end": 183, "topic_guess": "algebra", "text": "GMAT algebra note: This is particularly common in the following situations:. Try-It #4-1 What is the value of ab?", "topic": "algebra"}
{"id": "book 2_p183_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 183, "page_end": 183, "topic_guess": "algebra", "text": "GMAT algebra note: Multiple equations are involved—particularly if they are non-linear. A complicated inequality expression is present.", "topic": "algebra"}
{"id": "book 2_p183_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 183, "page_end": 183, "topic_guess": "algebra", "text": "GMAT algebra note: Variables hidden within a Geometry problem are related. A = b + 1 (1) a2 = b + 1 (2).", "topic": "algebra"}
{"id": "book 2_p184_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 184, "page_end": 184, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating. Evaluating both statements together is trickier, however:.", "topic": "word_problems"}
{"id": "book 2_p184_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 184, "page_end": 184, "topic_guess": "general", "text": "GMAT quant note: Based on this work, either a = 0 or a = 1. It would be tempting at this stage to decide that since a can have two different values, statements (1) and (2) together are insufficient.", "topic": "word_problems"}
{"id": "book 2_p184_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 184, "page_end": 184, "topic_guess": "general", "text": "GMAT quant note: Look at the values b can hold in these two scenarios: a b ab 1 a − 1 = 0 0 0 a − 1 = −1 0 While it is true that a can take on different values, ab is equal to zero in either case. When a = 1, b = 0, and ab = 0.", "topic": "word_problems"}
{"id": "book 2_p184_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 184, "page_end": 184, "topic_guess": "general", "text": "GMAT quant note: When a = 0, b = −1, and ab = 0.", "topic": "word_problems"}
{"id": "book 2_p185_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 185, "page_end": 185, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Therefore, (1) and (2) combined are SUFFICIENT to answer this specific question:.", "topic": "word_problems"}
{"id": "book 2_p185_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 185, "page_end": 185, "topic_guess": "word_problems", "text": "GMAT word problem note: In a multiple-scenario problem, be sure to compute for the specific question asked (in this case, ab) in order to determine whether the end result for each scenario is actually different. EXTRACT THE EQUATION For Word Problems, setting up an algebraic representation of the question is essential.", "topic": "word_problems"}
{"id": "book 2_p185_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 185, "page_end": 185, "topic_guess": "word_problems", "text": "GMAT word problem note: It is very easy to get intellectually lazy and miss an embedded trick in the problem. These tricks are usually much easier to spot if you are looking at the underlying algebra behind the problem.", "topic": "word_problems"}
{"id": "book 2_p185_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 185, "page_end": 185, "topic_guess": "word_problems", "text": "GMAT word problem note: Try-It #4-2 A store sells two types of bird feeders:. Alphas feed 1 bird at a time, whereas Bravos feed 2 birds at a time.", "topic": "word_problems"}
{"id": "book 2_p185_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 185, "page_end": 185, "topic_guess": "word_problems", "text": "GMAT word problem note: The total number of birds that can be fed at one time by bird feeders sold last month is 50. What is the total revenue generated by birdfeeders sold last month?", "topic": "word_problems"}
{"id": "book 2_p185_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 185, "page_end": 185, "topic_guess": "word_problems", "text": "GMAT word problem note: Last month, the price of each Alpha was $15 and the price of each Bravo was $30. 40 Alphas were sold last month.(2).", "topic": "word_problems"}
{"id": "book 2_p186_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 186, "page_end": 186, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. From the words in the question stem, Extract the Equation.", "topic": "algebra"}
{"id": "book 2_p186_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 186, "page_end": 186, "topic_guess": "algebra", "text": "GMAT algebra note: The problem indicates that Alphas can feed 1 bird at a time and Bravos 2 birds at a time. The problem also indicates that last month 50 birds could be fed at a time, so you have this:.", "topic": "algebra"}
{"id": "book 2_p186_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 186, "page_end": 186, "topic_guess": "algebra", "text": "GMAT algebra note: To calculate the revenue, it seems you will need the prices of the bird feeders and the number of bird feeders, A and B. Again extract the equation from the wording of the question:.", "topic": "algebra"}
{"id": "book 2_p186_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 186, "page_end": 186, "topic_guess": "algebra", "text": "GMAT algebra note: It turns out that you don’t need to know A and B individually, since the question stem equation indicated that A + 2B = 50. Therefore, total revenue equals $15(A + 2B) = $15(50) = $750.", "topic": "algebra"}
{"id": "book 2_p186_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 186, "page_end": 186, "topic_guess": "algebra", "text": "GMAT algebra note: It’s true that there are many possible values of A and B that satisfy the condition that A + 2B = 50. However, mathematically every possible combination that satisfies this equation would lead to the same revenue of $750.", "topic": "algebra"}
{"id": "book 2_p186_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 186, "page_end": 186, "topic_guess": "algebra", "text": "GMAT algebra note: The number of each type of bird feeder sold is irrelevant. In this sense, Extract the Equation can be similar to the Compute to Completion strategy because once the equation has been extracted, you may find that the multiple possibilities for the variables might converge to a single answer to the specific question that’s been asked.", "topic": "algebra"}
{"id": "book 2_p186_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 186, "page_end": 186, "topic_guess": "algebra", "text": "GMAT algebra note: Total number of birds fed = A + 2B = 50, where A and B represent the number of birdfeeders of each type that have been sol.", "topic": "algebra"}
{"id": "book 2_p187_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 187, "page_end": 187, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If there were 40 Alphas sold, there were 5 Bravos sold.", "topic": "number_properties"}
{"id": "book 2_p187_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 187, "page_end": 187, "topic_guess": "number_theory", "text": "GMAT number properties note: But you still don’t know the prices, so you can’t compute revenue. Be sure to translate all Word Problems into math so they can be properly evaluated.", "topic": "number_properties"}
{"id": "book 2_p187_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 187, "page_end": 187, "topic_guess": "number_theory", "text": "GMAT number properties note: These issues are sensitive to the exact numbers given. KNOW THE CODE Many DS questions and statements are intentionally written in a complicated way.", "topic": "number_properties"}
{"id": "book 2_p187_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 187, "page_end": 187, "topic_guess": "number_theory", "text": "GMAT number properties note: Take a look at the two statements below:. These statements look very different, but from the perspective of someone who’s solving a DS problem, they say exactly the same thing.", "topic": "number_properties"}
{"id": "book 2_p187_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 187, "page_end": 187, "topic_guess": "number_theory", "text": "GMAT number properties note: In other words, the first statement is just saying that a and b are both odd—but it’s saying it in code to keep you from noticing. There are two clues that the statement is really about odd and even numbers.", "topic": "number_properties"}
{"id": "book 2_p187_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 187, "page_end": 187, "topic_guess": "number_theory", "text": "GMAT number properties note: First, it specifies that a and b are integers. Second, if 2 is not a factor of a certain integer, that’s just another way of saying that the integer is odd.", "topic": "number_properties"}
{"id": "book 2_p187_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 187, "page_end": 187, "topic_guess": "number_theory", "text": "GMAT number properties note: You can write that out as follows: a and b are integers such that 2 is not a factor of 7a2b3.1. A and b are both odd integers.2.", "topic": "number_properties"}
{"id": "book 2_p188_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 188, "page_end": 188, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The next thing to notice is that multiplying a number by 7 has no effect on whether it’s odd or even.", "topic": "number_properties"}
{"id": "book 2_p188_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 188, "page_end": 188, "topic_guess": "number_theory", "text": "GMAT number properties note: Multiplying an even number by 7 gives you an even number, and multiplying an odd number by 7 gives you an odd number. So the 7 can be removed entirely:.", "topic": "number_properties"}
{"id": "book 2_p188_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 188, "page_end": 188, "topic_guess": "number_theory", "text": "GMAT number properties note: The exponents also don’t matter. Squaring or cubing an odd integer gives you an odd integer, and squaring or cubing an even integer gives you an even integer.", "topic": "number_properties"}
{"id": "book 2_p188_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 188, "page_end": 188, "topic_guess": "number_theory", "text": "GMAT number properties note: So saying that a2b3 is odd is no different from saying that ab is odd. Finally, if the product of two integers is odd, both of those integers have to be odd.", "topic": "number_properties"}
{"id": "book 2_p188_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 188, "page_end": 188, "topic_guess": "number_theory", "text": "GMAT number properties note: Multiplying an even integer by another integer always results in an even value. Here’s what the statement really says:.", "topic": "number_properties"}
{"id": "book 2_p188_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 188, "page_end": 188, "topic_guess": "number_theory", "text": "GMAT number properties note: The two statements were really the same all along. If you see something in a DS problem that looks like it could be GMAT code, try to crack the code before you go further.", "topic": "number_properties"}
{"id": "book 2_p188_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 188, "page_end": 188, "topic_guess": "number_theory", "text": "GMAT number properties note: You may find that a complicated statement is disguising something much easier to work with. USE THE CONSTRAINTS a and b are integers, and 7a2b3 is odd.", "topic": "number_properties"}
{"id": "book 2_p188_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 188, "page_end": 188, "topic_guess": "number_theory", "text": "GMAT number properties note: A and b are integers, and a2b3 is odd.", "topic": "number_properties"}
{"id": "book 2_p189_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 189, "page_end": 189, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. When this information is given, it is usually essential to the problem.", "topic": "algebra"}
{"id": "book 2_p189_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 189, "page_end": 189, "topic_guess": "algebra", "text": "GMAT algebra note: Try-It #4-3 If 8x > 3x + 4x, what is the value of the integer x? On top of that, there is also another constraint given: x must be an integer.", "topic": "algebra"}
{"id": "book 2_p189_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 189, "page_end": 189, "topic_guess": "algebra", "text": "GMAT algebra note: This limits the scope of the potential values of x even further. Make note of this type of constraint in your work on paper.", "topic": "algebra"}
{"id": "book 2_p189_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 189, "page_end": 189, "topic_guess": "algebra", "text": "GMAT algebra note: Write “x = int” or something similar. You could also incorporate this information by rephrasing the question to include the constraint: “If the integer x is positive, what is the value of x?” (1) SUFFICIENT:.", "topic": "algebra"}
{"id": "book 2_p189_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 189, "page_end": 189, "topic_guess": "algebra", "text": "GMAT algebra note: Solve this inequality for x: 8x can only be greater than 7x when x is positive.", "topic": "algebra"}
{"id": "book 2_p190_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 190, "page_end": 190, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Since you know from the question stem that x > 0, you can conclude that 0 < x < 2.", "topic": "word_problems"}
{"id": "book 2_p190_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 190, "page_end": 190, "topic_guess": "word_problems", "text": "GMAT word problem note: The only integer between 0 and 2 is 1. Manipulate this compound inequality as follows:.", "topic": "word_problems"}
{"id": "book 2_p190_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 190, "page_end": 190, "topic_guess": "word_problems", "text": "GMAT word problem note: Note that it’s fine to manipulate all the parts of the compound inequality at the same time as long as you perform each manipulation to all three parts of the inequality. Since the question stem indicates that x > 0, it must be the case that 0 < x ≤ 1.", "topic": "word_problems"}
{"id": "book 2_p190_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 190, "page_end": 190, "topic_guess": "word_problems", "text": "GMAT word problem note: The only integer that fits this criteria is 1. If you had overlooked the fact that x is an integer, you would have determined that there are many values between 0 and 1 or between 0 and 2.", "topic": "word_problems"}
{"id": "book 2_p191_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 191, "page_end": 191, "topic_guess": "word_problems", "text": "GMAT word problem note: Watch divisibility, factors, remainders, parity, and prime structure. Make a note of any additional information given to you in the question stem (e.g., “x is positive” or “x is an integer”).", "topic": "number_properties"}
{"id": "book 2_p191_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 191, "page_end": 191, "topic_guess": "word_problems", "text": "GMAT word problem note: You o en will have to use this information properly to get the correct answer. Integer constraints in particular are very potent: they o en limit the possible solutions for a problem to a small set.", "topic": "number_properties"}
{"id": "book 2_p191_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 191, "page_end": 191, "topic_guess": "word_problems", "text": "GMAT word problem note: Sometimes this set is so small that it contains only one item. Constraints will not always be explicitly given.", "topic": "number_properties"}
{"id": "book 2_p191_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 191, "page_end": 191, "topic_guess": "word_problems", "text": "GMAT word problem note: The ones the GMAT doesn’t explicitly give you can be called hidden constraints. Hidden constraints are most prevalent in Word Problems and Geometry questions.", "topic": "number_properties"}
{"id": "book 2_p191_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 191, "page_end": 191, "topic_guess": "word_problems", "text": "GMAT word problem note: The number of countable items must be a non-negative integer. Note that zero is only a possibility if it is possible for the items not to exist at all—if the problem clearly assumes that the items exist, then the number of items must be positive.", "topic": "number_properties"}
{"id": "book 2_p191_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 191, "page_end": 191, "topic_guess": "word_problems", "text": "GMAT word problem note: Number of people Number of yachts Number of books Many non-countable quantities must be non-negative numbers, though not necessarily integers. Again, zero is only an option if the underlying object might not exist.", "topic": "number_properties"}
{"id": "book 2_p191_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 191, "page_end": 191, "topic_guess": "word_problems", "text": "GMAT word problem note: If the problem clearly assumes the existence and typical definition of an object, then these quantities must be positive.", "topic": "number_properties"}
{"id": "book 2_p192_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 192, "page_end": 192, "topic_guess": "geometry", "text": "GMAT geometry note: If these constraints are important in a P S problem, then failing to take a constraint into account may make you unable to solve the problem. That will alert you to the existence of the constraints, since every P S problem must be solvable.", "topic": "geometry"}
{"id": "book 2_p192_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 192, "page_end": 192, "topic_guess": "geometry", "text": "GMAT geometry note: In contrast, you will get no such signal on a DS problem. A er all, solvability is the very issue that D S tests!", "topic": "geometry"}
{"id": "book 2_p192_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 192, "page_end": 192, "topic_guess": "geometry", "text": "GMAT geometry note: BEWARE OF INEQUALITIES Whenever a D S question involves inequality symbols, be especially careful. The GMAT can employ a variety of different inequality-specific tricks.", "topic": "geometry"}
{"id": "book 2_p192_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 192, "page_end": 192, "topic_guess": "geometry", "text": "GMAT geometry note: The side of a triangle must have a positive length. The only exception is negative coordinates in a coordinate plane problem.) The weight of a shipment of products must be positive in any unit.", "topic": "geometry"}
{"id": "book 2_p192_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 192, "page_end": 192, "topic_guess": "geometry", "text": "GMAT geometry note: The height of a person must be positive in any unit. Many other non-countable quantities are theoretically allowed to take on negative values.", "topic": "geometry"}
{"id": "book 2_p192_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 192, "page_end": 192, "topic_guess": "geometry", "text": "GMAT geometry note: The profit of a company (However, if a company made a profit, then that profit is positive!) The growth rate of a population The change in the value of essentially a.", "topic": "geometry"}
{"id": "book 2_p193_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 193, "page_end": 193, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. One inequality can imply another seemingly unrelated inequality, depending on the situation.", "topic": "word_problems"}
{"id": "book 2_p193_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 193, "page_end": 193, "topic_guess": "word_problems", "text": "GMAT word problem note: If x is greater than 5, then it must be positive, thus x > 0. However, the opposite is not the case.", "topic": "word_problems"}
{"id": "book 2_p193_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 193, "page_end": 193, "topic_guess": "word_problems", "text": "GMAT word problem note: If you knew that x < 5, then you would not be able to determine whether x > 0. A er all, x could be positive but less than 5, or x could be negative.", "topic": "word_problems"}
{"id": "book 2_p193_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 193, "page_end": 193, "topic_guess": "word_problems", "text": "GMAT word problem note: Inequalities can combine with integer constraints to produce a single value. Some Word Problems can create a hidden constraint involving inequalities.", "topic": "word_problems"}
{"id": "book 2_p193_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 193, "page_end": 193, "topic_guess": "word_problems", "text": "GMAT word problem note: These inequalities may come into play in determining the correct answer. The next oldest student in the class...", "topic": "word_problems"}
{"id": "book 2_p193_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 193, "page_end": 193, "topic_guess": "word_problems", "text": "GMAT word problem note: The youngest student in the class...” This can be translated to the following inequality: youngest < middle < oldest. Inequalities involving a variable in a denominator o en involve two possibilities: a positive and a negative one.", "topic": "word_problems"}
{"id": "book 2_p193_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 193, "page_end": 193, "topic_guess": "word_problems", "text": "GMAT word problem note: However, this may not be correct. It depends on whether y is a positive or negative number.", "topic": "word_problems"}
{"id": "book 2_p193_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 193, "page_end": 193, "topic_guess": "word_problems", "text": "GMAT word problem note: If y > 0, then it is correct to infer t.", "topic": "word_problems"}
{"id": "book 2_p194_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 194, "page_end": 194, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Try-It #4-4 If is a prime number, what is the value of x?", "topic": "number_properties"}
{"id": "book 2_p194_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 194, "page_end": 194, "topic_guess": "number_theory", "text": "GMAT number properties note: If is a prime number, then possible values are 2, 3, 5, and so on. Therefore, x must be a perfect square of a prime.", "topic": "number_properties"}
{"id": "book 2_p194_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 194, "page_end": 194, "topic_guess": "number_theory", "text": "GMAT number properties note: Possible values include 4, 9, 25, and so on. Manipulate the inequality to isolate x:.", "topic": "number_properties"}
{"id": "book 2_p194_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 194, "page_end": 194, "topic_guess": "number_theory", "text": "GMAT number properties note: Many questions involving inequalities are actually disguised positive/negative questions. What is interesting is that the product is positive, meaning both x and y are positive or both x and y are negative.", "topic": "number_properties"}
{"id": "book 2_p194_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 194, "page_end": 194, "topic_guess": "number_theory", "text": "GMAT number properties note: Thus, x and y have the same sign. Here, the inequality symbol is used to disguise the fact that x and y have the same sign.", "topic": "number_properties"}
{"id": "book 2_p194_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 194, "page_end": 194, "topic_guess": "number_theory", "text": "GMAT number properties note: −16 < −3x + 5<22(1) x2 is a two-digit number(2).", "topic": "number_properties"}
{"id": "book 2_p195_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 195, "page_end": 195, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Since x is the square of a prime, it can’t be negative or zero.", "topic": "number_properties"}
{"id": "book 2_p195_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 195, "page_end": 195, "topic_guess": "number_theory", "text": "GMAT number properties note: The smallest possible square of a prime is 4 and the next smallest possible square of a prime is 9. This inequality allows just one possible value: x = 4.", "topic": "number_properties"}
{"id": "book 2_p195_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 195, "page_end": 195, "topic_guess": "number_theory", "text": "GMAT number properties note: In addition, from the question stem, is a prime number. Determine the possible value(s) for x:.", "topic": "number_properties"}
{"id": "book 2_p195_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 195, "page_end": 195, "topic_guess": "number_theory", "text": "GMAT number properties note: Because there are two possible values for x, this statement is not sufficient. Try-It #4-5 If mn ≠ 0, is m > n?", "topic": "number_properties"}
{"id": "book 2_p195_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 195, "page_end": 195, "topic_guess": "number_theory", "text": "GMAT number properties note: If x = 4, then is a prime number, 2, and x2 is a two-digit number, 16. If x = 9, then is a prime number, 3, and x2 is a two-digit number, 81.", "topic": "number_properties"}
{"id": "book 2_p196_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 196, "page_end": 196, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. The constraint in the question stem indicates that neither m nor n equals zero.", "topic": "algebra"}
{"id": "book 2_p196_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 196, "page_end": 196, "topic_guess": "algebra", "text": "GMAT algebra note: You can solve algebraically/theoretically or you can test cases. If you solve algebraically, be careful: you have to account for multiplying the inequality by a negative:.", "topic": "algebra"}
{"id": "book 2_p196_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 196, "page_end": 196, "topic_guess": "algebra", "text": "GMAT algebra note: Alternatively, test cases: (2) INSUFFICIENT:. This statement indicates nothing about the signs of the two variables.", "topic": "algebra"}
{"id": "book 2_p196_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 196, "page_end": 196, "topic_guess": "algebra", "text": "GMAT algebra note: Either one could be positive or negative. If m and n are both positive, then m > n.", "topic": "algebra"}
{"id": "book 2_p196_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 196, "page_end": 196, "topic_guess": "algebra", "text": "GMAT algebra note: If m and n are both negative, the sign flips twice, so m > n again. If only one is negative, then the sign flips once and m < n.", "topic": "algebra"}
{"id": "book 2_p196_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 196, "page_end": 196, "topic_guess": "algebra", "text": "GMAT algebra note: In this case, m must be the negative number, since any positive is greater than any negative. If m and n are both positive, then m > n and m2 > n2.", "topic": "algebra"}
{"id": "book 2_p196_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 196, "page_end": 196, "topic_guess": "algebra", "text": "GMAT algebra note: Both statements allow this scenario. If m and n are both negative, then m > n but m2 is not greater than n2.", "topic": "algebra"}
{"id": "book 2_p197_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 197, "page_end": 197, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Because there are scenarios in which m > n and m < n, both statements together are still insufficient to answer the question.", "topic": "algebra"}
{"id": "book 2_p197_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 197, "page_end": 197, "topic_guess": "algebra", "text": "GMAT algebra note: One way to show that this is true is by Testing Cases. Write down several scenarios, or cases, in which the statement is true.", "topic": "algebra"}
{"id": "book 2_p197_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 197, "page_end": 197, "topic_guess": "algebra", "text": "GMAT algebra note: For instance, you might think of different possible values for the variables in the statement. For each of these cases, determine the answer to the question.", "topic": "algebra"}
{"id": "book 2_p197_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 197, "page_end": 197, "topic_guess": "algebra", "text": "GMAT algebra note: If m is negative and n is positive, then m < n. It could also be true that m2 > n2, as long as m’s magnitude is larger than n’s.", "topic": "algebra"}
{"id": "book 2_p197_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 197, "page_end": 197, "topic_guess": "algebra", "text": "GMAT algebra note: If you’re not sure, test cases (see below). If m = 3 and n = 2, then m > n and m2 > n2.", "topic": "algebra"}
{"id": "book 2_p197_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 197, "page_end": 197, "topic_guess": "algebra", "text": "GMAT algebra note: Both statements allow this scenario.", "topic": "algebra"}
{"id": "book 2_p198_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 198, "page_end": 198, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. If the question has different answers depending on which case you’re testing, the statement is insufficient.", "topic": "word_problems"}
{"id": "book 2_p198_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 198, "page_end": 198, "topic_guess": "word_problems", "text": "GMAT word problem note: If the question always has the same answer in every case you try, the statement is likely sufficient. However, the theory required to answer a question may be cumbersome to figure out in two minutes.", "topic": "word_problems"}
{"id": "book 2_p198_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 198, "page_end": 198, "topic_guess": "word_problems", "text": "GMAT word problem note: If implemented correctly, Testing Cases can be fast, easy, and accurate, so it should be part of your toolbox. Particularly, testing cases is o en the simplest way to show that a statement is not sufficient.", "topic": "word_problems"}
{"id": "book 2_p198_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 198, "page_end": 198, "topic_guess": "word_problems", "text": "GMAT word problem note: It’s easy to overlook a salient scenario. The key is to have a systematic approach to testing cases.", "topic": "word_problems"}
{"id": "book 2_p198_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 198, "page_end": 198, "topic_guess": "word_problems", "text": "GMAT word problem note: This relies on three approaches: the standard number set for testing, discrete number listing, and case testing with concepts. Sometimes testing enough cases will also allow you to notice a pattern to help you deduce whether the statement is sufficient.", "topic": "word_problems"}
{"id": "book 2_p198_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 198, "page_end": 198, "topic_guess": "word_problems", "text": "GMAT word problem note: Standard Number Set for Testing The GMAT o en tests.", "topic": "word_problems"}
{"id": "book 2_p199_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 199, "page_end": 199, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Odd Even Proper Fraction Improper Fraction Negative −1 −2 − Zero 0 Positive 1 2 This set includes integers, non-integers, positive and negative numbers, and numbers greater than and less than 1.", "topic": "number_properties"}
{"id": "book 2_p199_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 199, "page_end": 199, "topic_guess": "number_theory", "text": "GMAT number properties note: Thus, a comprehensive set of test numbers (to memorize and apply) would be as follows:. Remember this list as “every integer and half-integer between −2 and 2.” Not all of these numbers will be relevant or possible on every problem.", "topic": "number_properties"}
{"id": "book 2_p199_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 199, "page_end": 199, "topic_guess": "number_theory", "text": "GMAT number properties note: The question itself may suggest certain values to test, but always keep in mind the potential need to test a value of each relevant type—and if a problem really might entail testing nine different cases, consider whether that problem is really worth your time.", "topic": "number_properties"}
{"id": "book 2_p199_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 199, "page_end": 199, "topic_guess": "number_theory", "text": "GMAT number properties note: Using the Standard Set ensures that you don’t get tripped up by forgetting to try a particular type of value that will give you a different answer, thereby allowing you to prove definitively that the statement is not sufficient.", "topic": "number_properties"}
{"id": "book 2_p200_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 200, "page_end": 200, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. This problem presents inequalities, and the question asks whether a is negative.", "topic": "number_properties"}
{"id": "book 2_p200_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 200, "page_end": 200, "topic_guess": "number_theory", "text": "GMAT number properties note: Therefore, test different values of a to see which values fit the statements. The question stem doesn’t provide any constraints, so begin by testing some easier integers and then move to fractions if needed: (1) INSUFFICIENT:.", "topic": "number_properties"}
{"id": "book 2_p200_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 200, "page_end": 200, "topic_guess": "number_theory", "text": "GMAT number properties note: Test the possible integers from the standard number set: a a3 − a2 − 2a < 0 a < 0? −2 (−2)3 − (−2)2 − 2(−2) = −8 (valid) Yes 1 (1)3 − (1)2 − 2(1) = −2 (valid) No a3 < a2 + 2a(1) a2 > a3(2) −1 (−1)3 − (−1)2 − 2(−1) = 0 (invalid) 0 (0)3 − (0)2 − 2(0) = 0 (invalid).", "topic": "number_properties"}
{"id": "book 2_p201_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 201, "page_end": 201, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Stop when you find two valid cases that return different answers to the question.", "topic": "percent"}
{"id": "book 2_p201_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 201, "page_end": 201, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Try integers first: a a2 − a3 > 0 a < 0? −2 (−2)2 − (−2)3 = 12 (valid) Yes −1 (−1)2 − (−1)3 = 2 (valid) Yes Be careful:.", "topic": "percent"}
{"id": "book 2_p201_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 201, "page_end": 201, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: −2 (−2)2 − (−2)3 = 12 (valid) Yes −1 (−1)2 − (−1)3 = 2 (valid) Yes 0 (0)2 − (0)3 = 0 (invalid) 1 (1)2 − (1)3 = 0 (invalid) 2 (2)2 − (2)3 = -4 (invalid) 0 (0)2 − (0)3 = 0 (invalid) 1 (1)2 − (1)3 = 0 (invalid) 2 (2)2 − (2)3 = −4 (invalid).", "topic": "percent"}
{"id": "book 2_p202_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 202, "page_end": 202, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. When , the constraint is fulfilled, but a is positive.", "topic": "algebra"}
{"id": "book 2_p202_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 202, "page_end": 202, "topic_guess": "algebra", "text": "GMAT algebra note: Combining statements (1) and (2) shows that whenever a = −2, − , or both conditions are fulfilled. The variable a could thus be positive or negative.", "topic": "algebra"}
{"id": "book 2_p202_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 202, "page_end": 202, "topic_guess": "algebra", "text": "GMAT algebra note: Notice that you did not need to test every possible value for a. That means that testing − was unlikely to be necessary, since that value is not much different from −2.", "topic": "algebra"}
{"id": "book 2_p202_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 202, "page_end": 202, "topic_guess": "algebra", "text": "GMAT algebra note: Furthermore, when you do find a contradictory answer, you can stop testing. For a Yes/No question, all you need to do is find one valid Yes and one valid No to prove insufficiency.", "topic": "algebra"}
{"id": "book 2_p202_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 202, "page_end": 202, "topic_guess": "algebra", "text": "GMAT algebra note: This problem can also be solved algebraically, but it takes more conceptual work. First solve the inequality as though it were an equation, then map the solutions on a number line and test to see in which regions the inequality is true.", "topic": "algebra"}
{"id": "book 2_p203_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 203, "page_end": 203, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. This number line demonstrates that either a < −1 or 0 < a < 2.", "topic": "word_problems"}
{"id": "book 2_p203_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 203, "page_end": 203, "topic_guess": "word_problems", "text": "GMAT word problem note: Since a can’t be zero, and a2 must be positive, you can divide by a2. Overlapping the possible ranges, either a < −1 or 0 < a < 1.", "topic": "word_problems"}
{"id": "book 2_p203_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 203, "page_end": 203, "topic_guess": "word_problems", "text": "GMAT word problem note: This is still not enough information to tell whether a is negative. In some cases, the standard number testing list may not quite suffice.", "topic": "word_problems"}
{"id": "book 2_p204_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 204, "page_end": 204, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. However, x could be 1.1, in which case x2 = 1.21, which is less than 1.3.", "topic": "algebra"}
{"id": "book 2_p204_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 204, "page_end": 204, "topic_guess": "algebra", "text": "GMAT algebra note: You could figure out that x could be greater than 1 upon a quick inspection of this problem. You might think to try a number slightly larger than 1.", "topic": "algebra"}
{"id": "book 2_p204_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 204, "page_end": 204, "topic_guess": "algebra", "text": "GMAT algebra note: However, if the problem were more complicated, it might not be so obvious. In cases like these, use the boundary principle: test values that are close to boundaries given in the problem.", "topic": "algebra"}
{"id": "book 2_p204_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 204, "page_end": 204, "topic_guess": "algebra", "text": "GMAT algebra note: In this case, the boundary value is 1, so add 0.9 and 1.1 to your list of numbers to test. You might even try 0.99 or 1.01.", "topic": "algebra"}
{"id": "book 2_p204_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 204, "page_end": 204, "topic_guess": "algebra", "text": "GMAT algebra note: You should know that −1, 0, and 1 are natural boundaries, because numbers behave differently on either side of them (that’s why the standard list contains numbers in the ranges defined by −1, 0, and 1). Discrete Number Listing In the previous problems, the variables a and x were not constrained, so you had to test a series of different numbers to solve the problem.", "topic": "algebra"}
{"id": "book 2_p204_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 204, "page_end": 204, "topic_guess": "algebra", "text": "GMAT algebra note: By contrast, many questions suggest.", "topic": "algebra"}
{"id": "book 2_p205_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 205, "page_end": 205, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. By contrast, some problems describe a smooth range of potential values for a variable or expression (e.g., 0 < x < 1 or x must be negative).", "topic": "number_properties"}
{"id": "book 2_p205_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 205, "page_end": 205, "topic_guess": "number_theory", "text": "GMAT number properties note: In these cases, don’t list consecutive values to test because the set of possible values is not discrete. If the variable can take on any real number in a range, then rely on the standard number testing list, potentially with some modifications, as described in the previous section.", "topic": "number_properties"}
{"id": "book 2_p205_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 205, "page_end": 205, "topic_guess": "number_theory", "text": "GMAT number properties note: A key to the discrete number listing process is to test consecutive values that fit the criteria—it would be too easy to leave out the one exception that proves insufficiency. Never skip numbers that fit the constraint.", "topic": "number_properties"}
{"id": "book 2_p205_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 205, "page_end": 205, "topic_guess": "number_theory", "text": "GMAT number properties note: This is especially important if you are listing discrete numbers to equal an expression, not just a variable. By the way, remember to work from the facts to the question, not the other way around!", "topic": "number_properties"}
{"id": "book 2_p205_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 205, "page_end": 205, "topic_guess": "number_theory", "text": "GMAT number properties note: Don’t assume that the question should be answered Yes and only test values that make it so. Try-It #4-8 Is x a multiple of 12?", "topic": "number_properties"}
{"id": "book 2_p205_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 205, "page_end": 205, "topic_guess": "number_theory", "text": "GMAT number properties note: Any set that is “integer-like,” with well-defined, separated values is odd.(1) x is a multiple of 3.(2).", "topic": "number_properties"}
{"id": "book 2_p206_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 206, "page_end": 206, "topic_guess": "geometry", "text": "GMAT geometry note: Watch divisibility, factors, remainders, parity, and prime structure. Notice that you’re picking values for , not x.", "topic": "number_properties"}
{"id": "book 2_p206_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 206, "page_end": 206, "topic_guess": "geometry", "text": "GMAT geometry note: It would be far too much work to test different values for x to determine which make odd, and you could potentially miss some values that fit the statement. Do not plug in numbers for x here!", "topic": "number_properties"}
{"id": "book 2_p206_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 206, "page_end": 206, "topic_guess": "geometry", "text": "GMAT geometry note: Instead, list consecutive odd values for , a quick and easy process. Then, solve for x in each case.", "topic": "number_properties"}
{"id": "book 2_p206_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 206, "page_end": 206, "topic_guess": "geometry", "text": "GMAT geometry note: For this problem, your work on paper may look something like this: (1) AND (2) SUFFICIENT:. Combine these statements by selecting only the values for x that are in both lists.", "topic": "number_properties"}
{"id": "book 2_p206_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 206, "page_end": 206, "topic_guess": "geometry", "text": "GMAT geometry note: On your paper, circle the following values: x = 12 and x = 84. This seems to be SUFFICIENT—the values for x that fit both statements are multiples of 12.", "topic": "number_properties"}
{"id": "book 2_p206_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 206, "page_end": 206, "topic_guess": "geometry", "text": "GMAT geometry note: At this point, if you wanted to check another value, you could, or you could go with the trend, which is likely going to be correct a er testing this many cases. INSUFFICIENT: = odds = 1, 3, 5, 7, 9, etc.", "topic": "number_properties"}
{"id": "book 2_p206_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 206, "page_end": 206, "topic_guess": "geometry", "text": "GMAT geometry note: X − 3 = 1, 9, 25, 49, 81, etc. INSUFFICIENT: x = multiples of 3 = 3, 6, 9, 12, 15, etc.", "topic": "number_properties"}
{"id": "book 2_p207_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 207, "page_end": 207, "topic_guess": "algebra", "text": "GMAT algebra note: Watch divisibility, factors, remainders, parity, and prime structure. Thus, x is an odd perfect square plus 3.", "topic": "number_properties"}
{"id": "book 2_p207_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 207, "page_end": 207, "topic_guess": "algebra", "text": "GMAT algebra note: One might conclude that x is even, which is a true but incomplete description! Listing numbers is an easy way to see that these numbers are all multiples of 4.", "topic": "number_properties"}
{"id": "book 2_p207_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 207, "page_end": 207, "topic_guess": "algebra", "text": "GMAT algebra note: Yes, it’s worth knowing how to do this algebra. The point is that a discrete number testing process is quick and simple, so it’s also worth knowing how to do.", "topic": "number_properties"}
{"id": "book 2_p207_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 207, "page_end": 207, "topic_guess": "algebra", "text": "GMAT algebra note: Case Testing with Concepts You don’t need to use actual numbers to do case testing. In fact, it’s sometimes more efficient to case test using concepts, rather than numbers.", "topic": "number_properties"}
{"id": "book 2_p207_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 207, "page_end": 207, "topic_guess": "algebra", "text": "GMAT algebra note: (1) INSUFFICIENT: where k is an integer. (2) INSUFFICIENT: x is a multiple of 3, so x must be divisible by 3.", "topic": "number_properties"}
{"id": "book 2_p207_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 207, "page_end": 207, "topic_guess": "algebra", "text": "GMAT algebra note: (1) AND (2) SUFFICIENT: x is divisible by 3 and by 4, so x is divisible by 12.", "topic": "number_properties"}
{"id": "book 2_p208_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 208, "page_end": 208, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Try-It #4-9 If a, b, and c are integers, is abc > 0?", "topic": "number_properties"}
{"id": "book 2_p208_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 208, "page_end": 208, "topic_guess": "number_theory", "text": "GMAT number properties note: You could test cases in the traditional way: find specific values for a, b, and/or c that fit one statement or the other, then multiply those three values together and check whether their product was greater than zero. However, doing that much actual multiplication would be overkill.", "topic": "number_properties"}
{"id": "book 2_p208_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 208, "page_end": 208, "topic_guess": "number_theory", "text": "GMAT number properties note: Instead of case testing by checking specific numbers, try case testing with concepts. Instead of choosing appropriate values for a, b, and c, think about whether they could be positive or negative.", "topic": "number_properties"}
{"id": "book 2_p208_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 208, "page_end": 208, "topic_guess": "number_theory", "text": "GMAT number properties note: For instance, one scenario that fits the statement is that a, b, and c are all positive. Another scenario is that a and b are positive, but c is negative.", "topic": "number_properties"}
{"id": "book 2_p208_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 208, "page_end": 208, "topic_guess": "number_theory", "text": "GMAT number properties note: You can put this information in a chart, just as you would with ordinary case testing: a b c ab > 0 abc > 0? + + + (+)(+) = + (valid) Yes + + — (+)(+) = + (valid) No (2) INSUFFICIENT: ab > 0(1) bc > 0(2).", "topic": "number_properties"}
{"id": "book 2_p209_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 209, "page_end": 209, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. A b c bc > 0 abc > 0?a b c bc > 0 abc > 0?", "topic": "word_problems"}
{"id": "book 2_p209_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 209, "page_end": 209, "topic_guess": "word_problems", "text": "GMAT word problem note: + + + (+)(+) = + (valid) Yes — + + (+)(+) = + (valid) No Neither statement is sufficient on its own, so try them together. (1) AND (2) INSUFFICIENT: a b c (1): ab > 0 (2): bc > 0 abc > 0?", "topic": "word_problems"}
{"id": "book 2_p209_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 209, "page_end": 209, "topic_guess": "word_problems", "text": "GMAT word problem note: It was never necessary to work with specific numbers to solve this problem. This strategy doesn’t only work on positive/negative or odd/even problems.", "topic": "word_problems"}
{"id": "book 2_p209_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 209, "page_end": 209, "topic_guess": "word_problems", "text": "GMAT word problem note: Before reading the explanation, try to figure out how to solve the following problem using concepts (not specific numbers) to test cases. Try-It #4-10 At a certain company, the bonus pool is divided among a group of employees consisting entirely of engineers and managers so that.", "topic": "word_problems"}
{"id": "book 2_p210_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 210, "page_end": 210, "topic_guess": "statistics", "text": "GMAT statistics note: Percent problems usually relate part, whole, rate, and percent change. Each engineer receives a bonus of e dollars and each manager receives a bonus of m dollars.", "topic": "percent"}
{"id": "book 2_p210_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 210, "page_end": 210, "topic_guess": "statistics", "text": "GMAT statistics note: Is m at least 20% greater than e? At least 20% of the employees in the group are managers, but the amount earned by each employee is unknown.", "topic": "percent"}
{"id": "book 2_p210_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 210, "page_end": 210, "topic_guess": "statistics", "text": "GMAT statistics note: There could be a lot of managers who each earn a large bonus or there could be a lot of managers who each earn a small bonus. Think about scenarios on the extreme end of the range, ones that could result in a different answer to the question.", "topic": "percent"}
{"id": "book 2_p210_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 210, "page_end": 210, "topic_guess": "statistics", "text": "GMAT statistics note: # of Engineers # of Managers e m At least 20% managers? Few Many Small Large Lots of managers, and each manager earns a large bonus: (valid) Yes Few Many Large Small Lots of managers, but each manager earns almost no bonus: (valid) No (2) INSUFFICIENT:.", "topic": "percent"}
{"id": "book 2_p210_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 210, "page_end": 210, "topic_guess": "statistics", "text": "GMAT statistics note: This statement tells you about the total amount awarded to the managers. However, this total amount could be received by only a single manager or split across a huge number of managers.", "topic": "percent"}
{"id": "book 2_p210_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 210, "page_end": 210, "topic_guess": "statistics", "text": "GMAT statistics note: At least 20% of the employees in the group are managers.(1) The total amount awarded to managers is at least 20% greater than the total amount awarded to engineers.", "topic": "percent"}
{"id": "book 2_p211_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 211, "page_end": 211, "topic_guess": "general", "text": "GMAT quant note: Percent problems usually relate part, whole, rate, and percent change. # of Engineers # of Managers e m m total 20% + greater than e total?", "topic": "percent"}
{"id": "book 2_p211_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 211, "page_end": 211, "topic_guess": "general", "text": "GMAT quant note: Many One Very small Very large Only one manager, but that manager gets most of the bonus pool: (valid) Yes Few Many Equal Equal Everybody gets the same bonus, but there are significantly more managers, so most of the pool goes to managers: (valid) No (1) AND (2) INSUFFICIENT:. Even considering both statements, there are multiple possible situations.", "topic": "percent"}
{"id": "book 2_p211_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 211, "page_end": 211, "topic_guess": "general", "text": "GMAT quant note: For instance, the group could be composed mostly of managers. In this case, since e = m, the answer to the question is No.", "topic": "percent"}
{"id": "book 2_p211_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 211, "page_end": 211, "topic_guess": "general", "text": "GMAT quant note: Another possibility is that the group contains an equal number of managers and engineers. Finally, since m is much greater than e, the answer to the question is Yes.", "topic": "percent"}
{"id": "book 2_p211_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 211, "page_end": 211, "topic_guess": "general", "text": "GMAT quant note: # of Engineers # of Managers e m (1) 20% + managers? (2) m total 20% + greater than e total?", "topic": "percent"}
{"id": "book 2_p211_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 211, "page_end": 211, "topic_guess": "general", "text": "GMAT quant note: Few Many Equal Equal Yes: (valid) Yes: (valid) No Equal Equal Very small Very large Yes: (valid) Yes: (valid) Yes.", "topic": "percent"}
{"id": "book 2_p212_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 212, "page_end": 212, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Try one more problem, this time Testing Cases based on odd and even numbers.", "topic": "number_properties"}
{"id": "book 2_p212_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 212, "page_end": 212, "topic_guess": "number_theory", "text": "GMAT number properties note: Try-It #4-11 If a, b, and c are integers, is abc divisible by 4? Evaluate the possible odd/even combinations of a, b, and c without testing specific numbers.", "topic": "number_properties"}
{"id": "book 2_p212_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 212, "page_end": 212, "topic_guess": "number_theory", "text": "GMAT number properties note: (1) INSUFFICIENT: 2c must be even because c is an integer. This statement implies that a + b is even, which occurs when a and b have the same odd/even parity.", "topic": "number_properties"}
{"id": "book 2_p212_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 212, "page_end": 212, "topic_guess": "number_theory", "text": "GMAT number properties note: (There is no constraint on c.) (2) INSUFFICIENT: 2b must be even because b is an integer. Thus, this statement implies that a + c is odd, which occurs when a and c have opposite odd/even parity.", "topic": "number_properties"}
{"id": "book 2_p212_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 212, "page_end": 212, "topic_guess": "number_theory", "text": "GMAT number properties note: (There is no constraint on b.) a + b + 2c is even.(1) a + 2b + c is odd.(2).", "topic": "number_properties"}
{"id": "book 2_p213_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 213, "page_end": 213, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. From (1) you know that a and b have the same odd/even parity, while from (2) you know that a and c have opposite odd/even parity.", "topic": "number_properties"}
{"id": "book 2_p213_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 213, "page_end": 213, "topic_guess": "general", "text": "GMAT quant note: A b c (1) a + b + 2c = Even (2) a + 2b + c = Odd Is abc divisible by 4? Even Even Odd ✓ (valid) ✓ Yes Odd Odd Even ✓ (valid) ✓ (valid) Maybe Even with these constraints, you do not have a definitive answer to the question.", "topic": "number_properties"}
{"id": "book 2_p213_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 213, "page_end": 213, "topic_guess": "general", "text": "GMAT quant note: Notice that in evaluating statements (1) and (2) together, you would not need to test completely new cases. Know what case you’re considering.", "topic": "number_properties"}
{"id": "book 2_p214_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 214, "page_end": 214, "topic_guess": "strategy", "text": "GMAT strategy note: Translate the wording into variables, relationships, and equations before calculating. There will be fewer opportunities to apply these tactics than the strategies mentioned previously.", "topic": "word_problems"}
{"id": "book 2_p214_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 214, "page_end": 214, "topic_guess": "strategy", "text": "GMAT strategy note: However, when all else fails, these tactics may be your only friend. The tactics are listed according to their reliability: the earlier tactics nearly always work, while the later tactics provide only a modest improvement over random guessing.", "topic": "word_problems"}
{"id": "book 2_p214_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 214, "page_end": 214, "topic_guess": "strategy", "text": "GMAT strategy note: Spot Identical Statements Certainty:. Very High IF the two statements tell you exactly the same thing (a er rephrasing)...", "topic": "word_problems"}
{"id": "book 2_p214_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 214, "page_end": 214, "topic_guess": "strategy", "text": "GMAT strategy note: Depending on (1) 3y – 6 = 2x.", "topic": "word_problems"}
{"id": "book 2_p215_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 215, "page_end": 215, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. The question, either each statement will be sufficient or each will not— because they are identical, there cannot be any benefit from looking at the statements together.", "topic": "algebra"}
{"id": "book 2_p215_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 215, "page_end": 215, "topic_guess": "algebra", "text": "GMAT algebra note: Spot Clear Sufficiency Certainty:. Very High IF the two statements are clearly sufficient together...", "topic": "algebra"}
{"id": "book 2_p215_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 215, "page_end": 215, "topic_guess": "algebra", "text": "GMAT algebra note: Try-It #4-12 If and mn ≠ 0, what is the value of Z? Quite o en, some algebraic work will m = 5(2).", "topic": "algebra"}
{"id": "book 2_p216_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 216, "page_end": 216, "topic_guess": "general", "text": "GMAT quant note: Reveal that one or both of the statements will be sufficient on their own. Spot One Statement Inside the Other Certainty:.", "topic": "general"}
{"id": "book 2_p216_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 216, "page_end": 216, "topic_guess": "general", "text": "GMAT quant note: Very High IF one statement is “contained within” (i.e., is a subset of) the other... However, the reverse is not true.", "topic": "general"}
{"id": "book 2_p216_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 216, "page_end": 216, "topic_guess": "general", "text": "GMAT quant note: If (2) is insufficient, (1) could still possibly be sufficient on its own. Either way, it is impossible for both of the statements to be required together to answer the question.", "topic": "general"}
{"id": "book 2_p216_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 216, "page_end": 216, "topic_guess": "general", "text": "GMAT quant note: So you can x > 50!(1) x > 10!(2).", "topic": "general"}
{"id": "book 2_p217_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 217, "page_end": 217, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. This situation can occur with inequalities.", "topic": "number_properties"}
{"id": "book 2_p217_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 217, "page_end": 217, "topic_guess": "number_theory", "text": "GMAT number properties note: Be careful with this tactic, though. It can be easy to think—incorrectly— that one statement is a subset of the other.", "topic": "number_properties"}
{"id": "book 2_p217_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 217, "page_end": 217, "topic_guess": "number_theory", "text": "GMAT number properties note: Almost every prime number is odd, but not all. Even if just one value escapes, you cannot use this tactic.", "topic": "number_properties"}
{"id": "book 2_p217_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 217, "page_end": 217, "topic_guess": "number_theory", "text": "GMAT number properties note: Spot One Statement Adding Nothing Certainty:. High IF one statement adds no information to the other...", "topic": "number_properties"}
{"id": "book 2_p217_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 217, "page_end": 217, "topic_guess": "number_theory", "text": "GMAT number properties note: (Some question involving x, y, and z) x is an odd number.(1) x is a prime number.(2).", "topic": "number_properties"}
{"id": "book 2_p218_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 218, "page_end": 218, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. This tactic may seem identical to the previous one, but it is not.", "topic": "algebra"}
{"id": "book 2_p218_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 218, "page_end": 218, "topic_guess": "algebra", "text": "GMAT algebra note: Therefore, neither statement is a subset of the other. If y = 4, then you’d get 4x + (−4)x = z.", "topic": "algebra"}
{"id": "book 2_p218_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 218, "page_end": 218, "topic_guess": "algebra", "text": "GMAT algebra note: Those equations are the same! The sign of y doesn’t matter because y and −y are symmetric.", "topic": "algebra"}
{"id": "book 2_p218_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 218, "page_end": 218, "topic_guess": "algebra", "text": "GMAT algebra note: Moderate IF it is very obvious that the combined statements would be sufficient, but you can’t eliminate the possibility that one statement alone is sufficient...", "topic": "algebra"}
{"id": "book 2_p219_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 219, "page_end": 219, "topic_guess": "general", "text": "GMAT quant note: Focus on equations, expressions, variables, and step-by-step manipulation. It does this by giving you two statements that are obviously sufficient when you put them together.", "topic": "algebra"}
{"id": "book 2_p219_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 219, "page_end": 219, "topic_guess": "general", "text": "GMAT quant note: The test writers hope that you’ll jump to putting the two statements together, without testing each statement individually first. Try-It #4-13 If and ab ≠ 0, what is the value of K?", "topic": "algebra"}
{"id": "book 2_p219_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 219, "page_end": 219, "topic_guess": "general", "text": "GMAT quant note: You can prove that the two statements combined are sufficient without actually doing any math. Then, you could plug those values into the question to solve for K.", "topic": "algebra"}
{"id": "book 2_p219_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 219, "page_end": 219, "topic_guess": "general", "text": "GMAT quant note: A er all, you didn’t do anything to the question stem. Your next thought should be that one of the statements alone may be sufficient.", "topic": "algebra"}
{"id": "book 2_p219_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 219, "page_end": 219, "topic_guess": "general", "text": "GMAT quant note: In some problems, the individual statements may a = 3(1) b − 3(5 − a) = 0(2).", "topic": "algebra"}
{"id": "book 2_p220_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 220, "page_end": 220, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. In these cases, the more complicated statement may be enough.", "topic": "algebra"}
{"id": "book 2_p221_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 221, "page_end": 221, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. This is not a tactic to use at lower levels of the test.", "topic": "algebra"}
{"id": "book 2_p221_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 221, "page_end": 221, "topic_guess": "algebra", "text": "GMAT algebra note: This is one of many reasons why the material in this book is only appropriate if you have achieved a certain level of proficiency on the math side of the GMAT. Use Basic Algebraic Reasoning Certainty:.", "topic": "algebra"}
{"id": "book 2_p221_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 221, "page_end": 221, "topic_guess": "algebra", "text": "GMAT algebra note: Rely on basic algebraic reasoning. Quite o en, basic intuition about algebra can lead you to the correct answer.", "topic": "algebra"}
{"id": "book 2_p221_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 221, "page_end": 221, "topic_guess": "algebra", "text": "GMAT algebra note: Try-It #4-14 A sales manager at an industrial company has an opportunity to switch to a new, higher-paying job in another state. If his current annual salary is $50,000 and his current state tax rate is 5%, how much income after state tax would he make at the new job?", "topic": "algebra"}
{"id": "book 2_p221_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 221, "page_end": 221, "topic_guess": "algebra", "text": "GMAT algebra note: His new salary will be 10% higher than his old salary.(1) His annual state taxes will total $2,200 in the new state at his new job.", "topic": "algebra"}
{"id": "book 2_p222_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 222, "page_end": 222, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. You may reason that in order to answer this question, you need to know how much his new salary will be and how much his taxes will increase by.", "topic": "algebra"}
{"id": "book 2_p222_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 222, "page_end": 222, "topic_guess": "algebra", "text": "GMAT algebra note: Of course, you need to be very careful about using this tactic. Many problems on the GMAT are designed to hoodwink your algebraic reasoning —usually to make you think that you need to know every value precisely to answer the question.", "topic": "algebra"}
{"id": "book 2_p222_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 222, "page_end": 222, "topic_guess": "algebra", "text": "GMAT algebra note: The problem explicitly states that the new job is higher-paying, so you only need to check whether the change in state tax might lead to a lower a er-tax compensation. You do not need to find out how much higher his pre-tax salary would be to answer this question.", "topic": "algebra"}
{"id": "book 2_p222_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 222, "page_end": 222, "topic_guess": "algebra", "text": "GMAT algebra note: Because the GMAT frequently uses traps involving basic algebraic reasoning, you should only resort to using it if you are truly stuck. Spot Cross-Multiplied Inequalities Certaint.", "topic": "algebra"}
{"id": "book 2_p223_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 223, "page_end": 223, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. This might be the hidden trick.", "topic": "algebra"}
{"id": "book 2_p223_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 223, "page_end": 223, "topic_guess": "algebra", "text": "GMAT algebra note: If it is negative, you would need to flip the sign: . On some GMAT problems, you can assume the sign of the variables because the variables represent countable quantities of physical x > 0(2).", "topic": "algebra"}
{"id": "book 2_p224_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 224, "page_end": 224, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. In those cases, you can assume that the variables are positive, and you can cross-multiply inequalities involving those variables at will.", "topic": "algebra"}
{"id": "book 2_p224_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 224, "page_end": 224, "topic_guess": "algebra", "text": "GMAT algebra note: Judge by Appearance Certainty:. Low IF a statement is tricky and you don’t know what to do with it...", "topic": "algebra"}
{"id": "book 2_p224_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 224, "page_end": 224, "topic_guess": "algebra", "text": "GMAT algebra note: THEN take a guess by “judging by appearance.” Sometimes a statement will leave you completely bewildered. In that case, the best (and only!) tactic is to guess whether it will be sufficient judging by how it looks:.", "topic": "algebra"}
{"id": "book 2_p224_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 224, "page_end": 224, "topic_guess": "algebra", "text": "GMAT algebra note: The general rule is this: if the information in a statement has a structure and complexity similar to the question, and has the right ingredients (variables, coefficients, etc.), it’s more likely to be sufficient than otherwise. This won’t crack every case by any means, but you’d be surprised at how much mileage you can get from this tactic.", "topic": "algebra"}
{"id": "book 2_p224_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 224, "page_end": 224, "topic_guess": "algebra", "text": "GMAT algebra note: Try-It #4-16 Does it look like it might be sufficient, even if you can’t see how? Does it use the variables you are looking for?", "topic": "algebra"}
{"id": "book 2_p224_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 224, "page_end": 224, "topic_guess": "algebra", "text": "GMAT algebra note: Could it likely be manipulated into a form similar to that of the question?", "topic": "algebra"}
{"id": "book 2_p225_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 225, "page_end": 225, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. However, what if you find yourself at a temporary loss on the test?", "topic": "algebra"}
{"id": "book 2_p225_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 225, "page_end": 225, "topic_guess": "algebra", "text": "GMAT algebra note: Your exponent engine may shut down for a problem or two. It’s a very complicated expression, but it seems to have the right ingredients.", "topic": "algebra"}
{"id": "book 2_p225_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 225, "page_end": 225, "topic_guess": "algebra", "text": "GMAT algebra note: It contains a and b and uses a as an exponent.", "topic": "algebra"}
{"id": "book 2_p226_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 226, "page_end": 226, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. I’ll bet if I manipulate this equation right, it will answer the question.", "topic": "algebra"}
{"id": "book 2_p226_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 226, "page_end": 226, "topic_guess": "algebra", "text": "GMAT algebra note: The GMAT never needs to know that you guessed.", "topic": "algebra"}
{"id": "book 2_p227_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 227, "page_end": 227, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Summary That was a lot of information!", "topic": "algebra"}
{"id": "book 2_p227_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 227, "page_end": 227, "topic_guess": "algebra", "text": "GMAT algebra note: If you can’t tell for certain whether the answer can be calculated in theory, keep going on the calculations all the way. Represent Word Problems with algebraic equations to avoid embedded tricks that can be difficult to find otherwise.", "topic": "algebra"}
{"id": "book 2_p227_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 227, "page_end": 227, "topic_guess": "algebra", "text": "GMAT algebra note: Translate complicated questions and statements into simpler language. Bring explicit and implicit constraints to the surface.", "topic": "algebra"}
{"id": "book 2_p227_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 227, "page_end": 227, "topic_guess": "algebra", "text": "GMAT algebra note: These constraints will o en be necessary to determine the correct answer. Whenever a problem involves inequality symbols, be careful—there are many ways in which inequalities can be used to trick you.", "topic": "algebra"}
{"id": "book 2_p227_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 227, "page_end": 227, "topic_guess": "algebra", "text": "GMAT algebra note: There may be many possible scenarios in which a particular statement is true. Methodically write down these cases and check to see whether they result in different answers to the question.", "topic": "algebra"}
{"id": "book 2_p228_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 228, "page_end": 228, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Spot Clear Sufficiency (High):.", "topic": "algebra"}
{"id": "book 2_p228_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 228, "page_end": 228, "topic_guess": "algebra", "text": "GMAT algebra note: Spot One Statement Inside the Other (High):. Spot One Statement Adding Nothing (High):.", "topic": "algebra"}
{"id": "book 2_p228_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 228, "page_end": 228, "topic_guess": "algebra", "text": "GMAT algebra note: Use Basic Algebraic Reasoning (Moderate):. Apply basic knowledge of algebra, such as considering the number of unknowns relative to the number of known equations, to guide your thought process.", "topic": "algebra"}
{"id": "book 2_p228_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 228, "page_end": 228, "topic_guess": "algebra", "text": "GMAT algebra note: Spot Cross-Multiplied Inequalities (Low):. If you’re completely unsure what to do, then make your best guess as to whether it appears to be sufficient.", "topic": "algebra"}
{"id": "book 2_p229_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 229, "page_end": 229, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Let’s call the first type of error a Type 1 error: you thought that a statement was sufficient, but it wasn’t.", "topic": "percent"}
{"id": "book 2_p229_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 229, "page_end": 229, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: In other words, you thought that it gave you an exact answer to the question, but it was actually possible to get at least two different answers to the question. Here’s why that might happen:.", "topic": "percent"}
{"id": "book 2_p229_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 229, "page_end": 229, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: The second type of DS error is a Type 2 error. If you made this type of error, you thought that a statement wasn’t sufficient, but it actually was.", "topic": "percent"}
{"id": "book 2_p229_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 229, "page_end": 229, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: It You thought that a statement or both statements together were sufficient, but they were actually insufficient. You thought that a statement or both statements together were insufficient, but they were actually sufficient.", "topic": "percent"}
{"id": "book 2_p229_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 229, "page_end": 229, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: If you tested cases, you might have missed a critical case. For instance, maybe the question has a different answer when you plug in a fraction or a negative value.", "topic": "percent"}
{"id": "book 2_p229_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 229, "page_end": 229, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: You might have used information from the other statement without meaning to. This can make it seem as though you have enough information to answer the question, when you actually don’t.", "topic": "percent"}
{"id": "book 2_p229_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 229, "page_end": 229, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Keep the two statements completely separate on your paper and test them individually before you put them together.", "topic": "percent"}
{"id": "book 2_p230_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 230, "page_end": 230, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Looked like you didn’t have enough information to answer the question, but you actually had more information than you thought:.", "topic": "word_problems"}
{"id": "book 2_p230_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 230, "page_end": 230, "topic_guess": "word_problems", "text": "GMAT word problem note: The common thread in many of these errors is that they stem from assumptions. The easiest way to miss a DS problem is to assume that a statement is or isn’t sufficient, rather than proving it.", "topic": "word_problems"}
{"id": "book 2_p230_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 230, "page_end": 230, "topic_guess": "word_problems", "text": "GMAT word problem note: You won’t always have time on test day to meticulously prove whether a statement is sufficient. But neither should you decide whether a statement is sufficient just by glancing at it.", "topic": "word_problems"}
{"id": "book 2_p230_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 230, "page_end": 230, "topic_guess": "word_problems", "text": "GMAT word problem note: At the very least, briefly think about how the statement bears on the specific question you’re being asked. And if you have time, prove your suspicions about each statement before moving on.", "topic": "word_problems"}
{"id": "book 2_p230_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 230, "page_end": 230, "topic_guess": "word_problems", "text": "GMAT word problem note: You may have assumed that you needed a specific value to answer a Yes/No question. The question might have asked something like “is x even?” If you treat this question as though you need to know the exact value of x, you’ll make Type 2 errors.", "topic": "word_problems"}
{"id": "book 2_p230_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 230, "page_end": 230, "topic_guess": "word_problems", "text": "GMAT word problem note: If you assume that you need both statements in order to answer the question, it’s easy to miss the fact that one statement is actually sufficient on its own as well. Not Testing Cases o en leads to this type of error.", "topic": "word_problems"}
{"id": "book 2_p230_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 230, "page_end": 230, "topic_guess": "word_problems", "text": "GMAT word problem note: Many statements look insufficient, because they don’t appear to include a lot of information.", "topic": "word_problems"}
{"id": "book 2_p231_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 231, "page_end": 231, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Problem Set For problems 1−3, list five values that satisfy each of the following constraints.", "topic": "number_properties"}
{"id": "book 2_p231_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 231, "page_end": 231, "topic_guess": "number_theory", "text": "GMAT number properties note: X2 > 0 , where M and N are positive integers. For problems 4–6, solve the problem by Testing Cases.", "topic": "number_properties"}
{"id": "book 2_p231_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 231, "page_end": 231, "topic_guess": "number_theory", "text": "GMAT number properties note: Note which method is appropriate for each problem: testing the standard number set, listing discrete numbers, or case testing with concepts rather than numbers.", "topic": "number_properties"}
{"id": "book 2_p232_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 232, "page_end": 232, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. At a certain florist shop, roses can be purchased either individually or as a bouquet of 12 at a discount of p percent.", "topic": "number_properties"}
{"id": "book 2_p232_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 232, "page_end": 232, "topic_guess": "number_theory", "text": "GMAT number properties note: What is the greatest number of roses that can be purchased with $45? The greatest number of roses that can be purchased with $30 is p = 20(2) 6.", "topic": "number_properties"}
{"id": "book 2_p232_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 232, "page_end": 232, "topic_guess": "number_theory", "text": "GMAT number properties note: For problems 7–9, Test cases to show that one or both statements are insufficient. If a, b, and c are positive integers, is abc an even integer?", "topic": "number_properties"}
{"id": "book 2_p232_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 232, "page_end": 232, "topic_guess": "number_theory", "text": "GMAT number properties note: A + b is even.(1) b + c is odd.(2) 8. Twenty-eight students were each assigned to one of five classes:.", "topic": "number_properties"}
{"id": "book 2_p232_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 232, "page_end": 232, "topic_guess": "number_theory", "text": "GMAT number properties note: Anthropology, Biology, Geology, Musicology, and Sociology. Was at least one of the classes assigned fewer than 5 students?", "topic": "number_properties"}
{"id": "book 2_p233_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 233, "page_end": 233, "topic_guess": "strategy", "text": "GMAT strategy note: Watch divisibility, factors, remainders, parity, and prime structure. At least 6 students were assigned to each of Biology, Geology, and Sociology.", "topic": "number_properties"}
{"id": "book 2_p233_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 233, "page_end": 233, "topic_guess": "strategy", "text": "GMAT strategy note: Geology and Sociology were not assigned the same number of students. Set A consists of the four integers x, x2, x3, and x4, and set B consists of the four integers x, 2x, 3x, and 4x.", "topic": "number_properties"}
{"id": "book 2_p233_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 233, "page_end": 233, "topic_guess": "strategy", "text": "GMAT strategy note: What is the probability that a randomly selected integer from 1 to 100, inclusive, is a member of neither set A nor set B? For problems 10–15, solve the problem.", "topic": "number_properties"}
{"id": "book 2_p233_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 233, "page_end": 233, "topic_guess": "strategy", "text": "GMAT strategy note: Also, which guessing tactics would not work, and why? The following advanced strategies and guessing tactics were discussed in this chapter:.", "topic": "number_properties"}
{"id": "book 2_p233_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 233, "page_end": 233, "topic_guess": "strategy", "text": "GMAT strategy note: Advanced Strategies Guessing Tactics x > 5(1) x < 10(2).", "topic": "number_properties"}
{"id": "book 2_p234_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 234, "page_end": 234, "topic_guess": "algebra", "text": "GMAT algebra note: Percent problems usually relate part, whole, rate, and percent change. The average (arithmetic mean) of the original six prices for six coats at a clothing store was $85.", "topic": "percent"}
{"id": "book 2_p234_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 234, "page_end": 234, "topic_guess": "algebra", "text": "GMAT algebra note: A er two of the six coats were each discounted by 20%, the average price of the six coats was $76. Was the coat with the lowest original price one of the two coats that were discounted?", "topic": "percent"}
{"id": "book 2_p234_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 234, "page_end": 234, "topic_guess": "algebra", "text": "GMAT algebra note: One of the discounted coats was the one with the highest original price. Before the discount, none of the coats had a price greater than 11.", "topic": "percent"}
{"id": "book 2_p234_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 234, "page_end": 234, "topic_guess": "algebra", "text": "GMAT algebra note: Amanda and Todd purchase candy, popcorn, and pretzels at the stadium. If a package of candy costs half as much as a bag of popcorn, how much more money did Amanda and Todd spend on the candy than on the popcorn and pretzels combined?", "topic": "percent"}
{"id": "book 2_p235_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 235, "page_end": 235, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The cost of a bag of popcorn is equal to the cost of a pretzel.(1) Amanda and Todd purchased 24 packages of candy, 6 pretzels, and 6 bags of popcorn.", "topic": "number_properties"}
{"id": "book 2_p235_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 235, "page_end": 235, "topic_guess": "number_theory", "text": "GMAT number properties note: What is the value of |x + 4|? X2 + 8x + 12 = 0(1) x2 + 6x = 0(2) 13.", "topic": "number_properties"}
{"id": "book 2_p235_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 235, "page_end": 235, "topic_guess": "number_theory", "text": "GMAT number properties note: If m and n are positive integers, is n a multiple of 24? N is a multiple of (m + 4).(2) 15.", "topic": "number_properties"}
{"id": "book 2_p235_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 235, "page_end": 235, "topic_guess": "number_theory", "text": "GMAT number properties note: What is the value of x(1 – y)(1 + y)?", "topic": "number_properties"}
{"id": "book 2_p236_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 236, "page_end": 236, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "book 2_p237_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 237, "page_end": 237, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. X = 0.5, 2, 3, 6.7, 0, −1, etc.", "topic": "number_properties"}
{"id": "book 2_p237_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 237, "page_end": 237, "topic_guess": "number_theory", "text": "GMAT number properties note: (Note that the list should include integers and non-integers, positive and negative values—although the items you choose do not need to match these exact values. Anything other than 0 will work!) 3.", "topic": "number_properties"}
{"id": "book 2_p237_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 237, "page_end": 237, "topic_guess": "number_theory", "text": "GMAT number properties note: M = 10, 17, 24, 31, 38, etc., and N = 1, 2, 3, 4, 5, etc.:. M must have a remainder of 3 when divided by 7:.", "topic": "number_properties"}
{"id": "book 2_p237_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 237, "page_end": 237, "topic_guess": "number_theory", "text": "GMAT number properties note: Note that successive values of M differ by 7 and the corresponding values of N are consecutive integers. One way to generate this list is to pick consecutive numbers for N and then calculate values for M:.", "topic": "number_properties"}
{"id": "book 2_p237_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 237, "page_end": 237, "topic_guess": "number_theory", "text": "GMAT number properties note: M = 7 + 3, 14 + 3, 21 + 3, 28 + 3, 35 + 3, etc. M = 10, 17, 24, 31, 38, etc.", "topic": "number_properties"}
{"id": "book 2_p238_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 238, "page_end": 238, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "book 2_p239_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 239, "page_end": 239, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. A Yes (1) AND (2) SUFFICIENT:.", "topic": "statistics"}
{"id": "book 2_p239_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 239, "page_end": 239, "topic_guess": "statistics", "text": "GMAT statistics note: Combining the two constraints, 0 < a < 1. Within the standard number set, only is within this range.", "topic": "statistics"}
{"id": "book 2_p239_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 239, "page_end": 239, "topic_guess": "statistics", "text": "GMAT statistics note: In addition, test numbers close to the boundaries of the range. A Yes Yes Yes It seems that at the extreme edges, the values are greater than 2, so statements (1) and (2) appear to be SUFFICIENT.", "topic": "statistics"}
{"id": "book 2_p239_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 239, "page_end": 239, "topic_guess": "statistics", "text": "GMAT statistics note: The question asks you to maximize the number of roses that can be purchased for $45. To do so, first purchase as many 12-rose bouquets as.", "topic": "statistics"}
{"id": "book 2_p240_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 240, "page_end": 240, "topic_guess": "general", "text": "GMAT quant note: Percent problems usually relate part, whole, rate, and percent change. Possible, then purchase as many individual roses as possible with the remaining money.", "topic": "percent"}
{"id": "book 2_p240_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 240, "page_end": 240, "topic_guess": "general", "text": "GMAT quant note: The greatest number of roses that can be purchased with $30 is 24. In other words, two bouquets can be purchased for $30, and any le over money is not enough to purchase one individual rose.", "topic": "percent"}
{"id": "book 2_p240_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 240, "page_end": 240, "topic_guess": "general", "text": "GMAT quant note: Test cases to determine how many roses could be purchased with $45. If a bouquet of 12 roses costs exactly $15, then exactly 24 roses can be purchased with $30, and exactly 36 roses can be purchased with Case 2:.", "topic": "percent"}
{"id": "book 2_p240_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 240, "page_end": 240, "topic_guess": "general", "text": "GMAT quant note: Try to find a case in which only 24 roses can be purchased with $30, but more than 36 roses can be purchased with $45. This will happen if two bouquets cost less than $30 and the le over money isn’t enough to purchase any more individual roses.", "topic": "percent"}
{"id": "book 2_p240_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 240, "page_end": 240, "topic_guess": "general", "text": "GMAT quant note: But the le over money from $45 when three bouquets are purchased is just enough to purchase an additional rose. For instance, suppose that a bouquet costs $14 and an individual rose costs $2.50 before the discount.", "topic": "percent"}
{"id": "book 2_p240_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 240, "page_end": 240, "topic_guess": "general", "text": "GMAT quant note: However, 37 roses could be purc.", "topic": "percent"}
{"id": "book 2_p241_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 241, "page_end": 241, "topic_guess": "general", "text": "GMAT quant note: Percent problems usually relate part, whole, rate, and percent change. The price of an individual rose is unknown.", "topic": "percent"}
{"id": "book 2_p241_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 241, "page_end": 241, "topic_guess": "general", "text": "GMAT quant note: Therefore, any number of roses might be purchased for $45. First, use the case in which exactly 36 roses can be purchased for $45.", "topic": "percent"}
{"id": "book 2_p241_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 241, "page_end": 241, "topic_guess": "general", "text": "GMAT quant note: Then, look for a case in which more than 36 roses can be purchased for $45. If a bouquet of roses costs exactly $15, then exactly 36 roses can be purchased for $45.", "topic": "percent"}
{"id": "book 2_p241_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 241, "page_end": 241, "topic_guess": "general", "text": "GMAT quant note: (For reference, this implies each rose in a bouquet cost = $1.25, so the individual [not discounted] rose price is Case 2:. Determine whether more than 36 roses could be purchased for $45.", "topic": "percent"}
{"id": "book 2_p241_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 241, "page_end": 241, "topic_guess": "general", "text": "GMAT quant note: If roses cost only slightly less than they did in Case 1, it might still only be possible to purchase 24 roses for $30, but it might also be possible to purchase one extra individual rose with the le over money. Try a cost of $1.50 per rose.", "topic": "percent"}
{"id": "book 2_p241_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 241, "page_end": 241, "topic_guess": "general", "text": "GMAT quant note: If a rose costs $1.50 individually, then a bouquet costs 12($1.50)(0.80) = $14.40, Since two bouquets cost 2($14.40) = $28.80, there isn’t enough money le over to purchase another single rose. Now three bouquets cost 3($14.40) = $43.20, and one more rose can be purchased for $1.50, bringing the total purchase to $44.70.", "topic": "percent"}
{"id": "book 2_p241_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 241, "page_end": 241, "topic_guess": "general", "text": "GMAT quant note: In this case, 37 roses can be purchased with $45.", "topic": "percent"}
{"id": "book 2_p242_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 242, "page_end": 242, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Watch divisibility, factors, remainders, parity, and prime structure. Since either 36 or 37 roses could be purchased for $45, the two statements are insufficient together.", "topic": "number_properties"}
{"id": "book 2_p242_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 242, "page_end": 242, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Because there are no constraints on y, use the standard number set, displayed here in number line graphic form:. Notice that you could use Guessing Tactic 3: spot one statement inside the other.", "topic": "number_properties"}
{"id": "book 2_p242_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 242, "page_end": 242, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: In order for abc to be even, what must be true? If just one of the numbers is even, then the product will be even.", "topic": "number_properties"}
{"id": "book 2_p242_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 242, "page_end": 242, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: The only case that will produce an odd is when all three numbers are odd. Keep this in mind when testing the statements.", "topic": "number_properties"}
{"id": "book 2_p243_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 243, "page_end": 243, "topic_guess": "algebra", "text": "GMAT algebra note: Watch divisibility, factors, remainders, parity, and prime structure. If a + b is even, then the two variables are either both even or both odd: a b c abc even?", "topic": "number_properties"}
{"id": "book 2_p243_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 243, "page_end": 243, "topic_guess": "algebra", "text": "GMAT algebra note: Even Even Even Yes Odd Odd Odd No (2) SUFFICIENT: b + c is odd, so one of b and c is odd and the other is even. In other words, it is impossible for all three variables to be odd.", "topic": "number_properties"}
{"id": "book 2_p243_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 243, "page_end": 243, "topic_guess": "algebra", "text": "GMAT algebra note: Therefore, the product abc must be even. Set up a table and try to find contradictory scenarios.", "topic": "number_properties"}
{"id": "book 2_p243_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 243, "page_end": 243, "topic_guess": "algebra", "text": "GMAT algebra note: 5 6 6 5 6 No 4 6 6 6 6 Yes (2) INSUFFICIENT.", "topic": "number_properties"}
{"id": "book 2_p244_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 244, "page_end": 244, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. A B G M S < 5 in one class?A B G M S < 5 in one class?", "topic": "probability"}
{"id": "book 2_p244_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 244, "page_end": 244, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: 6 6 4 6 6 Yes 6 5 5 6 6 No (1) AND (2) SUFFICIENT:. 4 6 6 5 7 Yes Since Geology and Sociology each have to have a minimum of 6 students, and they can’t have the same number of students, one of the two classes has to have at least 7 students.", "topic": "probability"}
{"id": "book 2_p244_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 244, "page_end": 244, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Between Biology, Geology, and Sociology, then, the minimum number of students is 6 + 6 + 7 = 19, leaving 9 students to be split among the other two classes. In this case, it’s impossible to have 5 or more students in both of those two remaining classes.", "topic": "probability"}
{"id": "book 2_p244_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 244, "page_end": 244, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: The question asks for the probability that a randomly selected integer, from 1 to 100, will be in neither one of the two sets. To calculate this probability, you need to know the total number of integers that are in one or both of the two sets, as well as within the 1 to 100 range.", "topic": "probability"}
{"id": "book 2_p245_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 245, "page_end": 245, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Jot down the definitions of the two sets.", "topic": "number_properties"}
{"id": "book 2_p245_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 245, "page_end": 245, "topic_guess": "number_theory", "text": "GMAT number properties note: Integer x is contained in both sets, so, at most, the two sets will contain seven distinct integers between them. However, some of these integers might be less than 1 or greater than 100, and there might be more integers that overlap between the two sets, decreasing the number of possible values that overlap with the selected integer to less than 7.", "topic": "number_properties"}
{"id": "book 2_p245_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 245, "page_end": 245, "topic_guess": "number_theory", "text": "GMAT number properties note: A statement is sufficient if every possible value of x yields the same number of distinct values between 1 and 100, inclusive. Test a straightforward case first, then test a case that might yield a different value.", "topic": "number_properties"}
{"id": "book 2_p245_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 245, "page_end": 245, "topic_guess": "number_theory", "text": "GMAT number properties note: The smallest possible value for x that fits the statement is 6. If x = 6, then set A consists of four integers: 6, 62, 63, and 64.", "topic": "number_properties"}
{"id": "book 2_p245_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 245, "page_end": 245, "topic_guess": "number_theory", "text": "GMAT number properties note: Two of those integers, 6 and 62 = 36, are smaller than 100. As long as you know that 63 and 64 are greater than 100, there is no need to calculate their exact values.", "topic": "number_properties"}
{"id": "book 2_p245_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 245, "page_end": 245, "topic_guess": "number_theory", "text": "GMAT number properties note: Set B also consists of four integers: 6, 12, 18, and 24. Therefore, the sets contain a total of five distinct integers between 1 and 100, inclusive.", "topic": "number_properties"}
{"id": "book 2_p245_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 245, "page_end": 245, "topic_guess": "number_theory", "text": "GMAT number properties note: Try an extreme case, such as x = 1,000. In this case, all of the integers in both set A and set B are much greater than 100.", "topic": "number_properties"}
{"id": "book 2_p245_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 245, "page_end": 245, "topic_guess": "number_theory", "text": "GMAT number properties note: Therefore, the sets contain no integers between 1 and 100.", "topic": "number_properties"}
{"id": "book 2_p246_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 246, "page_end": 246, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Try another extreme case, such as x = –2.", "topic": "number_properties"}
{"id": "book 2_p246_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 246, "page_end": 246, "topic_guess": "number_theory", "text": "GMAT number properties note: In this case, the only positive integers in either set will be (–2)2 = 4 and (–2)4 = 16, which are both between 1 and 100. Therefore, there are exactly two distinct integers within the range.", "topic": "number_properties"}
{"id": "book 2_p246_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 246, "page_end": 246, "topic_guess": "number_theory", "text": "GMAT number properties note: Since the set could contain five integers or two integers within the given range, this statement is also insufficient. (1) AND (2) SUFFICIENT: x = 6 yields five distinct integers that are within the range 1 to 100.", "topic": "number_properties"}
{"id": "book 2_p246_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 246, "page_end": 246, "topic_guess": "number_theory", "text": "GMAT number properties note: Check to see whether this is also true for x = 7, 8, and For x = 7, 8, or 9, set A contains two integers that are within the range 1 to 100: x and x2. In all three cases, x3 and x4 are outside of the range.", "topic": "number_properties"}
{"id": "book 2_p246_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 246, "page_end": 246, "topic_guess": "number_theory", "text": "GMAT number properties note: Also, set B contains four integers that are within the range, but one of them, x itself, is also contained in set A. So in every case in which x is between 5 and 10, there are exactly five distinct integers in the range that are contained in one or both sets.", "topic": "number_properties"}
{"id": "book 2_p246_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 246, "page_end": 246, "topic_guess": "number_theory", "text": "GMAT number properties note: The average price of six coats was originally $85. So the sum of the six prices was originally 6($85) = $510.", "topic": "number_properties"}
{"id": "book 2_p246_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 246, "page_end": 246, "topic_guess": "number_theory", "text": "GMAT number properties note: Two of the prices were discounted by 20%, resulting in a $9 reduction in the average price, or a 6($9) = $54 reduction in the overall price. Since 20% (or 1/5 of the cost of these two coats came to $54, the original price of these two coat.", "topic": "number_properties"}
{"id": "book 2_p247_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 247, "page_end": 247, "topic_guess": "general", "text": "GMAT quant note: Percent problems usually relate part, whole, rate, and percent change. From the question stem, the two discounted coats had a total price of $270, and according to this statement, one of those two coats was the most expensive.", "topic": "percent"}
{"id": "book 2_p247_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 247, "page_end": 247, "topic_guess": "general", "text": "GMAT quant note: Test cases to see whether the other coat could have been the least expensive. Note that the four non- discounted coats had a total price of $510 – $270 = $240.", "topic": "percent"}
{"id": "book 2_p247_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 247, "page_end": 247, "topic_guess": "general", "text": "GMAT quant note: If the discounted coats originally cost $1 and $269 and the other four coats each cost , then the least expensive coat was one of the two that was discounted, and the answer is Yes. If the discounted coats originally cost $130 and $140 and the other four coats each cost , then the least expensive coat was not one of the two that was discounted, and the answer is No.", "topic": "percent"}
{"id": "book 2_p247_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 247, "page_end": 247, "topic_guess": "general", "text": "GMAT quant note: This statement limits the possible original costs of the two discounted coats. Since neither of the coats could have cost more than $180, the less expensive of the two discounted coats must have cost at least $270 – $180 = $90.", "topic": "percent"}
{"id": "book 2_p247_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 247, "page_end": 247, "topic_guess": "general", "text": "GMAT quant note: However, this coat could not have been the one with the lowest original price, as $90 is greater than the original average price. The smallest number in any set cannot be greater than the average of th.", "topic": "percent"}
{"id": "book 2_p248_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 248, "page_end": 248, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Assign variables to the unknowns in the problem.", "topic": "algebra"}
{"id": "book 2_p248_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 248, "page_end": 248, "topic_guess": "algebra", "text": "GMAT algebra note: C = packages of candy P = bags of popcorn R = number of pretzels Pc = price of a package of candy Pp = price of a bag of popcorn Pr = price of a pretzel Algebraically, the question asks:. What is Pc × C − (Pp × P + Pr × R)?", "topic": "algebra"}
{"id": "book 2_p248_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 248, "page_end": 248, "topic_guess": "algebra", "text": "GMAT algebra note: The question stem indicates that (1) INSUFFICIENT:. Substitute and rephrase the question:.", "topic": "algebra"}
{"id": "book 2_p248_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 248, "page_end": 248, "topic_guess": "algebra", "text": "GMAT algebra note: What is What is What is Pc × C − (Pp × P + Pr × R)?", "topic": "algebra"}
{"id": "book 2_p249_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 249, "page_end": 249, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Given the relative prices of the candy, popcorn, and pretzels, and the quantity of each purchased, the cost of the candy will always equal the combined cost of the popcorn and pretzels, even though it’s impossible to calculate the exact prices.", "topic": "algebra"}
{"id": "book 2_p249_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 249, "page_end": 249, "topic_guess": "algebra", "text": "GMAT algebra note: Statements (1) and (2) combined are SUFFICIENT to answer the question. What is What is 12Pp − 6Pp − 6Pr?", "topic": "algebra"}
{"id": "book 2_p249_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 249, "page_end": 249, "topic_guess": "algebra", "text": "GMAT algebra note: It’s tempting to rephrase the question as “What is x?” but that is dangerous when the question contains absolute value symbols. It’s possible that two different values of x would resolve to the same value of |x + 4|.", "topic": "algebra"}
{"id": "book 2_p249_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 249, "page_end": 249, "topic_guess": "algebra", "text": "GMAT algebra note: Leave the question as it is. (1) SUFFICIENT: x2 + 8x + 12 = 0.", "topic": "algebra"}
{"id": "book 2_p250_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 250, "page_end": 250, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Plug these answers into the question stem:.", "topic": "algebra"}
{"id": "book 2_p250_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 250, "page_end": 250, "topic_guess": "algebra", "text": "GMAT algebra note: (2) INSUFFICIENT: x2 + 6x = 0. Plug both values into the question stem:.", "topic": "algebra"}
{"id": "book 2_p250_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 250, "page_end": 250, "topic_guess": "algebra", "text": "GMAT algebra note: If abcd ≠ 0, then none of the variables equals 0. In order for the product abcd to be negative, you would need to have an odd number of negatives in the mix.", "topic": "algebra"}
{"id": "book 2_p251_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 251, "page_end": 251, "topic_guess": "general", "text": "GMAT quant note: Scenario Product abcdScenario Product abcd All 4 positive or all 4 negative + 3 positive, 1 negative − 2 positive, 2 negative + 1 positive, 3 negative − (1) INSUFFICIENT:. If b and d are both positive or both negative, then ad > bc.", "topic": "general"}
{"id": "book 2_p251_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 251, "page_end": 251, "topic_guess": "general", "text": "GMAT quant note: If one is positive and one is negative, then ad < bc. This statement indicates nothing about a and c, though.", "topic": "general"}
{"id": "book 2_p251_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 251, "page_end": 251, "topic_guess": "general", "text": "GMAT quant note: If a and c are both positive or both negative, then bc > ad. If one is positive and one is negative, then bc < ad.", "topic": "general"}
{"id": "book 2_p251_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 251, "page_end": 251, "topic_guess": "general", "text": "GMAT quant note: This statement indicates nothing about b and d, though.", "topic": "general"}
{"id": "book 2_p252_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 252, "page_end": 252, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Here’s where the trick comes in.", "topic": "number_properties"}
{"id": "book 2_p252_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 252, "page_end": 252, "topic_guess": "number_theory", "text": "GMAT number properties note: The two statements allow two possible scenarios:. If ad > bc, then b and d have the same sign but a and c have opposite signs.", "topic": "number_properties"}
{"id": "book 2_p252_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 252, "page_end": 252, "topic_guess": "number_theory", "text": "GMAT number properties note: In this case, three signs are the same and one is not, so there are an odd number of negatives in the mix, and the product of all four variables must be negative. If ad < bc, b and d have opposite signs, but a and c have the same signs.", "topic": "number_properties"}
{"id": "book 2_p252_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 252, "page_end": 252, "topic_guess": "number_theory", "text": "GMAT number properties note: In this case, three signs are the same and one is not. As before, the product of all four variables must be negative.", "topic": "number_properties"}
{"id": "book 2_p252_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 252, "page_end": 252, "topic_guess": "number_theory", "text": "GMAT number properties note: Either way, the two statements together are sufficient to answer the question. In other words, n is the product of four consecutive integers.", "topic": "number_properties"}
{"id": "book 2_p252_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 252, "page_end": 252, "topic_guess": "number_theory", "text": "GMAT number properties note: In any four consecutive integers, one number must be divisible by 3 and two numbers must be even. Furthermore, of the two even numbers, one must be divisible by 4.", "topic": "number_properties"}
{"id": "book 2_p252_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 252, "page_end": 252, "topic_guess": "number_theory", "text": "GMAT number properties note: As a result, the product of any four consecutive integers is divisible by (3)(2)(4) = 24.", "topic": "number_properties"}
{"id": "book 2_p253_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 253, "page_end": 253, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Alternatively, you could figure this out by testing a few cases: (2) INSUFFICIENT:.", "topic": "algebra"}
{"id": "book 2_p253_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 253, "page_end": 253, "topic_guess": "algebra", "text": "GMAT algebra note: If m = 1, then n could equal 5, which is not a multiple of 24. If m = 20, then n could equal 24, which is a multiple of 24.", "topic": "algebra"}
{"id": "book 2_p253_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 253, "page_end": 253, "topic_guess": "algebra", "text": "GMAT algebra note: M = 1: (8)(7)(6)(5), which is divisible by (8)(3) = 24 m = 2: (9)(8)(7)(6), which is divisible by (8)(3) = 24 m = 3: (10)(9)(8)(7), which is divisible by (8)(3) = 24 15. This algebra problem includes a special quadratic: (1 – y)(1 + y) = 1 – y2.", "topic": "algebra"}
{"id": "book 2_p253_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 253, "page_end": 253, "topic_guess": "algebra", "text": "GMAT algebra note: Rephrase the question as “What is the value of x(1 – y2)?” (1) INSUFFICIENT:. This statement appears to provide little information.", "topic": "algebra"}
{"id": "book 2_p253_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 253, "page_end": 253, "topic_guess": "algebra", "text": "GMAT algebra note: However, it significantly limits the possible values of x. If x = 1, the answer to the question is the value of 1 – y2, which is unknown.", "topic": "algebra"}
{"id": "book 2_p253_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 253, "page_end": 253, "topic_guess": "algebra", "text": "GMAT algebra note: If x = 0, the answer to the question is 0. If y2 = x, then the question simplifies to “What is the value of x(1 – x)?” Different values of x produce different answers, so this statement is insufficient.", "topic": "algebra"}
{"id": "book 2_p254_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 254, "page_end": 254, "topic_guess": "general", "text": "GMAT quant note: In this case, the answer is x(1 – y2) = 0(1 – 0) = 0. In this case, the answer is x(1 – y2) = 1(1 – 1) = 0.", "topic": "general"}
{"id": "book 2_p254_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 254, "page_end": 254, "topic_guess": "general", "text": "GMAT quant note: Since the answer to the question is always 0, the two statements together are sufficient.", "topic": "general"}
{"id": "book 2_p255_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 255, "page_end": 255, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. PART TWO Strategies for All Problem Types.", "topic": "word_problems"}
{"id": "book 2_p256_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 256, "page_end": 256, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation.", "topic": "algebra"}
{"id": "book 2_p257_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 257, "page_end": 257, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Pattern Recognition Problems Sequence Problems Units (Ones) Digit Problems Remainder Problems Other Pattern Problems.", "topic": "algebra"}
{"id": "book 2_p258_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 258, "page_end": 258, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. If you can grasp the rule, you can predict numbers that appear later in the series.", "topic": "algebra"}
{"id": "book 2_p258_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 258, "page_end": 258, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The series may be part of a defined sequence or it may arise from a general list of possibilities. Either way, if you can spot the pattern, you can eliminate a lot of unnecessary calculation.", "topic": "algebra"}
{"id": "book 2_p258_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 258, "page_end": 258, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: In fact, o en the only feasible way to get the answer in two minutes (e.g., finding the 100th number in some series) is to recognize the underlying pattern. Try-It #5-1 Each number in a sequence is 3 more than the previous number.", "topic": "algebra"}
{"id": "book 2_p258_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 258, "page_end": 258, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: If the first number is 4, what is the value of the 1,000th term in the sequence?", "topic": "algebra"}
{"id": "book 2_p259_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 259, "page_end": 259, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Obviously, finding the 1,000th number the long way (by computing every intervening number) is impossible in the time allotted on the GMAT.", "topic": "algebra"}
{"id": "book 2_p259_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 259, "page_end": 259, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: You can solve this problem in several ways, but one powerful way is to compute the first several terms, spot the underlying pattern, figure out the rule, and then apply it. The first five terms of the sequence are 4, 7, 10, 13, 16.", "topic": "algebra"}
{"id": "book 2_p259_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 259, "page_end": 259, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Notice that you repeatedly add 3 to get the next value. Repeated addition is just multiplication, so match these numbers to the multiples of 3.", "topic": "algebra"}
{"id": "book 2_p259_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 259, "page_end": 259, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The first five multiples of 3 are 3, 6, 9, 12, 15—all 1 less than the numbers in the sequence. Thus, the rule for generating the sequence is “take the corresponding multiple of 3, and add 1.” Therefore, the 1,000th term of the sequence is 1 more than the 1,000th multiple of 3, which is 3,000 + 1 = 3,001.", "topic": "algebra"}
{"id": "book 2_p259_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 259, "page_end": 259, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Once you spot the pattern, you can skip over vast amounts of unnecessary work. The two most basic patterns are these: integers.", "topic": "algebra"}
{"id": "book 2_p259_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 259, "page_end": 259, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: As simple as this pattern may seem, it is the basis for many other patterns. For instance, the sequence of the multiples of 7 (7, You can write this sequence as Sn = 7n, where n is the basic sequence of positive integers.", "topic": "algebra"}
{"id": "book 2_p259_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 259, "page_end": 259, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: A repeating cycle of numbers. For instance, the sequence 4, 2, 6, 4, cycles can be derived in various ways from the counting numbers (e.", "topic": "algebra"}
{"id": "book 2_p260_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 260, "page_end": 260, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. When you are examining a string of numbers for a pattern, follow these steps: to think of repeating cycles on their own terms, separately from the counting numbers and related patterns.", "topic": "algebra"}
{"id": "book 2_p260_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 260, "page_end": 260, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Compute the first five to eight terms and try to match them to a pattern that you already know. The most basic pattern is the counting numbers, but you should also have these related patterns up your sleeve:.", "topic": "algebra"}
{"id": "book 2_p260_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 260, "page_end": 260, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: For each of these sequences, notice exactly where the counting numbers come into play. For instance, in the squares series, the counting numbers are the bases, but in the powers series, the counting numbers are the exponents.", "topic": "algebra"}
{"id": "book 2_p260_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 260, "page_end": 260, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Multiples (e.g., 7 × 1 = 7, 7 × 2 = 14, 7 × 3 = 21, 7 × 4 = 28, 7 × 5 = 35 a. As soon as you generate a repeated term, see whether the sequence will repeat itself from that point onward.", "topic": "algebra"}
{"id": "book 2_p260_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 260, "page_end": 260, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Repeating cycles on the GMAT typically begin repeating every four terms (or fewer), so five to eight terms should be sufficient to identify the pattern. Of course, some cycles repeat every single term —that is, it’s the same number over and over again!", "topic": "algebra"}
{"id": "book 2_p260_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 260, "page_end": 260, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: If you are stuck, look for patterns within differences between terms or sums across terms.", "topic": "algebra"}
{"id": "book 2_p261_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 261, "page_end": 261, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Look at the difference between consecutive terms.", "topic": "algebra"}
{"id": "book 2_p261_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 261, "page_end": 261, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: For instance, this process can help you spot linear sequences (sequences of multiples plus a constant): a. Also look at the cumulative sum of all the terms up to that point.", "topic": "algebra"}
{"id": "book 2_p261_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 261, "page_end": 261, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: This is helpful if the terms get closer to zero or alternate in sign:. Notice that the cumulative sum for this sequence approaches 1.", "topic": "algebra"}
{"id": "book 2_p261_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 261, "page_end": 261, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Some sums involve matching pairs that sum to the same number (or even cancel each other out). Be on the lookout for such matching pairs.", "topic": "algebra"}
{"id": "book 2_p261_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 261, "page_end": 261, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: You can, of course, sum these numbers in order, but look to make natural intermediate sums (subtotals) with matching pairs. Working outward, 10 and 12 sum to 22 as.", "topic": "algebra"}
{"id": "book 2_p262_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 262, "page_end": 262, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. O en, the pattern will be completely hidden until you actually compute the first several values of the sequence or other initial results.", "topic": "algebra"}
{"id": "book 2_p262_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 262, "page_end": 262, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Several types of problems frequently involve underlying patterns. When you see these types of problems on the GMAT, be ready to analyze the pattern so you can find the rule: well.", "topic": "algebra"}
{"id": "book 2_p262_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 262, "page_end": 262, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: So do 8 and 14, 5 and 17, and 1 and 21. In all, there are five subtotals of 22, for a grand total of 110.", "topic": "algebra"}
{"id": "book 2_p262_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 262, "page_end": 262, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Look at characteristics of the numbers: positive/negative, odd/even, integer/non-integer, etc. Once you have extended the pattern for several terms, these characteristics will generally repeat or alternate in some predictable way.", "topic": "algebra"}
{"id": "book 2_p262_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 262, "page_end": 262, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Nearly all sequence problems involve a pattern in the elements (or terms) of the sequence. Sequences can be defined either directly (i.e., each value in the series is a function of its location in the order of the sequence) or recursively (i.e., each value is a function of the previous items in the sequence).", "topic": "algebra"}
{"id": "book 2_p262_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 262, "page_end": 262, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Units (Ones) Digit Problems:. Questions involving the last digit (sometimes called the units or ones digit) of an integer almost always involve some sort of repeating cycle pattern that can be.", "topic": "algebra"}
{"id": "book 2_p263_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 263, "page_end": 263, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Integer is a special case of a remainder: it’s the remainder a er division by 10.", "topic": "number_properties"}
{"id": "book 2_p263_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 263, "page_end": 263, "topic_guess": "number_theory", "text": "GMAT number properties note: Some pattern problems do not involve deciphering a string of numbers and discovering the rule. For instance, you may have to count a set of numbers that all fit some constraint.", "topic": "number_properties"}
{"id": "book 2_p263_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 263, "page_end": 263, "topic_guess": "number_theory", "text": "GMAT number properties note: The point is to discover a simple rule or group of rules that let you account for all the numbers—and therefore count them —without having to generate each one. Break the problem into sub-problems.", "topic": "number_properties"}
{"id": "book 2_p263_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 263, "page_end": 263, "topic_guess": "number_theory", "text": "GMAT number properties note: For instance, a sum may be split into several smaller sums. Or you might count a larger total, then subtract items that do not fit the constraint.", "topic": "number_properties"}
{"id": "book 2_p263_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 263, "page_end": 263, "topic_guess": "number_theory", "text": "GMAT number properties note: You even might multiply a larger total by the proportion of suitable items, if that fraction is easy to calculate. Recall counting and summing methods from Manhattan Prep’s All the Quant guide:.", "topic": "number_properties"}
{"id": "book 2_p263_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 263, "page_end": 263, "topic_guess": "number_theory", "text": "GMAT number properties note: When you have a series of successive decisions, you multiply the number of choices you have at each stage to find the number of total choices you have. For instance, if you can choose 1 appetizer out of 6 possible appetizers and 1 main course out of 7 possible main courses, then you could have 6 × 7 = 42 possible meals.", "topic": "number_properties"}
{"id": "book 2_p263_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 263, "page_end": 263, "topic_guess": "number_theory", "text": "GMAT number properties note: Number of Items in a Consecutive Set of Integers:. The number of integers in a consecutive set of integers equals.", "topic": "number_properties"}
{"id": "book 2_p264_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 264, "page_end": 264, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Average of the largest integer and the smallest integer.", "topic": "statistics"}
{"id": "book 2_p264_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 264, "page_end": 264, "topic_guess": "statistics", "text": "GMAT statistics note: This is also equal to the median, or “middle,” number in the set. As you go, always check that the extreme cases are still valid.", "topic": "statistics"}
{"id": "book 2_p264_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 264, "page_end": 264, "topic_guess": "statistics", "text": "GMAT statistics note: Two or three constraints can interact in surprising ways, eliminating some of the values that would seem to work otherwise.", "topic": "statistics"}
{"id": "book 2_p265_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 265, "page_end": 265, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Sequence Problems Any question that involves the definition of a sequence (usually involving subscripted variables, such as An and Sn) is very likely to involve patterns.", "topic": "algebra"}
{"id": "book 2_p265_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 265, "page_end": 265, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: These patterns can range from relatively straightforward linear patterns to much more complicated ones. When you are given a sequence definition, list a few terms of the sequence, starting with any particular terms you are given, and look for a pattern.", "topic": "algebra"}
{"id": "book 2_p265_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 265, "page_end": 265, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Do not be intimidated by a recursive definition for a sequence, in which each term is defined using earlier terms. (By contrast, a direct definition defines each term using the position or index of the term.) To illustrate the difference, here are two ways to define the series of positive odd integers {1, 3, 5, 7, 9, etc.}:.", "topic": "algebra"}
{"id": "book 2_p265_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 265, "page_end": 265, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Recursive Definition Direct Definition An = An − 1 + 2 where n > 1 and A1 = 1 Translation: “This term = the previous term + 2, and the first term is 1.” An = 2n − 1, where n ≥ 1 Translation: “This term = the index number × 2, minus 1. Thus, the first term is (2)(1) − 1 = 1.”.", "topic": "algebra"}
{"id": "book 2_p266_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 266, "page_end": 266, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Try-It #5-2 The sequence Xn is defined as follows:.", "topic": "algebra"}
{"id": "book 2_p266_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 266, "page_end": 266, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Xn = 2Xn − 1 − 1 whenever n is an integer greater than 1. If X1 = 3, what is the value of X20 − X19?", "topic": "algebra"}
{"id": "book 2_p266_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 266, "page_end": 266, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The pattern underlying this sequence is not obvious, so begin computing a few of the terms in the set: n Xn 2 2(3) − 1 = 5 3 2(5) − 1 = 9 4 2(9) − 1 = 17 5 2(17) − 1 = 33 6 2(33) − 1 = 65 7 2(65) − 1 = 129.", "topic": "algebra"}
{"id": "book 2_p267_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 267, "page_end": 267, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. You might notice that there appears to be a repeating pattern among the not help to answer the question, which asks about the difference between two consecutive elements later in the set.", "topic": "algebra"}
{"id": "book 2_p267_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 267, "page_end": 267, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Instead, look at the differences between consecutive elements: n Xn Xn − Xn − 1 2 2(3) − 1 = 5 5 − 3 = 2 3 2(5) − 1 = 9 9 − 5 = 4 4 2(9) − 1 = 17 17 − 9 = 8 5 2(17) − 1 = 33 33 − 17 = 16 6 2(33) − 1 = 65 65 − 33 = 32 7 2(65) − 1 = 129 129 − 65 = 64 The pattern quickly emerges: the difference between consecutive terms in the sequence appears to always be a power of 2.", "topic": "algebra"}
{"id": "book 2_p267_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 267, "page_end": 267, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Specifically, X2 − X1 = 2 = 21, X3 − X2 = 4 = 22, X4 − X3 = 8 = 23, etc.", "topic": "algebra"}
{"id": "book 2_p268_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 268, "page_end": 268, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. You can determine the pattern for the sequence:.", "topic": "algebra"}
{"id": "book 2_p268_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 268, "page_end": 268, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: This is a difference pattern—a pattern or rule that exists among the differences between consecutive terms in the sequence. Be careful at this last step!", "topic": "algebra"}
{"id": "book 2_p268_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 268, "page_end": 268, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: When you extrapolate the pattern, you might accidentally think that the number you want is 220. Always explicitly match to the index, and realize that you might be slightly shi ed.", "topic": "algebra"}
{"id": "book 2_p268_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 268, "page_end": 268, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: In this case, the difference you want is not 2n. Try-It #5-3 If for all positive integers n, what is the sum of the first 100 elements of An?", "topic": "algebra"}
{"id": "book 2_p268_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 268, "page_end": 268, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Once again, compute the first few elements of An. Because you need to know the sum of the first 100 elements, also track the cumulative sum: n An Sum through An.", "topic": "algebra"}
{"id": "book 2_p269_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 269, "page_end": 269, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. N An Sum through An The sum of the first n terms of An equals .", "topic": "algebra"}
{"id": "book 2_p269_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 269, "page_end": 269, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Therefore, the sum of the first 100 terms is . This is a summing pattern—a pattern or rule that exists among the cumulative sum of the terms in the sequence.", "topic": "algebra"}
{"id": "book 2_p270_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 270, "page_end": 270, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Units (Ones) Digit Problems When you raise an integer to a power, the units digit always displays some kind of pattern as you increase the power.", "topic": "algebra"}
{"id": "book 2_p270_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 270, "page_end": 270, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Try-It #5-4 What is the units digit of 4674?", "topic": "algebra"}
{"id": "book 2_p270_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 270, "page_end": 270, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Observe what happens to the units digit of the consecutive powers of 4, starting with 41: last digit of 4 = 4 last digit of 4(4) = last digit of 16 = 6 last digit of 4(6) = last digit of 24 = 4 last digit of 4(4) = last digit of 16 = 6 Because the computations (4 × 4 = 16) and (4 × 6 = 24) keep repeating, the units digit will continue to alternate [4, 6].", "topic": "algebra"}
{"id": "book 2_p270_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 270, "page_end": 270, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Thus, 4x will have a units digit of 4 whenever x is odd and a units digit of 6 whenever x is even (assuming, of course, that x is positive). The units digit of 4674 is therefore 6.", "topic": "algebra"}
{"id": "book 2_p270_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 270, "page_end": 270, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Also notice that in determining the value of the units digit of a product, all of the other digits besides the units digit are irrelevant.", "topic": "algebra"}
{"id": "book 2_p271_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 271, "page_end": 271, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. And 3,184674 will both also have units digits of 6.", "topic": "algebra"}
{"id": "book 2_p271_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 271, "page_end": 271, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: This is also true for multiplication of any two integers, as well as the addition of any integers:. As mentioned earlier, every integer raised to different positive exponents has a units digit pattern.", "topic": "algebra"}
{"id": "book 2_p271_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 271, "page_end": 271, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: As an exercise, derive several of the patterns yourself for the units digits 2, 3, 5, 7, and 8. You can check your work using the table below:.", "topic": "algebra"}
{"id": "book 2_p271_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 271, "page_end": 271, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Series Consecutive Powers Units Digit Pattern 1x 1.", "topic": "algebra"}
{"id": "book 2_p272_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 272, "page_end": 272, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. Series Consecutive Powers Units Digit Pattern 7x 7.", "topic": "number_properties"}
{"id": "book 2_p272_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 272, "page_end": 272, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: [0] You can either memorize this chart or know how to regenerate these patterns quickly. The units digits 1, 5, 6, and 0 just repeat the same digit forever.", "topic": "number_properties"}
{"id": "book 2_p272_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 272, "page_end": 272, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: There’s no real pattern to memorize. That leaves you with six possibilities to memorize (or to re-create when you need them).", "topic": "number_properties"}
{"id": "book 2_p272_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 272, "page_end": 272, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Note that no pattern goes beyond four numbers before repeating, so you don’t have to check beyond the first four terms of any pattern. Try-It #5-5 What is the units digit of 1940?", "topic": "number_properties"}
{"id": "book 2_p272_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 272, "page_end": 272, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Therefore, the pattern is a two-term just like odd and even integers. Since 40 is an even number, the units digit of 1940 will equal 1.", "topic": "number_properties"}
{"id": "book 2_p273_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 273, "page_end": 273, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. What is the remainder when 1940 is divided by 10?", "topic": "algebra"}
{"id": "book 2_p273_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 273, "page_end": 273, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: This alternative question is asking the exact same thing as the original question. The remainder whenever an integer is divided by 10 will always be the same as the units digit of the original number.", "topic": "algebra"}
{"id": "book 2_p274_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 274, "page_end": 274, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Remainder Problems In general, remainders provide a means by which the GMAT can disguise an underlying pattern.", "topic": "number_properties"}
{"id": "book 2_p274_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 274, "page_end": 274, "topic_guess": "number_theory", "text": "GMAT number properties note: The number of terms in the repeat equals the divisor in this case. Conversely, you can calculate all of the numbers that have a certain remainder when divided by a certain value, because they appear at regular intervals as well.", "topic": "number_properties"}
{"id": "book 2_p274_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 274, "page_end": 274, "topic_guess": "number_theory", "text": "GMAT number properties note: For instance, the numbers that have a remainder of 3 when divided by 4 are 3, 7, 11, 15, 19, 23, etc. Notice that those numbers are evenly spaced exactly 4 apart.", "topic": "number_properties"}
{"id": "book 2_p274_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 274, "page_end": 274, "topic_guess": "number_theory", "text": "GMAT number properties note: When a problem discusses remainders, look for patterns and take advantage of them. Try-It #5-6 If x and y are positive integers, what is the remainder when 5x is divided by y?", "topic": "number_properties"}
{"id": "book 2_p275_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 275, "page_end": 275, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. The problem provides no information about the value of y, however.", "topic": "number_properties"}
{"id": "book 2_p275_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 275, "page_end": 275, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: If y = 6, then the remainder is 5. This statement may not initially appear to be sufficient, but test some different values for x: x 5x Remainder of The pattern is clear: no matter what exponent 5 is raised to, the remainder when divided by 4 will always equal 1—a fact that you probably did not expect before testing the rule for this problem.", "topic": "number_properties"}
{"id": "book 2_p276_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 276, "page_end": 276, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. If you used both statements, you could calculate the exact remainder.", "topic": "algebra"}
{"id": "book 2_p276_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 276, "page_end": 276, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: However, that would be too easy for a tough GMAT Quant problem. You don’t actually need both statements to determine that the answer to the question is 1.", "topic": "algebra"}
{"id": "book 2_p277_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 277, "page_end": 277, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Other Pattern Problems Many questions will not at first glance demonstrate an obvious pattern.", "topic": "number_properties"}
{"id": "book 2_p277_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 277, "page_end": 277, "topic_guess": "number_theory", "text": "GMAT number properties note: The point is to discover a simple rule or group of rules that let you account for all the possibilities—and therefore count or maximize them—without having to generate each possibility separately. Try-It #5-7 How many of the integers between 1 and 400, inclusive, are not divisible by 4 and do not contain any 4’s as a digit?", "topic": "number_properties"}
{"id": "book 2_p277_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 277, "page_end": 277, "topic_guess": "number_theory", "text": "GMAT number properties note: This problem involves a counting pattern. It’s clear that there are 400 integers between 1 and 400, inclusive.", "topic": "number_properties"}
{"id": "book 2_p277_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 277, "page_end": 277, "topic_guess": "number_theory", "text": "GMAT number properties note: You’ll need to subtract the integers that are divisible by 4 or contain a 4 as a digit. The tricky part is the overlap: some numbers, such as 64 and 124, violate both constraints.", "topic": "number_properties"}
{"id": "book 2_p277_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 277, "page_end": 277, "topic_guess": "number_theory", "text": "GMAT number properties note: It is easier to determine the number of multiples of 4. Since there are 400 integers in the set, and those 400 integers are consecutive, there must be integers that are divisible by 4.", "topic": "number_properties"}
{"id": "book 2_p277_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 277, "page_end": 277, "topic_guess": "number_theory", "text": "GMAT number properties note: There are 400 − 100 = 300 integers remaining.", "topic": "number_properties"}
{"id": "book 2_p278_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 278, "page_end": 278, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Next, consider the remaining numbers that have a 4 among their digits.", "topic": "number_properties"}
{"id": "book 2_p278_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 278, "page_end": 278, "topic_guess": "number_theory", "text": "GMAT number properties note: These numbers have a 4 among their digits but are not themselves multiples of 4. Only one of the integers between 1 and 400, inclusive, has a 4 in the hundreds place: 400.", "topic": "number_properties"}
{"id": "book 2_p278_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 278, "page_end": 278, "topic_guess": "number_theory", "text": "GMAT number properties note: You have already eliminated that one from the count because it is a multiple of 4. To eliminate integers with a 4 in the tens place, of the form x4y, count only those that are not multiples of 4.", "topic": "number_properties"}
{"id": "book 2_p278_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 278, "page_end": 278, "topic_guess": "number_theory", "text": "GMAT number properties note: These are the numbers whose last two digits are 41, 42, 43, 45, 46, 47, or 49. There are seven such numbers in each set of “hundreds,” that is, the 300’s, the 200’s, the 100’s, and the no hundreds.", "topic": "number_properties"}
{"id": "book 2_p278_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 278, "page_end": 278, "topic_guess": "number_theory", "text": "GMAT number properties note: That is a total of 7 × 4 = 28 terms. There are 300 − 28 = 272 integers remaining.", "topic": "number_properties"}
{"id": "book 2_p278_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 278, "page_end": 278, "topic_guess": "number_theory", "text": "GMAT number properties note: Last, count and eliminate the integers with 4 in the units digit, of the form xy4, that have not already been subtracted. These are the numbers whose last two digits are 14, 34, 54, 74, or 94 (numbers with an even integer in the tens place and 4 in the units are all divisible by 4, so they have been eliminated already).", "topic": "number_properties"}
{"id": "book 2_p278_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 278, "page_end": 278, "topic_guess": "number_theory", "text": "GMAT number properties note: There are 5 such numbers in each set of “hundreds,” so that is a total of 5 × 4 = 20 terms. There are 272 − 20 = 252 integers remaining.", "topic": "number_properties"}
{"id": "book 2_p278_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 278, "page_end": 278, "topic_guess": "number_theory", "text": "GMAT number properties note: Try-It #5-8 x = 1010 − z, where z is a two-digit integer.", "topic": "number_properties"}
{"id": "book 2_p279_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 279, "page_end": 279, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. The first thing to do in solving this problem is to subtract any two-digit number from 1010 and look for a pattern.", "topic": "number_properties"}
{"id": "book 2_p279_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 279, "page_end": 279, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Notice the pattern: the first eight digits of x are all 9’s, so those digits sum to 72. This will be true no matter which two-digit integer you try for z.", "topic": "number_properties"}
{"id": "book 2_p279_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 279, "page_end": 279, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Therefore, the final two digits of x must sum to 84 − 72 = 12. What possibilities would work for these final two digits of x?", "topic": "number_properties"}
{"id": "book 2_p279_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 279, "page_end": 279, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: 39, 48, 57, 66, 75, 84, and 93 all add to 12. When subtracted from 100, these numbers will have to produce a two-digit integer.", "topic": "number_properties"}
{"id": "book 2_p279_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 279, "page_end": 279, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Try subtracting from 100 to find the pattern:. Each successive number is larger, so it will result in a smaller two- digit integer.", "topic": "number_properties"}
{"id": "book 2_p279_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 279, "page_end": 279, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Do the numbers actually drop to one digit at some point? Jump to the other end of the scale and try 93:.", "topic": "number_properties"}
{"id": "book 2_p279_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 279, "page_end": 279, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: In order for x to end in 93, z would have to be 7, which is not a two-digit integer. According to the GMAT, a two-digit integer must have a nonzero.", "topic": "number_properties"}
{"id": "book 2_p280_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 280, "page_end": 280, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Track mean, median, range, distribution, and how operations change spread. Tens digit and zeros for all higher places.", "topic": "statistics"}
{"id": "book 2_p280_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 280, "page_end": 280, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: What about the next number up? If x ends in 84, then it would result in the two-digit integer 100 − 84 = 16.", "topic": "statistics"}
{"id": "book 2_p280_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 280, "page_end": 280, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: All of the other numbers are two-digit integers. Therefore, the final two digits of x can only be 39, 48, 57, 66, 75, and 84, resulting in six possible values for z.", "topic": "statistics"}
{"id": "book 2_p280_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 280, "page_end": 280, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: As the two previous problems demonstrate, unusual patterns can appear in problems on the GMAT. Sometimes you must “think outside the box” to identify the wanted pattern.", "topic": "statistics"}
{"id": "book 2_p280_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 280, "page_end": 280, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: O en, these repeating patterns can only be identified by listing out a few values in the pattern. An = n + k, where n and k are integers.", "topic": "statistics"}
{"id": "book 2_p280_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 280, "page_end": 280, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Note that the average term = the median term = (First + Last).", "topic": "statistics"}
{"id": "book 2_p281_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 281, "page_end": 281, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Track mean, median, range, distribution, and how operations change spread. An = 7n, where n is a set of consecutive integers.", "topic": "statistics"}
{"id": "book 2_p281_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 281, "page_end": 281, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The evens are just a special case (multiples of 2, or 2n). Note that the average term = the median term = × (First + Last).", "topic": "statistics"}
{"id": "book 2_p281_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 281, "page_end": 281, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Evenly Spaced Sets 9, 16, 23, 30, etc. (Constant difference of 7 between consecutive terms) When dividing this series by 7, each of the terms leaves a remainder of 2.", "topic": "statistics"}
{"id": "book 2_p281_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 281, "page_end": 281, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Can be defined as a multiple plus/minus a constant:. The odds are a special case (multiples of 2, plus 1, or 2n + 1).", "topic": "statistics"}
{"id": "book 2_p281_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 281, "page_end": 281, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Note that the average term = the median term = × (First + Last), as for consecutive multiples. Non- Uniform Spacing that Itself Follows a Pattern 0, 1, 3, 6, 10, 15, etc.", "topic": "statistics"}
{"id": "book 2_p281_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 281, "page_end": 281, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: (Spacing between terms follows 1, 2, 3, 4, 5, etc. (Spacing between squares = 1, 3, 5, 7, 9, etc.", "topic": "statistics"}
{"id": "book 2_p281_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 281, "page_end": 281, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: = the odd integers!) Alternating Sign −3, 3, etc. Can result from a (−1)n term in a direct sequence definition, or a (−(An − 1)) term in a recursive sequence definition.", "topic": "statistics"}
{"id": "book 2_p282_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 282, "page_end": 282, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation.", "topic": "algebra"}
{"id": "book 2_p283_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 283, "page_end": 283, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. What is the remainder when the 75th term is divided by 9?", "topic": "algebra"}
{"id": "book 2_p283_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 283, "page_end": 283, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: If x and y are integers between 0 and 9, inclusive, and the units digit of xy is 5, what are the possible values of x and y? What is the remainder when 1317 + 1713 is divided by 10?", "topic": "algebra"}
{"id": "book 2_p283_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 283, "page_end": 283, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: If y is a positive integer, what is the units digit of y? The units digit of y2 equals 6.(1) The units digit of (y + 1)2 equals 5.(2) 5.", "topic": "algebra"}
{"id": "book 2_p284_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 284, "page_end": 284, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. The units digit of y2 = 1.(1) The units digit of y does not equal 1.(2) 6.", "topic": "number_properties"}
{"id": "book 2_p284_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 284, "page_end": 284, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: In sequence A, A1 = 1, A2 = 100, and the value of An is strictly between the values of An – 1 and An – 2 for all n ≥ 3. Which of the following must be true?", "topic": "number_properties"}
{"id": "book 2_p284_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 284, "page_end": 284, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: If x is an integer, what is the remainder when x is divided by 5? X2 has a remainder of 4 when divided by 5.(1) x3 has a remainder of 2 when divided by 5.(2) 8.", "topic": "number_properties"}
{"id": "book 2_p284_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 284, "page_end": 284, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: If x and y are positive integers, what is the remainder when 5x is divided by y? X is an even integer.(1) y = 3(2).", "topic": "number_properties"}
{"id": "book 2_p285_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 285, "page_end": 285, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. A, b, c, and d are positive integers.", "topic": "algebra"}
{"id": "book 2_p285_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 285, "page_end": 285, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: If has a remainder of 9 and has a remainder of 10, what is the minimum possible value for bd? What is the sum of the numbers in the grid below?", "topic": "algebra"}
{"id": "book 2_p285_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 285, "page_end": 285, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: 7 −8 −4 5 12 9 −10 −5 6 15 11 −12 −6 7 18 all n > 1. If a1 = 10, what is the value of a11?", "topic": "algebra"}
{"id": "book 2_p286_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 286, "page_end": 286, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. In sequence Q, the first number is 3, and each subsequent number in the sequence is determined by doubling the previous number and then adding 2.", "topic": "algebra"}
{"id": "book 2_p286_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 286, "page_end": 286, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: How many times does the digit 8 appear in the units digit of the first 10 terms of the sequence? If ♦ K♦ is divisible by 411, what is the smallest possible value for K?", "topic": "algebra"}
{"id": "book 2_p287_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 287, "page_end": 287, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Mitchell plans to work at a day camp over the summer.", "topic": "percent"}
{"id": "book 2_p287_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 287, "page_end": 287, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Each week, he will be paid according to the following schedule:. At the end of the first week, he will receive $1.", "topic": "percent"}
{"id": "book 2_p287_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 287, "page_end": 287, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: At the end of each subsequent week, he will receive $1, plus an additional amount equal to the sum of all payments he’s received in previous weeks. How much money will Mitchell be paid in total during the summer, if he works for the entire duration of the 8-week-long camp?", "topic": "percent"}
{"id": "book 2_p288_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 288, "page_end": 288, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. The sequence starts with the number 4, then adds 5 to each subsequent term.", "topic": "number_properties"}
{"id": "book 2_p288_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 288, "page_end": 288, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The math plays out in this way:. The sequence pattern is 4 + 5(n − 1), where n is the number of the term.", "topic": "number_properties"}
{"id": "book 2_p288_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 288, "page_end": 288, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The 75th term in the sequence is therefore 4 + 5(75 − 1) = 4 + 5(74) = 374. In order to find the remainder, first find the multiple of 9 closest to 374 but smaller than that number.", "topic": "number_properties"}
{"id": "book 2_p288_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 288, "page_end": 288, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: 360 is a multiple of 9, and so is 369. Therefore, has a remainder of 0, so has a remainder of 5.", "topic": "number_properties"}
{"id": "book 2_p288_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 288, "page_end": 288, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Only the integer 5 can be raised to a power to result in a units digit of 5. Any power of 5 will have a units digit of 5, other than zero, because 50 = 1.", "topic": "number_properties"}
{"id": "book 2_p288_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 288, "page_end": 288, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The integer y, on the other hand, can have any value except for 0. The remainder when dividing an integer by 10 always equals the units digit.", "topic": "number_properties"}
{"id": "book 2_p288_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 288, "page_end": 288, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Ignore all but the units digits and rephrase the question: “What is the units digit of 317 + 713?”.", "topic": "number_properties"}
{"id": "book 2_p289_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 289, "page_end": 289, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. The pattern for the units digits of 3 is [3, 9, 7, 1].", "topic": "number_properties"}
{"id": "book 2_p289_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 289, "page_end": 289, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Every fourth term is the same. The 17th power is 1 past the end of the repeat.", "topic": "number_properties"}
{"id": "book 2_p289_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 289, "page_end": 289, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Since 316 ends in 1, 317 must end in 3. The pattern for the units digits of 7 is [7, 9, 3, 1].", "topic": "number_properties"}
{"id": "book 2_p289_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 289, "page_end": 289, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The 13th power is 1 past the end of the repeat. Since 712 ends in 1, 713 must end in 7.", "topic": "number_properties"}
{"id": "book 2_p289_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 289, "page_end": 289, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The sum of these units digits is 3 + 7 = 10. For problems 4 and 5, reference the following chart:.", "topic": "number_properties"}
{"id": "book 2_p289_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 289, "page_end": 289, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: For integers ending in a certain digit, the table shows the units digit pattern. Integers ending in 9 have a units digit pattern of 9, 1.", "topic": "number_properties"}
{"id": "book 2_p289_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 289, "page_end": 289, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: 1 2 3 4 5 6 7 8 9 0 y1 1 2 3 4 5 6 7 8 9 0 y2 1 4 9 6 5 6 9 4 1 0 y3 1 8 7 4 5 6 3 2 9 0 y4 1 6 1 6 5 6 1 6 1 0 4. According to the Units Digit Patterns chart above, both 4 and 6 yield a units digit of 6 when raised to the second power.", "topic": "number_properties"}
{"id": "book 2_p289_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 289, "page_end": 289, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: (Notice that you would only need to check even numbers for this.", "topic": "number_properties"}
{"id": "book 2_p290_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 290, "page_end": 290, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Statement, as odd numbers to any power cannot end in a 6, an even number.) (2) SUFFICIENT:.", "topic": "algebra"}
{"id": "book 2_p290_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 290, "page_end": 290, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Only 5 yields a units digit of 5 when raised to any power. Since the units digit of y + 1 is 5, the units digit of y must be 4.", "topic": "algebra"}
{"id": "book 2_p290_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 290, "page_end": 290, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The statement indicates only that the units digit does not equal 1, but it could still be any other digit. Because both 1 and 9 yield a units digit of 1 when raised to the second power, there are two possible values for the units digit.", "topic": "algebra"}
{"id": "book 2_p290_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 290, "page_end": 290, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The two statements together indicate that the units digit of y must be 9. Since there isn’t time to calculate that many terms, there must be a pattern.", "topic": "algebra"}
{"id": "book 2_p290_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 290, "page_end": 290, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Write down what you know about the first few terms in order to identify that pattern. Each term in the sequence is strictly between the two terms that come immediately before it.", "topic": "algebra"}
{"id": "book 2_p290_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 290, "page_end": 290, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: To keep track of this, draw a number line on your.", "topic": "algebra"}
{"id": "book 2_p291_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 291, "page_end": 291, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Put the third term of the sequence somewhere between the first two terms:.", "topic": "algebra"}
{"id": "book 2_p291_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 291, "page_end": 291, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Continue to add terms to the number line, making sure to put each term between the two that came before it. The first eight terms have been plotted on the number line shown below:.", "topic": "algebra"}
{"id": "book 2_p291_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 291, "page_end": 291, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The odd-numbered terms of the sequence appear in increasing order, so that A1 < A3 < A5 < A7, and so on. The even-numbered terms are in decreasing order so that A2 > A4 > A6 > A8, etc.", "topic": "algebra"}
{"id": "book 2_p291_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 291, "page_end": 291, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Because the question is asking about the remainder when x is divided by 5, you only need to check the units digits 0 through 4. The pattern will recycle for the next set of 5 (5 through 9) and for every set of 5 a er that.", "topic": "algebra"}
{"id": "book 2_p292_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 292, "page_end": 292, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. X x2 = remainder of 4 = remainder of?", "topic": "number_properties"}
{"id": "book 2_p292_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 292, "page_end": 292, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: 2 4 Yes = 0 remainder 2 3 9 Yes = 0 remainder 3 There are at least two possible values for the remainder. (2) SUFFICIENT: x x3 = remainder of 2 = remainder of?", "topic": "number_properties"}
{"id": "book 2_p292_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 292, "page_end": 292, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: 3 27 Yes = 0 remainder 3 There is only one possible remainder: 3. If you aren’t sure that you only need to test these five cases, try the next five values for x.", "topic": "number_properties"}
{"id": "book 2_p292_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 292, "page_end": 292, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Note the repeat of the remainder pattern. 0 0 No (invalid) 1 1 No (invalid) 0 0 No (invalid) 1 1 No (invalid) 2 8 No (invalid) 4 64 No (invalid).", "topic": "number_properties"}
{"id": "book 2_p293_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 293, "page_end": 293, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. If x is even, then 5x = 25, 625, and so on.", "topic": "number_properties"}
{"id": "book 2_p293_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 293, "page_end": 293, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: This statement provides no information about y, though. Test some different values for x.", "topic": "number_properties"}
{"id": "book 2_p293_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 293, "page_end": 293, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Because you will have to test the two statements together, continue testing another couple of numbers to see whether there is a pattern. When 5 is raised to an odd power, the remainder is 2, but when 5 is raised to an even power, the remainder is 1.", "topic": "number_properties"}
{"id": "book 2_p294_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 294, "page_end": 294, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. When 5even integer is divided by 3, the remainder is always 1.", "topic": "number_properties"}
{"id": "book 2_p294_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 294, "page_end": 294, "topic_guess": "number_theory", "text": "GMAT number properties note: The remainder must always be smaller than the divisor. Thus, b must be at least 10, and d must be at least 11.", "topic": "number_properties"}
{"id": "book 2_p294_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 294, "page_end": 294, "topic_guess": "number_theory", "text": "GMAT number properties note: Therefore, bd must be at least 110. The purpose of this problem is to remind you of these constraints on remainders.", "topic": "number_properties"}
{"id": "book 2_p294_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 294, "page_end": 294, "topic_guess": "number_theory", "text": "GMAT number properties note: Each column contains a series of evenly spaced integers. To avoid calculation, cancel out as many values as possible before summing the remaining values.", "topic": "number_properties"}
{"id": "book 2_p294_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 294, "page_end": 294, "topic_guess": "number_theory", "text": "GMAT number properties note: In each row, the sum of the integers in the first two columns is −1, and the sum of the integers in the third and fourth columns is 1. Therefore, the sum of the integers in the first four columns in each row is −1 + 1 = 0.", "topic": "number_properties"}
{"id": "book 2_p294_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 294, "page_end": 294, "topic_guess": "number_theory", "text": "GMAT number properties note: Only the integers in the fi h column need to be considered. The integers in the fi h column can be rewritten as follows:.", "topic": "number_properties"}
{"id": "book 2_p294_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 294, "page_end": 294, "topic_guess": "number_theory", "text": "GMAT number properties note: The sum of the integers is therefore 3(21) = 63. Write out the first few terms of the sequence to find the pattern.", "topic": "number_properties"}
{"id": "book 2_p295_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 295, "page_end": 295, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. To simplify the calculation, these terms can be rewritten as follows:.", "topic": "number_properties"}
{"id": "book 2_p295_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 295, "page_end": 295, "topic_guess": "number_theory", "text": "GMAT number properties note: Therefore, the nth term in the sequence can be found by adding 10 a total of n times, then adding the sum of the integers from 1 to n – 1, which equals 10(11) + 55 = 165. Compute the first few elements of Sn.", "topic": "number_properties"}
{"id": "book 2_p296_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 296, "page_end": 296, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. N Sn Use a number line to track the sum:.", "topic": "word_problems"}
{"id": "book 2_p296_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 296, "page_end": 296, "topic_guess": "word_problems", "text": "GMAT word problem note: Place the first term, , on the number line. The second term is + so the sum will move to the right (closer to 0) on the number line.", "topic": "word_problems"}
{"id": "book 2_p296_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 296, "page_end": 296, "topic_guess": "word_problems", "text": "GMAT word problem note: The third term is so the sum will move to the le, but it can’t go as far as again, because you’re only subtracting this time—the distance is smaller than the first hop that you made. Each subsequent hop flips back and forth between positive and negative but also keeps getting smaller and smaller, so you’ll never “break out” of the range to 0.", "topic": "word_problems"}
{"id": "book 2_p297_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 297, "page_end": 297, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. Calculate the first several terms of the sequence to find the pattern:.", "topic": "number_properties"}
{"id": "book 2_p297_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 297, "page_end": 297, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The pattern should continue, so 8 will be the units digit 9 out of the first 10 times. This is a counting pattern problem.", "topic": "number_properties"}
{"id": "book 2_p297_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 297, "page_end": 297, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: In order for ♦ K♦ to be divisible by 411, it must be divisible by (22)11 = 222. Thus, ♦ K♦ must contain 22 twos in its prime factorization.", "topic": "number_properties"}
{"id": "book 2_p297_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 297, "page_end": 297, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The number 2 has one 2 in its prime factorization. This amounts to a total of only eight 2’s:.", "topic": "number_properties"}
{"id": "book 2_p297_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 297, "page_end": 297, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Number 2 4 6 8 10 Prime Factor(s) 2 2, 2 2, 3 2, 2, 2 2, 5 Total 2’s in PF 1 2 1 3 1 Cumulative 2’s 1 3 4 7 8.", "topic": "number_properties"}
{"id": "book 2_p298_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 298, "page_end": 298, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Keep adding even numbers to the result until you get to 22 twos in total:.", "topic": "number_properties"}
{"id": "book 2_p298_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 298, "page_end": 298, "topic_guess": "number_theory", "text": "GMAT number properties note: Number 2 4 6 8 10 12 14 16 18 20 22 24 Prime Factor(s) Total 2’s in PF 1 2 1 3 1 2 1 4 1 2 1 3 Cumulative 2’s 1 3 4 7 8 10 11 15 16 18 19 22 Thus, the smallest possible value for K is 24. At the end of the first week, Mitchell receives $1.", "topic": "number_properties"}
{"id": "book 2_p298_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 298, "page_end": 298, "topic_guess": "number_theory", "text": "GMAT number properties note: At the end of the second week, he gets $1, plus $1 for the total he had been paid up to that point, for a total of $2. At the end of the third week, he gets $1, plus ($1 + $2), or $3, for the total he had been paid up to that point, so this third week’s total is $4.", "topic": "number_properties"}
{"id": "book 2_p298_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 298, "page_end": 298, "topic_guess": "number_theory", "text": "GMAT number properties note: Week # Paid This Week($) Cumulative Pay Including This Week ($).", "topic": "number_properties"}
{"id": "book 2_p299_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 299, "page_end": 299, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation.", "topic": "algebra"}
{"id": "book 2_p299_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 299, "page_end": 299, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Week # Paid This Week($) Cumulative Pay Including This Week ($) 2 1 + 1 = 2 1 + 2 = 3 3 1 + 3 = 4 3 + 4 = 7 4 1 + 7 = 8 7 + 8 = 15 5 1 + 15 = 16 15 + 16 = 31 6 1 + 31 = 32 31 + 32 = 63 7 1 + 63 = 64 63 + 64 = 127 8 1 + 127 = 128 127 + 128 = 255 This calculation is not so bad, but you may notice that this payment schedule is a geometric sequence, 2n − 1, where n is the number of the week in which Mitchell is being paid.", "topic": "algebra"}
{"id": "book 2_p299_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 299, "page_end": 299, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Summing that sequence is equivalent to 2t − 1, where t is the total number of weeks. In other words, the cumulative pay is one less than the next power of 2.", "topic": "algebra"}
{"id": "book 2_p300_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 300, "page_end": 300, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation.", "topic": "algebra"}
{"id": "book 2_p301_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 301, "page_end": 301, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Common Terms Quadratic Templates Quadratic Templates in Disguise.", "topic": "algebra"}
{"id": "book 2_p302_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 302, "page_end": 302, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. But of all the steps you could take, how do you decide which steps you should take?", "topic": "algebra"}
{"id": "book 2_p302_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 302, "page_end": 302, "topic_guess": "algebra", "text": "GMAT algebra note: Two indicators can o en help you on the GMAT:. You can substitute one expression for another if they are equal.", "topic": "algebra"}
{"id": "book 2_p302_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 302, "page_end": 302, "topic_guess": "algebra", "text": "GMAT algebra note: You can add some number to one side of an equation as long as you do the same on the other side of the equation. You can cross-multiply to simplify equations with fractions on each side of the equals sign.", "topic": "algebra"}
{"id": "book 2_p303_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 303, "page_end": 303, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Common Terms If you spot common terms, you can o en spot the path all the way to the solution.", "topic": "algebra"}
{"id": "book 2_p303_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 303, "page_end": 303, "topic_guess": "algebra", "text": "GMAT algebra note: Common terms appear on the GMAT in three typical ways. ALGEBRA Look for terms that appear in the same form more than once.", "topic": "algebra"}
{"id": "book 2_p303_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 303, "page_end": 303, "topic_guess": "algebra", "text": "GMAT algebra note: Those recurring expressions might also appear in slightly modified form such as reciprocal, negative, or raised to a power: is the reciprocal of is negative Is ? Where and 3y appear both squared and multiplied together.", "topic": "algebra"}
{"id": "book 2_p303_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 303, "page_end": 303, "topic_guess": "algebra", "text": "GMAT algebra note: This expression is of the form: (More on Quadratic Templates later in this chapter).", "topic": "algebra"}
{"id": "book 2_p304_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 304, "page_end": 304, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Try-It #6-1 If y ≠ 3, simplify as much as possible:.", "topic": "number_properties"}
{"id": "book 2_p304_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 304, "page_end": 304, "topic_guess": "number_theory", "text": "GMAT number properties note: Spot the common term (3 − y). Note that (−3 + y) is −(3 − y), or −1 × (3 − y).", "topic": "number_properties"}
{"id": "book 2_p304_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 304, "page_end": 304, "topic_guess": "number_theory", "text": "GMAT number properties note: Factor out the common term (3 − y) and cancel:. By the way, the condition that y could not equal 3 just prevented you from dividing by 0 and ending up with an undefined number.", "topic": "number_properties"}
{"id": "book 2_p304_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 304, "page_end": 304, "topic_guess": "number_theory", "text": "GMAT number properties note: EXPONENTS Exponents can be manipulated when either bases or exponents are common. Also look for bases that have common factors, such as 3 and 12 (common factor of 3).", "topic": "number_properties"}
{"id": "book 2_p304_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 304, "page_end": 304, "topic_guess": "number_theory", "text": "GMAT number properties note: You can o en create a common base. Similarly, 4, 16, and 256 are all powers of 4.", "topic": "number_properties"}
{"id": "book 2_p305_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 305, "page_end": 305, "topic_guess": "algebra", "text": "GMAT algebra note: Watch divisibility, factors, remainders, parity, and prime structure. Try-It #6-2 If 3x + 243 = 2(3x), what is the value of x?", "topic": "number_properties"}
{"id": "book 2_p305_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 305, "page_end": 305, "topic_guess": "algebra", "text": "GMAT algebra note: Note the common term 3x, and note the fact that 243 = 35: 3. FACTORS AND MULTIPLES When many terms share a factor, pull that shared factor out to the side.", "topic": "number_properties"}
{"id": "book 2_p305_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 305, "page_end": 305, "topic_guess": "algebra", "text": "GMAT algebra note: These can appear in algebraic or numerical expressions: → get common denominators, then cross them all off → x18 + 2x16 + x14 → x14 is a factor of each term → x14(x4 + 2x2 + 1) = x14(x2 +.", "topic": "number_properties"}
{"id": "book 2_p306_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 306, "page_end": 306, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Factorials are particularly noteworthy, as they o en have an abundance of shared factors.", "topic": "number_properties"}
{"id": "book 2_p306_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 306, "page_end": 306, "topic_guess": "number_theory", "text": "GMAT number properties note: For any integer n, the factorial n! Is calculated as follows: n!", "topic": "number_properties"}
{"id": "book 2_p306_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 306, "page_end": 306, "topic_guess": "number_theory", "text": "GMAT number properties note: = (6)(5)(4)(3)(2)(1) = (6)(5)(4!). More generally, factorials are “super multiples.” Without ever computing their precise value, you can tell that they’re divisible by all sorts of numbers.", "topic": "number_properties"}
{"id": "book 2_p306_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 306, "page_end": 306, "topic_guess": "number_theory", "text": "GMAT number properties note: If x is an integer between 7! X is one of the following integers: x has one of the following factors, if x is: 7!", "topic": "number_properties"}
{"id": "book 2_p306_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 306, "page_end": 306, "topic_guess": "number_theory", "text": "GMAT number properties note: + 3 = 3 × Integer not prime! Sometimes a common factor is just a random number buried inside a couple of larger numbers.", "topic": "number_properties"}
{"id": "book 2_p306_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 306, "page_end": 306, "topic_guess": "number_theory", "text": "GMAT number properties note: Find it and pull it out: 10’s in the numerator line up with 5’s in the corresponding digit place of the denominator.", "topic": "number_properties"}
{"id": "book 2_p307_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 307, "page_end": 307, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Try-It #6-3 If n is a positive integer and is a positive integer, what is the value of n?", "topic": "number_properties"}
{"id": "book 2_p307_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 307, "page_end": 307, "topic_guess": "number_theory", "text": "GMAT number properties note: What would need to be true in order for the square root to be a positive integer? The number under the square root symbol would have to be a perfect square.", "topic": "number_properties"}
{"id": "book 2_p307_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 307, "page_end": 307, "topic_guess": "number_theory", "text": "GMAT number properties note: Rearrange the expression to determine whether there are any restrictions that could help narrow down the possibilities before going to the statements. Try to break the numbers down into primes to locate and pull out any existing perfect squares: n is prime.(1) n < 3(2).", "topic": "number_properties"}
{"id": "book 2_p308_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 308, "page_end": 308, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. It turns out that you can also pull out the term 7n − 1.", "topic": "number_properties"}
{"id": "book 2_p308_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 308, "page_end": 308, "topic_guess": "general", "text": "GMAT quant note: The term 7n + 1 = (7n − 1) The first two terms are both perfect squares and can be pulled out of the square root sign:. What would need to be true in order for 7n − 1 to be pulled out of the square root sign?", "topic": "number_properties"}
{"id": "book 2_p308_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 308, "page_end": 308, "topic_guess": "general", "text": "GMAT quant note: It would also have to be a perfect square, so n – 1 must be even and n itself must be odd.", "topic": "number_properties"}
{"id": "book 2_p309_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 309, "page_end": 309, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. The question is: “If n is odd, what is the value of n?” (1) INSUFFICIENT:.", "topic": "number_properties"}
{"id": "book 2_p309_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 309, "page_end": 309, "topic_guess": "general", "text": "GMAT quant note: This statement allows multiple possible odd values of n. The question stem indicates that n is a positive integer, and this statement specifies that n < 3.", "topic": "number_properties"}
{"id": "book 2_p309_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 309, "page_end": 309, "topic_guess": "general", "text": "GMAT quant note: The only odd, positive integer less than 3 is the number 1.", "topic": "number_properties"}
{"id": "book 2_p310_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 310, "page_end": 310, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Quadratic Templates On the GMAT, quadratic expressions take three common forms called the Quadratic Templates.", "topic": "algebra"}
{"id": "book 2_p310_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 310, "page_end": 310, "topic_guess": "algebra", "text": "GMAT algebra note: Memorize these templates, and get comfortable transforming back and forth between factored and distributed form:.", "topic": "algebra"}
{"id": "book 2_p310_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 310, "page_end": 310, "topic_guess": "algebra", "text": "GMAT algebra note: Factored Distributed Square of a Sum (a + b)2 = a2 + 2ab + b2 Square of a Difference (a − b)2 = a2 − 2ab + b2 Difference of Two Squares (a + b) (a − b) = a2 − b2 QUICK MANIPULATION Expressions with both squared and non-squared common terms should make you suspect that you are looking at a Quadratic Template.", "topic": "algebra"}
{"id": "book 2_p311_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 311, "page_end": 311, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. This problem requires you to manipulate a rather complicated expression.", "topic": "algebra"}
{"id": "book 2_p311_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 311, "page_end": 311, "topic_guess": "algebra", "text": "GMAT algebra note: However, by using the common terms, you can put the problem in the more basic template form to solve:. Once you are comfortable with Quadratic Templates, you can manipulate even complicated expressions quickly, as in the middle box above.", "topic": "algebra"}
{"id": "book 2_p311_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 311, "page_end": 311, "topic_guess": "algebra", "text": "GMAT algebra note: Until then, write down the templates and the substitution of the common terms, as in the box on the right. The very same problem could have been presented in disguise:.", "topic": "algebra"}
{"id": "book 2_p311_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 311, "page_end": 311, "topic_guess": "algebra", "text": "GMAT algebra note: Factor The common terms are slightly harder to spot in this form. In such a case, start with the squared terms, and 25y2.", "topic": "algebra"}
{"id": "book 2_p311_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 311, "page_end": 311, "topic_guess": "algebra", "text": "GMAT algebra note: Then, try to untangle their square roots, and 5y, from the remaining term.", "topic": "algebra"}
{"id": "book 2_p312_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 312, "page_end": 312, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. THE MIDDLE TERM: 2AB The square of a sum and square of a difference templates have something in common: the middle term is ±2ab.", "topic": "algebra"}
{"id": "book 2_p312_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 312, "page_end": 312, "topic_guess": "algebra", "text": "GMAT algebra note: The only difference is the sign of that middle term. When you add these two templates, the middle terms cancel, leaving the end terms:.", "topic": "algebra"}
{"id": "book 2_p312_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 312, "page_end": 312, "topic_guess": "algebra", "text": "GMAT algebra note: Factored Distributed Square of a Sum (a + b)2 = a2 + 2ab + b2 + Square of a Difference (a − b)2 = a2 − 2ab + b2 Addition: (a + b)2 + (a − b)2 = 2a2 + 0 + 2b2 2(a2 + b2).", "topic": "algebra"}
{"id": "book 2_p313_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 313, "page_end": 313, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. In contrast, when you subtract these two templates, the end terms cancel, leaving the middle term:.", "topic": "algebra"}
{"id": "book 2_p313_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 313, "page_end": 313, "topic_guess": "algebra", "text": "GMAT algebra note: Factored Distributed Square of a Sum (a + b)2 = a2 + 2ab + b2 + Square of a Difference (a − b)2 = a2 − 2ab + b2 Subtraction: (a + b)2 − (a − b)2 = 0 + 4ab + 0 This is handy for simplification. Also, whenever you see the sum of two squares (a2 + b2), which is not itself a Quadratic Template, remember that it can be derived from this sum of two templates.", "topic": "algebra"}
{"id": "book 2_p313_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 313, "page_end": 313, "topic_guess": "algebra", "text": "GMAT algebra note: Try It #6-5 What is the sum of 9,9992 and 10,0012?", "topic": "algebra"}
{"id": "book 2_p314_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 314, "page_end": 314, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. These numbers all look seriously cumbersome.", "topic": "algebra"}
{"id": "book 2_p314_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 314, "page_end": 314, "topic_guess": "algebra", "text": "GMAT algebra note: If only they had given the When wishful thinking pops up, try to use it to make the problem easier. If you changed both of these numbers to the form 10,000 plus or minus a number, what would that be?", "topic": "algebra"}
{"id": "book 2_p314_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 314, "page_end": 314, "topic_guess": "algebra", "text": "GMAT algebra note: These are the Quadratic Templates! In this case, a = 10,000 and b = 1.", "topic": "algebra"}
{"id": "book 2_p314_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 314, "page_end": 314, "topic_guess": "algebra", "text": "GMAT algebra note: If you’ve memorized the sum term, plug these in:. Now, notice something: the part in the parentheses is going to have a units digit of 1.", "topic": "algebra"}
{"id": "book 2_p314_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 314, "page_end": 314, "topic_guess": "algebra", "text": "GMAT algebra note: Multiply the number by 2, and the end result will have a units digit of 2. Even if you don’t memorize the sum term of the Quadratic Templates, the math is still far easier to do in this rewritten form.", "topic": "algebra"}
{"id": "book 2_p316_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 316, "page_end": 316, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Quadratic Templates in Disguise Quadratic Templates can be disguised in arithmetic computations.", "topic": "algebra"}
{"id": "book 2_p316_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 316, "page_end": 316, "topic_guess": "algebra", "text": "GMAT algebra note: Try-It #6-6 What is 198 × 202? You can round each number and quickly estimate the result to be about 2002.", "topic": "algebra"}
{"id": "book 2_p316_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 316, "page_end": 316, "topic_guess": "algebra", "text": "GMAT algebra note: Or, you could laboriously multiply two 3-digit numbers by hand to get an exact result. But if you need an exact result quickly, you can use a Quadratic Template, as shown in the previous section.", "topic": "algebra"}
{"id": "book 2_p316_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 316, "page_end": 316, "topic_guess": "algebra", "text": "GMAT algebra note: You just need to turn 198 into (200 − 2) and 202 into (200 + 2) as shown here:. Another place to hide a Quadratic Template is in an advanced right-triangle problem.", "topic": "algebra"}
{"id": "book 2_p317_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 317, "page_end": 317, "topic_guess": "geometry", "text": "GMAT geometry note: You know that and d2 = b2 + h2 (by the Pythagorean theorem). Do the common terms b2, h2, and bh look familiar?", "topic": "geometry"}
{"id": "book 2_p317_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 317, "page_end": 317, "topic_guess": "geometry", "text": "GMAT geometry note: Use the Square of a Sum template:. Likewise, there is a similar relationship based on the Square of a Difference template:.", "topic": "geometry"}
{"id": "book 2_p317_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 317, "page_end": 317, "topic_guess": "geometry", "text": "GMAT geometry note: An advanced GMAT problem can draw on these complicated relationships. For instance, you can compute the area of a right triangle directly from the sum of the shorter sides and the hypotenuse.", "topic": "geometry"}
{"id": "book 2_p318_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 318, "page_end": 318, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. You should absolutely not memorize these particular formulas.", "topic": "algebra"}
{"id": "book 2_p318_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 318, "page_end": 318, "topic_guess": "algebra", "text": "GMAT algebra note: Rather, be able to recognize when the GMAT is indirectly testing these generic Quadratic Templates.", "topic": "algebra"}
{"id": "book 2_p319_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 319, "page_end": 319, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. For problems 2–4, if x < –1, which of the following inequalities must be true?", "topic": "percent"}
{"id": "book 2_p319_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 319, "page_end": 319, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: If, and |x| ≠ |y|, what is the ratio of x to y?", "topic": "percent"}
{"id": "book 2_p320_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 320, "page_end": 320, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If , what is the value of 7.", "topic": "number_properties"}
{"id": "book 2_p320_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 320, "page_end": 320, "topic_guess": "number_theory", "text": "GMAT number properties note: If (ax + by)2 = (bx + ay)2, what is the value of a – b? X2 > y2(1) a and b are positive integers.(2) 9.", "topic": "number_properties"}
{"id": "book 2_p320_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 320, "page_end": 320, "topic_guess": "number_theory", "text": "GMAT number properties note: If x and k are both integers, x > k, and x–k = 625, what is x? |k| is a prime number.(1) x + k > 20(2).", "topic": "number_properties"}
{"id": "book 2_p321_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 321, "page_end": 321, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Distribute the expression in problems 10–14 without FOILing (doing the math the long way).", "topic": "algebra"}
{"id": "book 2_p321_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 321, "page_end": 321, "topic_guess": "algebra", "text": "GMAT algebra note: Use the Quadratic Templates. Factor problems 15–19 according to the Quadratic Templates.", "topic": "algebra"}
{"id": "book 2_p322_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 322, "page_end": 322, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation.", "topic": "algebra"}
{"id": "book 2_p323_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 323, "page_end": 323, "topic_guess": "geometry", "text": "GMAT geometry note: In the right triangle below, side a is 7 inches longer than side b. If the area of the triangle is 30 inches2, what is the length of hypotenuse c?", "topic": "geometry"}
{"id": "book 2_p324_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 324, "page_end": 324, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. From xy > 0, you know that neither x nor y equals 0, and they must have the same sign (++ or − −).", "topic": "algebra"}
{"id": "book 2_p324_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 324, "page_end": 324, "topic_guess": "algebra", "text": "GMAT algebra note: Combine the exponent in the question to get What is the significance of the inequality in the question stem? If the exponent equals 2, then 52 = 25.", "topic": "algebra"}
{"id": "book 2_p324_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 324, "page_end": 324, "topic_guess": "algebra", "text": "GMAT algebra note: In order to be greater than 25, the exponent has to be greater than 2. If, on the other hand, y is negative, then (1) INSUFFICIENT:.", "topic": "algebra"}
{"id": "book 2_p325_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 325, "page_end": 325, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Therefore, y3 must equal 8, so y = 2.", "topic": "algebra"}
{"id": "book 2_p325_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 325, "page_end": 325, "topic_guess": "algebra", "text": "GMAT algebra note: This provides no information about x, however. For questions 2–4: since x ≠ 0, divide by the common terms, making sure to flip the inequality sign if the common term is negative.", "topic": "algebra"}
{"id": "book 2_p325_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 325, "page_end": 325, "topic_guess": "algebra", "text": "GMAT algebra note: Divide both sides by x2, leaving the sign as it is. The square of a number smaller than −1 will be greater than positive 1.", "topic": "algebra"}
{"id": "book 2_p325_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 325, "page_end": 325, "topic_guess": "algebra", "text": "GMAT algebra note: (Or, you might have subtracted x3 immediately.) TRUE:. Divide both sides by x5, flipping the inequality sign.", "topic": "algebra"}
{"id": "book 2_p326_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 326, "page_end": 326, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Try to find like terms in order to simplify the le -hand side of the equation:.", "topic": "algebra"}
{"id": "book 2_p326_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 326, "page_end": 326, "topic_guess": "algebra", "text": "GMAT algebra note: The ratio of x to y is 2: 1. Since n is an integer, 4n is even.", "topic": "algebra"}
{"id": "book 2_p326_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 326, "page_end": 326, "topic_guess": "algebra", "text": "GMAT algebra note: An even exponent “hides the sign” of the base, so you can treat the (−3) base as a (3).", "topic": "algebra"}
{"id": "book 2_p327_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 327, "page_end": 327, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Begin by simplifying the equation in the question stem, making it look as similar as possible to the question itself:.", "topic": "algebra"}
{"id": "book 2_p327_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 327, "page_end": 327, "topic_guess": "algebra", "text": "GMAT algebra note: Do not divide by x2 – y2, since it may equal 0. Instead, subtract b2(x2 – y2) from both sides of the equation, then factor:.", "topic": "algebra"}
{"id": "book 2_p327_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 327, "page_end": 327, "topic_guess": "algebra", "text": "GMAT algebra note: So either a2 – b2 = 0 (in which case a2 = b2), or x2 – y2 = 0 (in which case x2 = y2). One or both of these must be true.", "topic": "algebra"}
{"id": "book 2_p328_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 328, "page_end": 328, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If x2 > y2, then x2 cannot equal y2.", "topic": "number_properties"}
{"id": "book 2_p328_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 328, "page_end": 328, "topic_guess": "number_theory", "text": "GMAT number properties note: It is possible that a and b are equal, in which case a – b = 0, so the answer is 0. It is also possible that a and b are equal in absolute value but have opposite signs.", "topic": "number_properties"}
{"id": "book 2_p328_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 328, "page_end": 328, "topic_guess": "number_theory", "text": "GMAT number properties note: In this case, the answer is a – b = –4. More than one answer is possible, so this statement is not sufficient.", "topic": "number_properties"}
{"id": "book 2_p328_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 328, "page_end": 328, "topic_guess": "number_theory", "text": "GMAT number properties note: This statement only says that a and b are positive integers, but does not give further indication of what they could be. There are many possible answers for the value of a – b, so this statement is insufficient.", "topic": "number_properties"}
{"id": "book 2_p328_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 328, "page_end": 328, "topic_guess": "number_theory", "text": "GMAT number properties note: Thus, either a = b or a = –b. Therefore, a = b, and the answer to the question is a – b = 0.", "topic": "number_properties"}
{"id": "book 2_p328_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 328, "page_end": 328, "topic_guess": "number_theory", "text": "GMAT number properties note: The statements together are sufficient. The fact that x and k are both integers (and that x > k) significantly limits the possible values for x and k.", "topic": "number_properties"}
{"id": "book 2_p328_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 328, "page_end": 328, "topic_guess": "number_theory", "text": "GMAT number properties note: The possible pairings are as follows: x k.", "topic": "number_properties"}
{"id": "book 2_p329_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 329, "page_end": 329, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If the absolute value of k is prime, then only the second possibility works: x = 25 and k = −2.", "topic": "number_properties"}
{"id": "book 2_p329_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 329, "page_end": 329, "topic_guess": "number_theory", "text": "GMAT number properties note: The second and third possibilities both make this statement true, so it isn’t possible to determine a single value for x.", "topic": "number_properties"}
{"id": "book 2_p330_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 330, "page_end": 330, "topic_guess": "general", "text": "GMAT quant note: From the Pythagorean theorem, a2 + b2 = c2. Use the Square of a Difference template.", "topic": "general"}
{"id": "book 2_p331_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 331, "page_end": 331, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "book 2_p332_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 332, "page_end": 332, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "book 2_p333_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 333, "page_end": 333, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Representing Objects with Pictures Rubber Band Geometry Baseline Calculations for Averages Number Line Techniques for Statistics Problems.", "topic": "statistics"}
{"id": "book 2_p334_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 334, "page_end": 334, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Still, visual thinking is a powerful tool.", "topic": "word_problems"}
{"id": "book 2_p334_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 334, "page_end": 334, "topic_guess": "word_problems", "text": "GMAT word problem note: It can expand your comprehension of a topic. It may enable you to solve particular problem types more easily or “break through” on a difficult problem.", "topic": "word_problems"}
{"id": "book 2_p334_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 334, "page_end": 334, "topic_guess": "word_problems", "text": "GMAT word problem note: In fact, visualization is the only realistic way to approach certain problems. So it’s worth trying your hand with visual approaches.", "topic": "word_problems"}
{"id": "book 2_p334_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 334, "page_end": 334, "topic_guess": "word_problems", "text": "GMAT word problem note: In this chapter, we will discuss the following Visual Solution techniques:. Representing Objects with Pictures:.", "topic": "word_problems"}
{"id": "book 2_p334_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 334, "page_end": 334, "topic_guess": "word_problems", "text": "GMAT word problem note: Drawing a good picture will make the problem-solving process easier and less error-prone.", "topic": "word_problems"}
{"id": "book 2_p335_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 335, "page_end": 335, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Different “rubber band” scenarios according to those constraints and freedoms can o en help you solve the problem without doing any computation.", "topic": "statistics"}
{"id": "book 2_p335_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 335, "page_end": 335, "topic_guess": "statistics", "text": "GMAT statistics note: Baseline Calculations for Averages:. Visual techniques can help you compute averages (both basic and weighted) and can also foster a better understanding of those calculations.", "topic": "statistics"}
{"id": "book 2_p335_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 335, "page_end": 335, "topic_guess": "statistics", "text": "GMAT statistics note: Number Line Techniques for Statistics Problems:. You can solve a variety of common problems involving statistics by using a number line to visualize and manipulate the problem.", "topic": "statistics"}
{"id": "book 2_p336_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 336, "page_end": 336, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Sometimes it’s the only viable approach!", "topic": "word_problems"}
{"id": "book 2_p336_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 336, "page_end": 336, "topic_guess": "word_problems", "text": "GMAT word problem note: Even if you are good at visualizing objects in your head, draw the picture anyway. It’s just too easy to make a mistake on many of these questions.", "topic": "word_problems"}
{"id": "book 2_p336_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 336, "page_end": 336, "topic_guess": "word_problems", "text": "GMAT word problem note: Try-It #7-1 A rectangular wooden dowel measures 4 inches by 1 inch by 1 inch. If the dowel is painted on all surfaces and then cut into -inch cubes, what fraction of the resulting cube faces are painted?", "topic": "word_problems"}
{"id": "book 2_p337_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 337, "page_end": 337, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Total cubes = (4 inches × 2 cubes per inch) × (1 × 2) × (1 × 2) = 32 cubes Total cube faces = 32 cubes × 6 faces per cube = 192 faces total Now, consider the faces that were painted on the front and back of the dowel, the top and bottom of the dowel, and the ends of the dowel.", "topic": "percent"}
{"id": "book 2_p337_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 337, "page_end": 337, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Of course, there are other sides: the back, the bottom, and the other end. Now you can find the number of painted cubes:.", "topic": "percent"}
{"id": "book 2_p337_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 337, "page_end": 337, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Even if you can easily picture 3-D shapes and objects in your head, it is still better to draw a picture on your scrap paper.", "topic": "percent"}
{"id": "book 2_p338_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 338, "page_end": 338, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. The test will usually include among the incorrect answers numbers that you might get by losing track of your progress as you process the object in your mind.", "topic": "algebra"}
{"id": "book 2_p338_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 338, "page_end": 338, "topic_guess": "algebra", "text": "GMAT algebra note: This kind of process can also help you with questions that deal with the relative size of different objects. Try-It #7-2 Bucket A has twice the capacity of Bucket B, and Bucket A has the capacity of Bucket C.", "topic": "algebra"}
{"id": "book 2_p338_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 338, "page_end": 338, "topic_guess": "algebra", "text": "GMAT algebra note: Bucket B is full of water and Bucket C is half full of water. When the water from Bucket B is poured into Bucket C, Bucket C will be filled to what fraction of its capacity?", "topic": "algebra"}
{"id": "book 2_p338_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 338, "page_end": 338, "topic_guess": "algebra", "text": "GMAT algebra note: You could attempt to solve this problem algebraically, but the equations get messy very quickly. Instead, try drawing buckets A, B, and C in correct proportion to one another.", "topic": "algebra"}
{"id": "book 2_p338_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 338, "page_end": 338, "topic_guess": "algebra", "text": "GMAT algebra note: Then think through the problem.", "topic": "algebra"}
{"id": "book 2_p339_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 339, "page_end": 339, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Algebra Picture The algebra and the picture say the same thing, but the picture has several advantages:.", "topic": "algebra"}
{"id": "book 2_p339_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 339, "page_end": 339, "topic_guess": "algebra", "text": "GMAT algebra note: Based on this picture, you might pick a capacity of 1 for Bucket B, yielding a capacity of 2 for Bucket A and 6 for Bucket C. Bucket B would contain 1 unit of water and Bucket C, 3 units.", "topic": "algebra"}
{"id": "book 2_p339_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 339, "page_end": 339, "topic_guess": "algebra", "text": "GMAT algebra note: It’s much easier to comprehend at a glance. It’s harder to mistake relative sizes (e.g., accidentally thinking A is smallest).", "topic": "algebra"}
{"id": "book 2_p339_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 339, "page_end": 339, "topic_guess": "algebra", "text": "GMAT algebra note: You can easily represent both total capacity and amount of water visually. It prompts you to pursue the smartest, easiest solution: picking numbers for the capacities of the buckets.", "topic": "algebra"}
{"id": "book 2_p340_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 340, "page_end": 340, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. Notice that the buckets are not labeled in alphabetical order, even though that would be easy to incorrectly assume.", "topic": "number_properties"}
{"id": "book 2_p340_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 340, "page_end": 340, "topic_guess": "general", "text": "GMAT quant note: The GMAT frequently adds little layers of disguise and complexity such as this to induce you to make a mistake. By drawing the buckets carefully, you minimize the chance that you will fall into a trap on a problem such as this one.", "topic": "number_properties"}
{"id": "book 2_p341_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 341, "page_end": 341, "topic_guess": "geometry", "text": "GMAT geometry note: We call these questions Rubber Band Geometry problems, because they simultaneously involve constraints and flexibility. Your job is to figure out what specifics in the problem are constrained and what specifics are flexible.", "topic": "geometry"}
{"id": "book 2_p341_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 341, "page_end": 341, "topic_guess": "geometry", "text": "GMAT geometry note: In fact, it will always “rise” 2 units for every unit of “run.” However, you don’t know where the line will appear. The line is constrained in its slope, but it is flexible in that it can be moved up or down, right or le.", "topic": "geometry"}
{"id": "book 2_p341_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 341, "page_end": 341, "topic_guess": "geometry", "text": "GMAT geometry note: You can draw many different lines with a slope of 2 (these lines will all be parallel, of course). If, however, the problem only specifies that a line must go through the point (4, 0) in the coordinate plane, then the line is “fixed” at that point.", "topic": "geometry"}
{"id": "book 2_p341_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 341, "page_end": 341, "topic_guess": "geometry", "text": "GMAT geometry note: However, the slope of the line would now be flexible. You could draw many different lines with different slopes that run through that point.", "topic": "geometry"}
{"id": "book 2_p341_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 341, "page_end": 341, "topic_guess": "geometry", "text": "GMAT geometry note: If you knew both of these specifications—that the line must have a slope of 2 and must run through the point (.", "topic": "geometry"}
{"id": "book 2_p342_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 342, "page_end": 342, "topic_guess": "algebra", "text": "GMAT algebra note: Calculate the exact line that is being described. The slope of the line and a point that the line goes through specify the line precisely—there is no remaining flexibility for either the slope or the location of the line.", "topic": "geometry"}
{"id": "book 2_p342_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 342, "page_end": 342, "topic_guess": "algebra", "text": "GMAT algebra note: In a DS problem, you could answer any such questions about this line without actually calculating the answer. Therefore for these types of problems, your goal is to figure out what combination of information “cements” the problem in place—in other words, what combination of information removes all of the remaining flexibility.", "topic": "geometry"}
{"id": "book 2_p342_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 342, "page_end": 342, "topic_guess": "algebra", "text": "GMAT algebra note: No flexibility means sufficiency. And by using rubber band geometry thinking, you can o en do this without using any calculation or algebra at all.", "topic": "geometry"}
{"id": "book 2_p342_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 342, "page_end": 342, "topic_guess": "algebra", "text": "GMAT algebra note: For each piece of information that you’re given in this type of problem, think about what is fixed and what is flexible. Try to draw multiple versions of each object (if possible), testing the boundaries of this flexibility.", "topic": "geometry"}
{"id": "book 2_p342_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 342, "page_end": 342, "topic_guess": "algebra", "text": "GMAT algebra note: The following everyday objects may be useful as analogies in your thinking:.", "topic": "geometry"}
{"id": "book 2_p343_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 343, "page_end": 343, "topic_guess": "general", "text": "GMAT quant note: Constrained Flexible Analogous Object(s) Mental Picture/Simplified Sketch A line passes through a specified point. Slope of the line Drinking straw = line Thumbtack = point Line free to spin about a point.", "topic": "geometry"}
{"id": "book 2_p343_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 343, "page_end": 343, "topic_guess": "general", "text": "GMAT quant note: Two lines intersect at a specified point. Slope of the lines Drinking straw = line Thumbtack = point Both lines free to spin about a point.", "topic": "geometry"}
{"id": "book 2_p343_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 343, "page_end": 343, "topic_guess": "general", "text": "GMAT quant note: The angle is free to change. A line passes through two points.", "topic": "geometry"}
{"id": "book 2_p343_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 343, "page_end": 343, "topic_guess": "general", "text": "GMAT quant note: Nothing flexible Drinking straw = line Thumbtacks = points Two points pin down a line—no flexibility.", "topic": "geometry"}
{"id": "book 2_p344_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 344, "page_end": 344, "topic_guess": "word_problems", "text": "GMAT word problem note: Constrained Flexible Analogous Object(s) Mental Picture/Simplified Sketch Specified distance between two points Absolute or relative location of the points Thumbtacks = points Drinking straw = distance between points Fix one point temporarily. Line (straw) free to spin about one point, tracing the circle of possible locations of the other point.", "topic": "geometry"}
{"id": "book 2_p344_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 344, "page_end": 344, "topic_guess": "word_problems", "text": "GMAT word problem note: Slope of a line Location of the line, or points the line may pass through Drinking straw = line Line is free to “float around” but not rotate. Points are on a line (either in the coordinate plane or on a basic number line).", "topic": "geometry"}
{"id": "book 2_p344_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 344, "page_end": 344, "topic_guess": "word_problems", "text": "GMAT word problem note: Distance between the points Rubber band = stretched between points “Stretchy” distance between points.", "topic": "geometry"}
{"id": "book 2_p345_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 345, "page_end": 345, "topic_guess": "geometry", "text": "GMAT geometry note: Constrained Flexible Analogous Object(s) Mental Picture/Simplified Sketch Points on a line are a specified distance apart.", "topic": "geometry"}
{"id": "book 2_p345_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 345, "page_end": 345, "topic_guess": "geometry", "text": "GMAT geometry note: Order of the points (le -to- middle- to-right) Drinking straws = fixed lengths between points Manipulation will be determined by other stated constraints, but thinking of the points as the endpoints of rigid straws will ensure that you do not forget the distance constraint. Lines could be laid end- to-end, overlapping, separated, and flipped right-to-le.", "topic": "geometry"}
{"id": "book 2_p345_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 345, "page_end": 345, "topic_guess": "geometry", "text": "GMAT geometry note: Triangle with a fixed area and a fixed base (and therefore a fixed height) Position of the third vertex along a line parallel to the base Straw = fixed base Rubber bands = other sides of the triangle Thumbtacks = endpoints of the base This is not intended to be an exhaustive list. The idea is to show you a new way of thinking through some difficult Geometry problems.", "topic": "geometry"}
{"id": "book 2_p346_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 346, "page_end": 346, "topic_guess": "geometry", "text": "GMAT geometry note: Try-It #7-3 A circle in a coordinate plane has a center at point A and a diameter of 6. If points B and C also lie in the same coordinate plane, is point B inside the circle?", "topic": "geometry"}
{"id": "book 2_p346_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 346, "page_end": 346, "topic_guess": "geometry", "text": "GMAT geometry note: The exact locations of points A, B, and C do not matter—only the relative locations of the points matter. Therefore, you can arbitrarily assign point A to a specific location (when possible, choose the origin of the coordinate plane) and draw a circle with a radius of 3 units around it.", "topic": "geometry"}
{"id": "book 2_p346_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 346, "page_end": 346, "topic_guess": "geometry", "text": "GMAT geometry note: However, it does constrain point B to be exactly 2 units away from wherever point C is. You could imagine point C at the center of a circle of size 2, with point B somewhere on the The distance between point A and point C equals 2.(1) The distance between point B and point C equals 2.(2).", "topic": "geometry"}
{"id": "book 2_p347_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 347, "page_end": 347, "topic_guess": "geometry", "text": "GMAT geometry note: (1) The distance between A and C is (2) The distance between B and C is circle around it (and this circle could pick up and move anywhere on the coordinate plane). Finally, combine these two statements to see that depending on where point C is drawn, point B may be inside the dotted circle, and it may not be.", "topic": "geometry"}
{"id": "book 2_p347_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 347, "page_end": 347, "topic_guess": "geometry", "text": "GMAT geometry note: As long as you can achieve a visual proof of the answer, you don’t need to prove it algebraically. That’s what rubber band geometry is all about: testing scenarios for Geometry problems without the need to plug in numbers or use algebra.", "topic": "geometry"}
{"id": "book 2_p347_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 347, "page_end": 347, "topic_guess": "geometry", "text": "GMAT geometry note: All you need is a visual environment that can be manipulated—one that preserves all key constraints and freedoms in the problem and allows you to see and test them.", "topic": "geometry"}
{"id": "book 2_p348_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 348, "page_end": 348, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Baseline Calculations for Averages BASIC AVERAGES Try-It #7-4 What is the average (arithmetic mean) of 387, 388, and 389?", "topic": "statistics"}
{"id": "book 2_p348_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 348, "page_end": 348, "topic_guess": "statistics", "text": "GMAT statistics note: Without even calculating, you may be able to see that the average is 388. How did you arrive at that answer?", "topic": "statistics"}
{"id": "book 2_p348_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 348, "page_end": 348, "topic_guess": "statistics", "text": "GMAT statistics note: It’s very unlikely that you calculated the average the classical way:. You probably noticed that the numbers are very close together and evenly spaced: 387 is 1 less than 388, and 389 is 1 greater than 388.", "topic": "statistics"}
{"id": "book 2_p348_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 348, "page_end": 348, "topic_guess": "statistics", "text": "GMAT statistics note: Thus, the average must be 388—right in the middle. Whether you realize it or not, you’re using a relatively advanced technique to solve this problem: a baseline calculation.", "topic": "statistics"}
{"id": "book 2_p348_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 348, "page_end": 348, "topic_guess": "statistics", "text": "GMAT statistics note: The baseline in this case is 388—the middle number. This concept can be applied to more difficult calculations of averages, making the calculation process much easier.", "topic": "statistics"}
{"id": "book 2_p348_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 348, "page_end": 348, "topic_guess": "statistics", "text": "GMAT statistics note: Try-It #7-5 A consumer finds that five bags of popcorn contain 257, 261, 273, 280, and 259 corn kernels per bag, respectively. What is the average (arithmetic mean) number of corn kernels per bag of popcorn?", "topic": "statistics"}
{"id": "book 2_p349_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 349, "page_end": 349, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. First, note that all of the bags have at least 257 kernels, so the average must be greater than 257.", "topic": "statistics"}
{"id": "book 2_p349_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 349, "page_end": 349, "topic_guess": "statistics", "text": "GMAT statistics note: First, consider how much each term differs from 257. The biggest numbers rise the highest.", "topic": "statistics"}
{"id": "book 2_p349_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 349, "page_end": 349, "topic_guess": "statistics", "text": "GMAT statistics note: A number equal to the baseline has no height. The height of the column thus represents the difference between the number and the baseline value:.", "topic": "statistics"}
{"id": "book 2_p349_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 349, "page_end": 349, "topic_guess": "statistics", "text": "GMAT statistics note: Calculate the sum of the differences: 0 + 4 + 16 + 23 + 2 = 45. Divide by the number of terms: the average difference is 45 ÷ 5 = 9.", "topic": "statistics"}
{"id": "book 2_p349_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 349, "page_end": 349, "topic_guess": "statistics", "text": "GMAT statistics note: Therefore, the average number of kernels per bag equals baseline + average difference: 257 + 9 = 266. Simply put, a baseline picture is a column chart.", "topic": "statistics"}
{"id": "book 2_p349_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 349, "page_end": 349, "topic_guess": "statistics", "text": "GMAT statistics note: The columns don’t show the actual value of any number—rather, they show the difference between the baseline and the number. The baseline can be any convenient number.", "topic": "statistics"}
{"id": "book 2_p349_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 349, "page_end": 349, "topic_guess": "statistics", "text": "GMAT statistics note: The smallest term in the set The largest term in the set The median term in the set A round number near the range of values.", "topic": "statistics"}
{"id": "book 2_p350_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 350, "page_end": 350, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. For sets with apparent symmetry, choosing a baseline in the middle is a good way not only to confirm the symmetry, but also to compute the average.", "topic": "statistics"}
{"id": "book 2_p350_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 350, "page_end": 350, "topic_guess": "statistics", "text": "GMAT statistics note: In this scenario, represent numbers lower than the baseline with columns that drop below the baseline. As before, the size of the column represents the difference between the number and the baseline.", "topic": "statistics"}
{"id": "book 2_p350_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 350, "page_end": 350, "topic_guess": "statistics", "text": "GMAT statistics note: Use trial and error to pick a possible average baseline, then adjust the drawing and calculations if necessary. Try-It #7-6 If a small business paid quarterly taxes last year of $10,079, $10,121, $10,112, and $10,088, what was the average (arithmetic mean) quarterly tax payment last year?", "topic": "statistics"}
{"id": "book 2_p350_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 350, "page_end": 350, "topic_guess": "statistics", "text": "GMAT statistics note: In this case, some of the numbers are below $10,100 and others are above $10,100, so $10,100 is a natural first guess:. If the baseline is the average, then the sum of the differences from the baseline will be zero.", "topic": "statistics"}
{"id": "book 2_p350_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 350, "page_end": 350, "topic_guess": "statistics", "text": "GMAT statistics note: Since the differences from the baseline do in fact sum to zero, $10,100 is indeed the average of this set.", "topic": "statistics"}
{"id": "book 2_p351_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 351, "page_end": 351, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Some sets may have many terms, but each of those terms has one of only two possible values.", "topic": "statistics"}
{"id": "book 2_p351_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 351, "page_end": 351, "topic_guess": "statistics", "text": "GMAT statistics note: Rather than add each individual term together, simplify the calculations by using a weighted average calculation. For weighted averages, use visualization to advance your understanding of the math.", "topic": "statistics"}
{"id": "book 2_p351_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 351, "page_end": 351, "topic_guess": "statistics", "text": "GMAT statistics note: Two real-life analogies can make it easier to remember how the relative weights of high and low values determine where the weighted average falls. Teeter-Totter Hanging Scale These pictures represent the values in a set as horizontal positions—le -to-right, as on a number line, not as vertical columns.", "topic": "statistics"}
{"id": "book 2_p351_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 351, "page_end": 351, "topic_guess": "statistics", "text": "GMAT statistics note: (It may help to imagine the balance beam or lever marked off in equal units like a number line.) Each pin or weight corresponds to the presence of a value in a set. The X in each picture marks the equilibrium point—in other words, the weighted average.", "topic": "statistics"}
{"id": "book 2_p351_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 351, "page_end": 351, "topic_guess": "statistics", "text": "GMAT statistics note: Both visual interpretations regard weighted averages as a kind of balancing act: the weighted average will be closer to the end of the range that has more weight. In the pictures, there are more instances of the le -hand side number in the set, so the X is relatively closer to the le -hand side.", "topic": "statistics"}
{"id": "book 2_p351_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 351, "page_end": 351, "topic_guess": "statistics", "text": "GMAT statistics note: You can think of this point as the point where the weight would be “balanced” between the two sets. Try-It #7-7 A convenience store stocks soda in 12-ounce and.", "topic": "statistics"}
{"id": "book 2_p352_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 352, "page_end": 352, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Note that 22 is much closer to 24 than to 12.", "topic": "statistics"}
{"id": "book 2_p352_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 352, "page_end": 352, "topic_guess": "statistics", "text": "GMAT statistics note: This implies that there will be many more 24-ounce bottles than 12-ounce bottles. Because the question asks about 12-ounce bottles, you could strategically eliminate any answer greater than or equal to Use your understanding of weighted averages as a balancing act to work backwards from the weighted average to the ratio of high-to-low terms:.", "topic": "statistics"}
{"id": "book 2_p352_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 352, "page_end": 352, "topic_guess": "statistics", "text": "GMAT statistics note: The total range between the high and low values (24 and 12) is 12 units. Mark off the distance from each end to the average of 22.", "topic": "statistics"}
{"id": "book 2_p352_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 352, "page_end": 352, "topic_guess": "statistics", "text": "GMAT statistics note: Because the weighted average is closer to 24, that side of the teeter-totter is assigned the greater weight of 10 out of 12. The other side, 12, is assigned the smaller weight of 2 out of 12.", "topic": "statistics"}
{"id": "book 2_p352_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 352, "page_end": 352, "topic_guess": "statistics", "text": "GMAT statistics note: Therefore, the 24-ounce bottles constitute of the total number of bottles. The 12-ounce bottles constitute of the total number of bottles.", "topic": "statistics"}
{"id": "book 2_p352_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 352, "page_end": 352, "topic_guess": "statistics", "text": "GMAT statistics note: When using this technique, it is important to remember that the weighted average is closer to (i.e., fewer units away from) the side that has greater weight, so that side should always be assigned the higher fraction. It is easy to reverse this logic accidentally when solving a weighted average problem with this technique, so be very careful!", "topic": "statistics"}
{"id": "book 2_p352_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 352, "page_end": 352, "topic_guess": "statistics", "text": "GMAT statistics note: Just remember: the weighted average point will be closer to the side with more weight.", "topic": "statistics"}
{"id": "book 2_p353_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 353, "page_end": 353, "topic_guess": "word_problems", "text": "GMAT word problem note: Percent problems usually relate part, whole, rate, and percent change. The citrusade is more than 45% water.", "topic": "percent"}
{"id": "book 2_p353_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 353, "page_end": 353, "topic_guess": "word_problems", "text": "GMAT word problem note: The lemonade is 50% water and the limeade is 40% water, so the lemonade must be more heavily weighted:. Therefore, the lemonade must make up more than 50% of the mixture.", "topic": "percent"}
{"id": "book 2_p353_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 353, "page_end": 353, "topic_guess": "word_problems", "text": "GMAT word problem note: Next, the citrusade is more than 24% sugar. The lemonade is 20% sugar and the limeade is 32% sugar, so what else can you figure out about the relative weighting?", "topic": "percent"}
{"id": "book 2_p353_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 353, "page_end": 353, "topic_guess": "word_problems", "text": "GMAT word problem note: Since lemonade is more heavily weighted (the mixture is to the right of 50/50), the weighting of the mixture is somewhere between 24% and the 50/50 weighting of the sugar, 26%. What weighting does the 24% figure represent?", "topic": "percent"}
{"id": "book 2_p353_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 353, "page_end": 353, "topic_guess": "word_problems", "text": "GMAT word problem note: The weighting is at most lemonade, or . The weighting, then, must be between and lemonade and the rest limeade.", "topic": "percent"}
{"id": "book 2_p354_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 354, "page_end": 354, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread.", "topic": "statistics"}
{"id": "book 2_p355_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 355, "page_end": 355, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Number Line Techniques for Statistics Problems Several other common types of questions involving statistics can be solved with visualization.", "topic": "statistics"}
{"id": "book 2_p355_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 355, "page_end": 355, "topic_guess": "statistics", "text": "GMAT statistics note: Specifically, using a number line can help simplify the work for many of these problems. MEDIAN RELATIVE TO MEAN Most questions involving the term median are really asking about the order of terms in a set: you line up the terms in a set in order of size, then select the middle term.", "topic": "statistics"}
{"id": "book 2_p355_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 355, "page_end": 355, "topic_guess": "statistics", "text": "GMAT statistics note: By contrast, the average, or arithmetic mean, is the sum of all of the terms divided by the number of terms. It can be visualized as the balancing point of all the terms laid out on the number line, as in the discussion of the balancing point for weighted averages in the previous section.", "topic": "statistics"}
{"id": "book 2_p355_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 355, "page_end": 355, "topic_guess": "statistics", "text": "GMAT statistics note: This technique is similar to rubber band geometry discussed earlier in this chapter, except this technique applies to problems involving sets rather than problems involving the coordinate plane. In this technique, you must place fixed terms in order from least to greatest (as you would on a number line), then move variable terms around according to the constraints.", "topic": "statistics"}
{"id": "book 2_p355_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 355, "page_end": 355, "topic_guess": "statistics", "text": "GMAT statistics note: By doing so, you can visualize what impact these changes hav.", "topic": "statistics"}
{"id": "book 2_p356_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 356, "page_end": 356, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. The median depends on where n falls relative to −2 and 4: below −2, between −2 and 4, or above 4.", "topic": "statistics"}
{"id": "book 2_p356_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 356, "page_end": 356, "topic_guess": "statistics", "text": "GMAT statistics note: One approach to this question is to think through the potential answers for all possible n values in the likely range (you can glance at the statements and other values in the list to get a sense of the relevant range) and draw out the scenarios on a number line. Try numbers around the relevant numbers, including a smaller and larger one at the far ends of the range.", "topic": "statistics"}
{"id": "book 2_p356_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 356, "page_end": 356, "topic_guess": "statistics", "text": "GMAT statistics note: Note that, by definition, all numbers in a set are different, so n cannot be −2 or 4:. This requires a fair amount of up-front work, but evaluating the statements is fast as a result.", "topic": "statistics"}
{"id": "book 2_p356_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 356, "page_end": 356, "topic_guess": "statistics", "text": "GMAT statistics note: By contrast, if n = 10, the mean would equal 4 and the median would still equal 4. By contrast, if n = −8, the mean would equal −2 and the median would still equal −2.", "topic": "statistics"}
{"id": "book 2_p356_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 356, "page_end": 356, "topic_guess": "statistics", "text": "GMAT statistics note: Taken together, however, any number in the range of 2 < n < 3 would feature a median greater than the mean.", "topic": "statistics"}
{"id": "book 2_p357_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 357, "page_end": 357, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Notice from this problem that as you move the variable terms, the mean always changes when the value of the variable terms change, but the median typically changes in jumps.", "topic": "statistics"}
{"id": "book 2_p357_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 357, "page_end": 357, "topic_guess": "statistics", "text": "GMAT statistics note: The median can get stuck while the number you’re changing doesn’t affect which number is in the middle. CHANGES IN STANDARD DEVIATION The GMAT will rarely (if ever) ask you to calculate the standard deviation of a list of numbers.", "topic": "statistics"}
{"id": "book 2_p357_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 357, "page_end": 357, "topic_guess": "statistics", "text": "GMAT statistics note: However, the exam will expect you to have some intuition about standard deviations. One way in which the GMAT might test your intuitive knowledge of standard deviations is by changing numbers within a list and asking you what the impact on standard deviation would be.", "topic": "statistics"}
{"id": "book 2_p357_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 357, "page_end": 357, "topic_guess": "statistics", "text": "GMAT statistics note: The relationship is relatively straightforward:. You might also see the term variance, which is also a measure of the spread of numbers in a set or list.", "topic": "statistics"}
{"id": "book 2_p357_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 357, "page_end": 357, "topic_guess": "statistics", "text": "GMAT statistics note: Variance and standard deviation indicate the same information. A variance of 0 indicates that all of the numbers are identical, as does a standard deviation of 0.", "topic": "statistics"}
{"id": "book 2_p357_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 357, "page_end": 357, "topic_guess": "statistics", "text": "GMAT statistics note: The larger the variance, or standard deviation, the more the numbers are spread out. (Variance and standard deviation are never negative.) Try-It #7-10 Last Year 9 9.5 10 10 11 11 11 11 11 12.5 13 13 This Year 9 x 10 10 11 11 11 11 11 y 13 13 Moving terms away from the mean increases t.", "topic": "statistics"}
{"id": "book 2_p358_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 358, "page_end": 358, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. The monthly sales (in thousands of $) at a certain restaurant for the past two years are given in the chart above.", "topic": "statistics"}
{"id": "book 2_p358_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 358, "page_end": 358, "topic_guess": "statistics", "text": "GMAT statistics note: If the standard deviation of the monthly sales is greater this year than last year, which of the following are possible values for x and y? Except for x and y, the two lists of monthly sales numbers are identical, so focus exclusively on those terms that changed: 9.5 and 12.5 from last year were replaced by x and y this year.", "topic": "statistics"}
{"id": "book 2_p358_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 358, "page_end": 358, "topic_guess": "statistics", "text": "GMAT statistics note: If this year’s standard deviation is greater, then this year’s numbers must be more spread out from the mean than last year’s. The numbers are close enough together to indicate that the average should be somewhere around 11.", "topic": "statistics"}
{"id": "book 2_p358_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 358, "page_end": 358, "topic_guess": "statistics", "text": "GMAT statistics note: Visually, here are the interesting terms from last year:. This problem does not require actual computation of the standard deviation using the new x and y values.", "topic": "statistics"}
{"id": "book 2_p358_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 358, "page_end": 358, "topic_guess": "statistics", "text": "GMAT statistics note: The math would be too complex to complete in two minutes. Instead, determine visually which x and y values increase the standard deviation: the pair of x and y values that are farther from the mean than are 9.5 and 12.5 will increase the standard deviation.", "topic": "statistics"}
{"id": "book 2_p359_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 359, "page_end": 359, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Only in this case is the deviation greater.", "topic": "statistics"}
{"id": "book 2_p359_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 359, "page_end": 359, "topic_guess": "statistics", "text": "GMAT statistics note: The GMAT may also test you on standard deviations by adding numbers to a list. When new terms are added, the GMAT will o en ask you to compare the old list to the new list, or to compare various options for the new list, or to do both.", "topic": "statistics"}
{"id": "book 2_p359_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 359, "page_end": 359, "topic_guess": "statistics", "text": "GMAT statistics note: You must have a technique to evaluate the standard deviation of different lists relative to one another. Again, pictures make for great comparison tools!", "topic": "statistics"}
{"id": "book 2_p359_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 359, "page_end": 359, "topic_guess": "statistics", "text": "GMAT statistics note: Try-It #7-11 A list of 12 test scores has an average (arithmetic mean) of 500 and a standard deviation of 50. Which of the following lists of additional test scores, when combined with the original list of 12 test scores, must result in a combined list with a standard deviation less than 50?", "topic": "statistics"}
{"id": "book 2_p360_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 360, "page_end": 360, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. It is not generally true that all of the terms in a list are within 1 standard deviation of the mean.", "topic": "statistics"}
{"id": "book 2_p360_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 360, "page_end": 360, "topic_guess": "statistics", "text": "GMAT statistics note: However, standard deviation is a measure of the spread of the terms of a list, so you could represent the original list of scores this way:. The oval spans ±1 standard deviation from the mean, where many of the scores will likely be.", "topic": "statistics"}
{"id": "book 2_p360_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 360, "page_end": 360, "topic_guess": "statistics", "text": "GMAT statistics note: This simplification is acceptable as long as you represent all of the other lists the same way so that you can compare the relative effects of the new test scores systematically.", "topic": "statistics"}
{"id": "book 2_p361_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 361, "page_end": 361, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. In general, these are the rules for adding a single term to a list:.", "topic": "statistics"}
{"id": "book 2_p361_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 361, "page_end": 361, "topic_guess": "statistics", "text": "GMAT statistics note: Note that mathematically this is a slight oversimplification, but for the purpose of adding terms to a list of numbers on the GMAT, you can accept this simplification as true. FLOATING TERMS IN A SET On GMAT Statistics problems involving elements (i.e., terms) in a list, you can usually focus your attention on a single term or two.", "topic": "statistics"}
{"id": "book 2_p361_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 361, "page_end": 361, "topic_guess": "statistics", "text": "GMAT statistics note: These terms could be considered the floating terms—the terms that are unknown or not completely defined among a list of more clearly defined terms. As you approach a question of this type, try to rephrase the question quickly so that you focus on the unknown, or floating, terms rather than on the known terms.", "topic": "statistics"}
{"id": "book 2_p361_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 361, "page_end": 361, "topic_guess": "statistics", "text": "GMAT statistics note: Try-It #7-12 List A contains 5 positive integers, and the average (arithmetic mean) of the integers in the list is 7. If the integers 6, 7, and 8 are in List A, what is the range of List A?", "topic": "statistics"}
{"id": "book 2_p361_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 361, "page_end": 361, "topic_guess": "statistics", "text": "GMAT statistics note: Adding a new term more than 1 standard deviation from the mean generally increases the standard deviation. Adding a new term less than 1 standard deviation from the mea.", "topic": "statistics"}
{"id": "book 2_p362_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 362, "page_end": 362, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The average of all five integers in the list is 7.", "topic": "number_properties"}
{"id": "book 2_p362_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 362, "page_end": 362, "topic_guess": "number_theory", "text": "GMAT number properties note: Three of the integers in the list are given (6, 7, and 8), and they all have an average of 7. Therefore, the floating terms in this problem must also have an average of 7.", "topic": "number_properties"}
{"id": "book 2_p362_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 362, "page_end": 362, "topic_guess": "number_theory", "text": "GMAT number properties note: Assign x and y to represent these terms:. The rephrased question is thus, “Given that x + y = 14, what is either x or y?” Once you know one of the values, you can solve for the other and thereby determine the range of the list.", "topic": "number_properties"}
{"id": "book 2_p362_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 362, "page_end": 362, "topic_guess": "number_theory", "text": "GMAT number properties note: If 3 is one of the unknown integers, the other must be 11. The range is thus 11 − 3 = 8.", "topic": "number_properties"}
{"id": "book 2_p362_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 362, "page_end": 362, "topic_guess": "number_theory", "text": "GMAT number properties note: This statement might seem a little too vague to be sufficient, but by visually listing the possible pairs that add up to 14, you can rule out pairs that don’t fit the constraint from this statement:. Notice that the pairings represent the constraint x + y = 14.", "topic": "number_properties"}
{"id": "book 2_p362_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 362, "page_end": 362, "topic_guess": "number_theory", "text": "GMAT number properties note: Visually, this means that x and y are balanced around 7. The largest term in List A is greater than 3 times and less than 4 times the size of the smallest term.", "topic": "number_properties"}
{"id": "book 2_p363_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 363, "page_end": 363, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Only one pair of integers results in a ratio strictly between 3 and 4.", "topic": "word_problems"}
{"id": "book 2_p363_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 363, "page_end": 363, "topic_guess": "word_problems", "text": "GMAT word problem note: The unknown terms must therefore be 3 and 11, and the range is 11 − 3 = 8. In this problem, the constraint x + y = 14 is a fixed sum.", "topic": "word_problems"}
{"id": "book 2_p363_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 363, "page_end": 363, "topic_guess": "word_problems", "text": "GMAT word problem note: Another common constraint is a fixed difference, such as a − b = 2. A fixed difference can be represented visually as a fixed distance between a and b on the number line, with a to the right because it is larger.", "topic": "word_problems"}
{"id": "book 2_p363_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 363, "page_end": 363, "topic_guess": "word_problems", "text": "GMAT word problem note: That distance could move le or right:. MAXIMIZING (OR MINIMIZING) ONE TERM Another visual technique for statistics involves maximizing (or minimizing) the value of a term in a set or list of numbers, subject to some constraints.", "topic": "word_problems"}
{"id": "book 2_p363_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 363, "page_end": 363, "topic_guess": "word_problems", "text": "GMAT word problem note: Such problems will usually employ the word maximum or minimum. For these problems, you o en should maximize (or minimize) the term by minimizing (or maximizing) the other terms, because the constraints usually involve mathematical trade-offs.", "topic": "word_problems"}
{"id": "book 2_p363_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 363, "page_end": 363, "topic_guess": "word_problems", "text": "GMAT word problem note: Try-It #7-13 8 is 1.33 times the size of 6 (the ratio is too low). 9 is 1.8 times the size of 5 (the ratio is too low).", "topic": "word_problems"}
{"id": "book 2_p363_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 363, "page_end": 363, "topic_guess": "word_problems", "text": "GMAT word problem note: 10 is 2.5 times the size of 4 (the ratio is too low). 11 is 3.66 times the size of 3 (an acceptable ratio).", "topic": "word_problems"}
{"id": "book 2_p363_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 363, "page_end": 363, "topic_guess": "word_problems", "text": "GMAT word problem note: 12 is 6 times the size of 2 (the ratio is too high).", "topic": "word_problems"}
{"id": "book 2_p364_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 364, "page_end": 364, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. In a certain lottery drawing, five balls are selected from a tumbler in which each ball is printed with a different two-digit positive integer.", "topic": "statistics"}
{"id": "book 2_p364_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 364, "page_end": 364, "topic_guess": "statistics", "text": "GMAT statistics note: If the average (arithmetic mean) of the five numbers drawn is 56 and the median is 60, what is the greatest value that the lowest number selected could be? The goal is to maximize the value of the lowest-numbered ball.", "topic": "statistics"}
{"id": "book 2_p364_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 364, "page_end": 364, "topic_guess": "statistics", "text": "GMAT statistics note: All balls contain a two- digit positive integer, and none of the balls have the same number. The problem provides enough information to calculate the sum and to lay out a visual listing of the numbers:.", "topic": "statistics"}
{"id": "book 2_p364_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 364, "page_end": 364, "topic_guess": "statistics", "text": "GMAT statistics note: In order to maximize the value of the first (lowest) number in the set, what do you need to do to the other numbers? You’d want to minimize them.", "topic": "statistics"}
{"id": "book 2_p364_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 364, "page_end": 364, "topic_guess": "statistics", "text": "GMAT statistics note: Select the smallest numbers that you can for the remaining slots:. In some problems, you might actually make different slots equal to each other (e.g., the three largest numbers could be 60, 60, and 60).", "topic": "statistics"}
{"id": "book 2_p365_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 365, "page_end": 365, "topic_guess": "algebra", "text": "GMAT algebra note: Track mean, median, range, distribution, and how operations change spread. The five numbers must sum to 280, so you can set up an algebraic equation and solve.", "topic": "statistics"}
{"id": "book 2_p365_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 365, "page_end": 365, "topic_guess": "algebra", "text": "GMAT algebra note: Can you think of a way to minimize the arithmetic needed to solve in that way? The three numerical values, 60, 61, and 62, are all larger than the average of 56.", "topic": "statistics"}
{"id": "book 2_p365_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 365, "page_end": 365, "topic_guess": "algebra", "text": "GMAT algebra note: Specifically, they are +4, +5, and +6 away from that average. The other two numbers, then, need to make up for that overage of 4 + 5 + 6 = 15:.", "topic": "statistics"}
{"id": "book 2_p365_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 365, "page_end": 365, "topic_guess": "algebra", "text": "GMAT algebra note: The numbers x and x + 1 are also consecutive, so they need to be −8 and −7 away from the average of 56. The two remaining numbers are 48 and 49.", "topic": "statistics"}
{"id": "book 2_p365_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 365, "page_end": 365, "topic_guess": "algebra", "text": "GMAT algebra note: Try-It #7-14 The average (arithmetic mean) of six numbers is 18 and the median of the six numbers is 16. What is the minimum possible value for the greatest number in the list?", "topic": "statistics"}
{"id": "book 2_p366_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 366, "page_end": 366, "topic_guess": "word_problems", "text": "GMAT word problem note: Track mean, median, range, distribution, and how operations change spread. This time, the goal is to minimize the largest number:.", "topic": "statistics"}
{"id": "book 2_p366_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 366, "page_end": 366, "topic_guess": "word_problems", "text": "GMAT word problem note: In order to minimize the final term, you’d want to maximize all of the other terms. This time, though, the list contains an even number of terms, so you can’t just set the median to the middle number.", "topic": "statistics"}
{"id": "book 2_p366_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 366, "page_end": 366, "topic_guess": "word_problems", "text": "GMAT word problem note: The two middle numbers average to 16. The pair (16, 16) averages to 16, as does the pair (15, 17).", "topic": "statistics"}
{"id": "book 2_p366_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 366, "page_end": 366, "topic_guess": "word_problems", "text": "GMAT word problem note: The first pair, though, is better when the goal is to minimize the final term, since the terms to the right have to be equal to or greater than the terms to the le.", "topic": "statistics"}
{"id": "book 2_p366_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 366, "page_end": 366, "topic_guess": "word_problems", "text": "GMAT word problem note: In other words, when trying to minimize the final term, it’s true that you want to maximize the earlier terms, but you also have to think about how to do so in a way that doesn’t make the final term too large (since it has to be larger than the earlier terms). In this case, the first four terms are all 16:.", "topic": "statistics"}
{"id": "book 2_p366_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 366, "page_end": 366, "topic_guess": "word_problems", "text": "GMAT word problem note: In order to minimize the final term, set the last two terms equal to each other and solve algebraically:. Or you can use the over/under approach.", "topic": "statistics"}
{"id": "book 2_p367_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 367, "page_end": 367, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Those last two terms have to make up for the −8 on the other side, so each x must be 4 over the average, or 22.", "topic": "statistics"}
{"id": "book 2_p367_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 367, "page_end": 367, "topic_guess": "statistics", "text": "GMAT statistics note: TAKING AND GIVING Another common scenario involves taking value from one term in a list and giving it to another term. The relative value of the terms in the list will change, leading to some interesting results.", "topic": "statistics"}
{"id": "book 2_p367_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 367, "page_end": 367, "topic_guess": "statistics", "text": "GMAT statistics note: If an integer number of Keri’s points were taken from her and given to Luke, and the median score of the five players increased, how many points were transferred from Keri to Luke? The key to this problem is that by taking enough points from Keri and giving them to Luke, the median of the list can change.", "topic": "statistics"}
{"id": "book 2_p368_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 368, "page_end": 368, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Order the scores from low to high on a number line, and represent the change in Luke’s score with x:.", "topic": "statistics"}
{"id": "book 2_p368_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 368, "page_end": 368, "topic_guess": "statistics", "text": "GMAT statistics note: In order for the median to change, Luke’s score must leap-frog those of Nora and Mia, pushing Mia into the bottom two scores and making Luke’s score the median. You don’t want to decrease Keri’s score so much that Luke and Mia surpass her, leaving Mia once again in the median score position.", "topic": "statistics"}
{"id": "book 2_p368_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 368, "page_end": 368, "topic_guess": "statistics", "text": "GMAT statistics note: If 15 + x = 38, Luke would match the current median score. That is x = 23, and Keri’s new score would be 63 − 23 = 40.", "topic": "statistics"}
{"id": "book 2_p368_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 368, "page_end": 368, "topic_guess": "statistics", "text": "GMAT statistics note: However, the median score would remain 38, with both Luke and Mia having that score. Therefore, x must be greater than 23.", "topic": "statistics"}
{"id": "book 2_p368_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 368, "page_end": 368, "topic_guess": "statistics", "text": "GMAT statistics note: Luke’s new point value is 15 + 24 = 39. Keri’s new point value is 63 − 24 = 39.", "topic": "statistics"}
{"id": "book 2_p368_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 368, "page_end": 368, "topic_guess": "statistics", "text": "GMAT statistics note: Both Nora and Mia are below Luke (and Keri), so the new median is 39:.", "topic": "statistics"}
{"id": "book 2_p369_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 369, "page_end": 369, "topic_guess": "geometry", "text": "GMAT geometry note: Does a rectangular mirror have an area greater than 10 square centimeters? The perimeter of the mirror is 24 cm.(1) The diagonal of the mirror is less than 11 cm.(2) 2.", "topic": "geometry"}
{"id": "book 2_p369_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 369, "page_end": 369, "topic_guess": "geometry", "text": "GMAT geometry note: A number line is numbered with the integers from 0 to 50 inclusive. An ant walks along the number line as follows:.", "topic": "geometry"}
{"id": "book 2_p369_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 369, "page_end": 369, "topic_guess": "geometry", "text": "GMAT geometry note: First, it walks in the positive direction until it reaches a multiple of 5 that it hasn’t previously reached. Then, it walks in the negative direction for at least 1 unit, stopping when it reaches any multiple of 2.", "topic": "geometry"}
{"id": "book 2_p369_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 369, "page_end": 369, "topic_guess": "geometry", "text": "GMAT geometry note: The ant starts at 0 on the number line and repeats this process until it first reaches the point marked 50. How many units does it travel in total?", "topic": "geometry"}
{"id": "book 2_p370_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 370, "page_end": 370, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If a, b, and c are positive, is On the number line, a is closer to b than it is to c.(1) b > c(2) 4.", "topic": "number_properties"}
{"id": "book 2_p370_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 370, "page_end": 370, "topic_guess": "number_theory", "text": "GMAT number properties note: The length of one edge of a cube equals 4. What is the distance between the center of the cube and one of its vertices?", "topic": "number_properties"}
{"id": "book 2_p370_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 370, "page_end": 370, "topic_guess": "number_theory", "text": "GMAT number properties note: A test is taken by 100 people and possible scores are the integers between 0 and 50, inclusive. For each of the following scenarios, determine whether the average (arithmetic mean) score would be greater than 30 (answer Yes, No, or Uncertain).", "topic": "number_properties"}
{"id": "book 2_p370_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 370, "page_end": 370, "topic_guess": "number_theory", "text": "GMAT number properties note: More than 70 people scored 40 or higher.a) 75 people scored 40 or higher.b) Fewer than 10 people scored 50.c) No more than 2 test-takers scored any given score.d).", "topic": "number_properties"}
{"id": "book 2_p371_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 371, "page_end": 371, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: As part of an experiment, a student repeatedly tests the temperature of a light bulb. The bar graph below displays the number of readings the student recorded at various temperatures, measured in degrees Fahrenheit.", "topic": "geometry"}
{"id": "book 2_p371_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 371, "page_end": 371, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: What was the (arithmetic mean) temperature reading of the light bulb? In a certain dance troupe, there are 55 women and 33 men.", "topic": "geometry"}
{"id": "book 2_p371_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 371, "page_end": 371, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: If all of the women are 62 inches tall and all of the men are 70 inches tall, what is the average height of the dancers in the troupe?", "topic": "geometry"}
{"id": "book 2_p372_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 372, "page_end": 372, "topic_guess": "word_problems", "text": "GMAT word problem note: Track mean, median, range, distribution, and how operations change spread. Anton and Lena start at the same point on a circular track, measuring 10 meters in diameter, and begin walking counterclockwise at the same time, with Anton walking more quickly than Lena.", "topic": "statistics"}
{"id": "book 2_p372_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 372, "page_end": 372, "topic_guess": "word_problems", "text": "GMAT word problem note: When Anton has traveled exactly halfway around the track, they both stop walking. They then observe that the distance between them along a straight line measures exactly meters, as shown above.", "topic": "statistics"}
{"id": "book 2_p372_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 372, "page_end": 372, "topic_guess": "word_problems", "text": "GMAT word problem note: What fraction of the track has Lena covered? Eddy, Mario, and Peter have $32, $72, and $98, respectively.", "topic": "statistics"}
{"id": "book 2_p372_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 372, "page_end": 372, "topic_guess": "word_problems", "text": "GMAT word problem note: They pool their money and redistribute the entire amount among themselves. If Eddy now has the median amount of money in the group, what is the greatest amount of money that Eddy could now have?", "topic": "statistics"}
{"id": "book 2_p373_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 373, "page_end": 373, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: The portion of the coordinate plane that lies within the square, but above the line, is then shaded as shown above. If A is the area of the shaded region in square units, which of the following specifies all the possible values of A?", "topic": "geometry"}
{"id": "book 2_p374_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 374, "page_end": 374, "topic_guess": "geometry", "text": "GMAT geometry note: Because the formula for the area of a rectangle is A = lw, rephrase the question. The area of a quadrilateral is maximized when the quadrilateral is a square, so first try l = w = 6.", "topic": "geometry"}
{"id": "book 2_p374_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 374, "page_end": 374, "topic_guess": "geometry", "text": "GMAT geometry note: In this case, the area is 36 cm2 and the answer to the question is Yes. If, on the other hand, l = 11.5 and w = 0.5, the perimeter is still 24, but the area is (11.5)(0.5) = 5.75 cm2.", "topic": "geometry"}
{"id": "book 2_p374_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 374, "page_end": 374, "topic_guess": "geometry", "text": "GMAT geometry note: In this case, the answer to the question is No. The diagonal of the rectangle is less than 11.", "topic": "geometry"}
{"id": "book 2_p374_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 374, "page_end": 374, "topic_guess": "geometry", "text": "GMAT geometry note: If the diagonal is less than 11, then the sides must also be less than 11. If l = w = 6, then the diagonal is shorter than 11 and, as last time, the area is 36.", "topic": "geometry"}
{"id": "book 2_p374_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 374, "page_end": 374, "topic_guess": "geometry", "text": "GMAT geometry note: In this case, the answer to the question is Yes. If, on the other hand, l = 3 and w = 1, the diagonal is less than 11, but the area is (3)(1) = 3 cm2.", "topic": "geometry"}
{"id": "book 2_p374_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 374, "page_end": 374, "topic_guess": "geometry", "text": "GMAT geometry note: The sides must be less than 11 and the perimeter must be 24. The case of the square still maximizes the area: l = w = 6 and the area is 36.", "topic": "geometry"}
{"id": "book 2_p374_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 374, "page_end": 374, "topic_guess": "geometry", "text": "GMAT geometry note: The largest possible length is just under 11, making the width just over 1. (<11) (>1) = something larger than 10.", "topic": "geometry"}
{"id": "book 2_p374_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 374, "page_end": 374, "topic_guess": "geometry", "text": "GMAT geometry note: The area must be greater than 10.", "topic": "geometry"}
{"id": "book 2_p375_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 375, "page_end": 375, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. The problem describes an ant walking along a number line in a complex way.", "topic": "word_problems"}
{"id": "book 2_p375_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 375, "page_end": 375, "topic_guess": "word_problems", "text": "GMAT word problem note: It starts at 0, then walks to the right until it reaches a multiple of Then, it walks to the le until it reaches a multiple of 2. The problem specifies that the ant must walk to the le at least one unit.", "topic": "word_problems"}
{"id": "book 2_p375_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 375, "page_end": 375, "topic_guess": "word_problems", "text": "GMAT word problem note: Therefore, even if it is already standing on a multiple of 2, it must walk to the next one. In this case, it walks le until it reaches 4:.", "topic": "word_problems"}
{"id": "book 2_p375_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 375, "page_end": 375, "topic_guess": "word_problems", "text": "GMAT word problem note: Then it walks to the right again until it reaches a new multiple of 5:. The question asks how far the ant walks, in total, to reach the number 50 for the first time.", "topic": "word_problems"}
{"id": "book 2_p375_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 375, "page_end": 375, "topic_guess": "word_problems", "text": "GMAT word problem note: Since this is too many units to realistically count by hand, there must be an underlying pattern. Write down the first few distances that the ant walks, using your number line.", "topic": "word_problems"}
{"id": "book 2_p376_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 376, "page_end": 376, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. There is a repeating cycle of four numbers.", "topic": "statistics"}
{"id": "book 2_p376_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 376, "page_end": 376, "topic_guess": "statistics", "text": "GMAT statistics note: Starting at 5 on the number line, the ant moves le 1, then right 6, then le 2, then right 7, which takes it to 15 on the number line. That is, the ant walks a total of 1 + 6 + 2 + 7 = 16 units to move from 5 to 15.", "topic": "statistics"}
{"id": "book 2_p376_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 376, "page_end": 376, "topic_guess": "statistics", "text": "GMAT statistics note: Then it walks another 16 units to move from 15 to 25, from 25 to 35, and from 35 to 45. In total, the ant walks 4(16) = 64 units to move from 5 to 45.", "topic": "statistics"}
{"id": "book 2_p376_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 376, "page_end": 376, "topic_guess": "statistics", "text": "GMAT statistics note: To get from 45 to 50, the ant first walks one unit le to 44, then six units right to 50. In total, the ant walks 64 + 1 + 6 = 71 units to move from 5 to 50.", "topic": "statistics"}
{"id": "book 2_p376_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 376, "page_end": 376, "topic_guess": "statistics", "text": "GMAT statistics note: Finally, add the five units it walks from the beginning: 71 + 5 = 76. It is also possible to estimate the answer.", "topic": "statistics"}
{"id": "book 2_p376_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 376, "page_end": 376, "topic_guess": "statistics", "text": "GMAT statistics note: As established above, traveling 10 points on the number line (from 15 to 25, from 25 to 35, and so on) requires the ant to travel 16 units. To begin at 0 and end at 50, the ant would travel approximately 5(16) = 80 units.", "topic": "statistics"}
{"id": "book 2_p376_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 376, "page_end": 376, "topic_guess": "statistics", "text": "GMAT statistics note: The problem indicates that a, b, and c are positive and asks whether a is greater than , which is the average (or arithmetic mean) of b and c. Draw a picture and rephrase the question: “On the number line, is a positioned to the right of the m.", "topic": "statistics"}
{"id": "book 2_p377_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 377, "page_end": 377, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. It isn’t clear whether b or c is the larger value—the point on the right—so a could be closer to the smaller number (making a less than the average) or the larger number (making a greater than the average).", "topic": "word_problems"}
{"id": "book 2_p377_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 377, "page_end": 377, "topic_guess": "word_problems", "text": "GMAT word problem note: This statement indicates nothing about a, so it can’t be sufficient. Together, the two statements indicate that b is the point on the right, so a must be on the right side of the midpoint.", "topic": "word_problems"}
{"id": "book 2_p377_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 377, "page_end": 377, "topic_guess": "word_problems", "text": "GMAT word problem note: The length of any side of the cube is 4, and the problem asks for the distance between the center of the cube and any of its vertices (corners). Chop up the cube into eight smaller cubes to see that the distance from the center of the 4 × 4 × 4 cube to any corner is the diagonal of a 2 × 2 × 2 cube.", "topic": "word_problems"}
{"id": "book 2_p378_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 378, "page_end": 378, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. You can find the diagonal of a cube in a variety of ways.", "topic": "statistics"}
{"id": "book 2_p378_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 378, "page_end": 378, "topic_guess": "statistics", "text": "GMAT statistics note: Probably the fastest (besides applying a memorized formula) is to use the “Super Pythagorean” theorem, which extends to three dimensions:. In the special case when the three sides of the box are equal, as they are in a cube, use this equation:.", "topic": "statistics"}
{"id": "book 2_p378_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 378, "page_end": 378, "topic_guess": "statistics", "text": "GMAT statistics note: To visualize this set of weighted averages problems, imagine a teeter-totter that is 50 meters long, marked off from 0 to 50 to represent scores. One hundred people of equal weight sit on the teeter-totter at their respective scores.", "topic": "statistics"}
{"id": "book 2_p378_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 378, "page_end": 378, "topic_guess": "statistics", "text": "GMAT statistics note: The weighted average is the position where the teeter-totter would balance. Thus, to answer whether the average score is greater than 30, take extremes according to the given conditions and see whether you can swing the balance to either side of 30 (or whether you are forced to balance on one side of 30 only).", "topic": "statistics"}
{"id": "book 2_p379_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 379, "page_end": 379, "topic_guess": "general", "text": "GMAT quant note: Track mean, median, range, distribution, and how operations change spread. More than 70 people scored 40 or higher.", "topic": "statistics"}
{"id": "book 2_p379_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 379, "page_end": 379, "topic_guess": "general", "text": "GMAT quant note: If all 100 scored 40 (or higher), then the average is above 30. If, on the other hand, 71 people scored 40 and the other 29 scored 0, does the average drop to 30 or lower?", "topic": "statistics"}
{"id": "book 2_p379_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 379, "page_end": 379, "topic_guess": "general", "text": "GMAT quant note: Match up the 29 people who scored 0 with 29 of the people who scored 40. These 58 people together have an average score of 20.", "topic": "statistics"}
{"id": "book 2_p379_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 379, "page_end": 379, "topic_guess": "general", "text": "GMAT quant note: The other 42 people have a score of 40. If you average these two groups, the average must be below 30, since 58 is larger than 42.", "topic": "statistics"}
{"id": "book 2_p379_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 379, "page_end": 379, "topic_guess": "general", "text": "GMAT quant note: Seventy-five people scored 40 or higher. If all 100 scored 40, then the average is higher than 30.", "topic": "statistics"}
{"id": "book 2_p379_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 379, "page_end": 379, "topic_guess": "general", "text": "GMAT quant note: If 75 scored 40 and the other 25 scored 0, then what? Match the 25 people who scored 0 with 25 people who scored 40.", "topic": "statistics"}
{"id": "book 2_p379_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 379, "page_end": 379, "topic_guess": "general", "text": "GMAT quant note: These 50 people have an average score of 20. The remaining 50 people have an average score of 40, so the overall average is exactly 30.", "topic": "statistics"}
{"id": "book 2_p379_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 379, "page_end": 379, "topic_guess": "general", "text": "GMAT quant note: It is possible, therefore, to have an average that is higher than 30 or an average that is not higher than 30.", "topic": "statistics"}
{"id": "book 2_p380_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 380, "page_end": 380, "topic_guess": "number_theory", "text": "GMAT number properties note: Track mean, median, range, distribution, and how operations change spread. Fewer than 10 people scored 50.", "topic": "statistics"}
{"id": "book 2_p380_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 380, "page_end": 380, "topic_guess": "number_theory", "text": "GMAT number properties note: Say that 1 person scored 50 and the other 99 scored 0. The average is definitely below 30.", "topic": "statistics"}
{"id": "book 2_p380_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 380, "page_end": 380, "topic_guess": "number_theory", "text": "GMAT number properties note: If on the other hand, 1 person scored 50 and the other 99 scored 40, then the average is definitely above 30. Each score was achieved by no more than 2 people.", "topic": "statistics"}
{"id": "book 2_p380_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 380, "page_end": 380, "topic_guess": "number_theory", "text": "GMAT number properties note: There are 51 integers between 0 and 50, inclusive. In other words, there are 102 possible scores to spread among the test-takers.", "topic": "statistics"}
{"id": "book 2_p380_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 380, "page_end": 380, "topic_guess": "number_theory", "text": "GMAT number properties note: Since there are 100 test-takers, almost every score is taken. The highest possible average will occur when nobody scores 0 points.", "topic": "statistics"}
{"id": "book 2_p380_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 380, "page_end": 380, "topic_guess": "number_theory", "text": "GMAT number properties note: If 2 test- takers score 1, 2 test-takers score 2, and so on up to 50, then the average score will be approximately 25. As this is the highest allowable average for this scenario, it’s impossible for the average to be greater than 30.", "topic": "statistics"}
{"id": "book 2_p380_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 380, "page_end": 380, "topic_guess": "number_theory", "text": "GMAT number properties note: The average looks like it’s in the 112° range. Calculate the over/under with 112° as the assumed baseline.", "topic": "statistics"}
{"id": "book 2_p380_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 380, "page_end": 380, "topic_guess": "number_theory", "text": "GMAT number properties note: The four 110° readings are each 2° below the baseline, and the three 111° readings are each 1° below the baseline, for a total of 11° below baseline. On the other side, the 113°, 114°, and 115° readings are a total of 11° above the baseline.", "topic": "statistics"}
{"id": "book 2_p380_c1_r9", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 380, "page_end": 380, "topic_guess": "number_theory", "text": "GMAT number properties note: The average is exactly 112°.", "topic": "statistics"}
{"id": "book 2_p381_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 381, "page_end": 381, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Because Anton has walked exactly halfway around the track, the distance between Anton and the starting point is the diameter.", "topic": "word_problems"}
{"id": "book 2_p381_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 381, "page_end": 381, "topic_guess": "word_problems", "text": "GMAT word problem note: Since is approximately 5(1.7) = 8.5, the line between Anton and Lena is only slightly shorter than the diameter of the circle.", "topic": "word_problems"}
{"id": "book 2_p382_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 382, "page_end": 382, "topic_guess": "geometry", "text": "GMAT geometry note: To calculate the exact answer, observe that Anton, Lena, and the starting line form the three vertices of a triangle. Because one side of the triangle is the diameter of the circle, the triangle is a right triangle with hypotenuse 10.", "topic": "geometry"}
{"id": "book 2_p382_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 382, "page_end": 382, "topic_guess": "geometry", "text": "GMAT geometry note: The known leg is equal to You can either use the Pythagorean theorem to calculate the missing leg (the straight line between Lena and the starting point) or you can note that the known leg and the hypotenuse have a ratio of , so the triangle must be a 30–60–90 triangle and the remaining side must equal 5.", "topic": "geometry"}
{"id": "book 2_p382_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 382, "page_end": 382, "topic_guess": "geometry", "text": "GMAT geometry note: The arc of the circle that Lena traveled has a corresponding inscribed angle of 30 degrees (the vertex at point A). The corresponding central angle is twice the inscribed angle, or 60 degrees.", "topic": "geometry"}
{"id": "book 2_p382_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 382, "page_end": 382, "topic_guess": "geometry", "text": "GMAT geometry note: Alternatively, the distance between Lena and the starting line is equal to the radius of the circle. Therefore, the triangle with vertices at Lena, the starting line, and the center of the circle is an equilateral triangle, and all of its angles equal 60 degrees.", "topic": "geometry"}
{"id": "book 2_p383_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 383, "page_end": 383, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Track mean, median, range, distribution, and how operations change spread. The pool of money is $32 + $72 + $98 = $202.", "topic": "statistics"}
{"id": "book 2_p383_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 383, "page_end": 383, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: A er the redistribution, each person will have an amount between $0 and $202, inclusive. Call the amounts L, M, and H (low, median, high).", "topic": "statistics"}
{"id": "book 2_p383_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 383, "page_end": 383, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: To maximize M (Eddy’s share), minimize L and H. Minimum L = $0 Minimum H = M Maximum M = Total pool of money − Minimum L − Minimum H M = $202 − $0 − M 2M = $202 M = $101 10.", "topic": "statistics"}
{"id": "book 2_p383_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 383, "page_end": 383, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: On your paper, draw the square in the coordinate plane. The line passes through the point (0, 4), so label that point as well.", "topic": "statistics"}
{"id": "book 2_p383_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 383, "page_end": 383, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Since the slope of the line could vary, sketch several possible lines that pass through the point, being certain to test extreme cases. If the slope is very positive, then the line is nearly vertical, and the area above the line will be very close to 2(8) = 16.", "topic": "statistics"}
{"id": "book 2_p384_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 384, "page_end": 384, "topic_guess": "geometry", "text": "GMAT geometry note: As the slope decreases, the area above the line will decrease. If the slope is 1, the area above the line is the area of a triangle with base 4 and height 4, which is equal to Then, as the slope continues to decrease, the area above the line will once again increase steadily.", "topic": "geometry"}
{"id": "book 2_p384_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 384, "page_end": 384, "topic_guess": "geometry", "text": "GMAT geometry note: The greatest area is found when the slope is very negative, since the area will be slightly less than (6)(8) = 48.", "topic": "geometry"}
{"id": "book 2_p385_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 385, "page_end": 385, "topic_guess": "geometry", "text": "GMAT geometry note: The area of the shaded region can be a minimum of exactly 8 and can be a maximum of almost, but not equal to, 48 (since the line cannot be vertical).", "topic": "geometry"}
{"id": "book 2_p386_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 386, "page_end": 386, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "book 2_p387_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 387, "page_end": 387, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Hybrid Problems Identify and Sequence the Parts Where to Start Minor Hybrids.", "topic": "algebra"}
{"id": "book 2_p388_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 388, "page_end": 388, "topic_guess": "statistics", "text": "GMAT statistics note: Percent problems usually relate part, whole, rate, and percent change. Try these four hybrid problems.", "topic": "percent"}
{"id": "book 2_p388_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 388, "page_end": 388, "topic_guess": "statistics", "text": "GMAT statistics note: Try-It #8-1 Set A consists of four consecutive positive integers. Set B is constructed as follows: each integer in set A is randomly either increased by 10% or decreased by 10%, and the four resulting values compose set B.", "topic": "percent"}
{"id": "book 2_p388_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 388, "page_end": 388, "topic_guess": "statistics", "text": "GMAT statistics note: Try-It #8-2 Set S contains 100 consecutive integers. If the range of the negative elements of set S equals 80, what is the average (arithmetic mean) of the positive numbers in the set?", "topic": "percent"}
{"id": "book 2_p388_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 388, "page_end": 388, "topic_guess": "statistics", "text": "GMAT statistics note: The smallest integer in set A was increased by 10% when set B was constructed. The greatest integer in set A was increased by 10% when set B was constructed.", "topic": "percent"}
{"id": "book 2_p389_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 389, "page_end": 389, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Try-It #8-3 If a and b are consecutive positive integers, is ab divisible by 30?", "topic": "probability"}
{"id": "book 2_p389_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 389, "page_end": 389, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Try-It #8-4 A carnival card game gives the contestant a one in three probability of successfully choosing the correct card and thereby winning the game. If a contestant plays the game repeatedly, what is the minimum number of times that he must play the game so that the probability that he never loses is less than 1%?", "topic": "probability"}
{"id": "book 2_p389_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 389, "page_end": 389, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: A2 is divisible by 25.(1) 63 is a factor of b2.(2).", "topic": "probability"}
{"id": "book 2_p390_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 390, "page_end": 390, "topic_guess": "geometry", "text": "GMAT geometry note: Hybrid Problems Hybrid problems blend topics together. They contain two or more qualitatively different kinds of obstacles that you must surmount on the way to the answer.", "topic": "geometry"}
{"id": "book 2_p390_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 390, "page_end": 390, "topic_guess": "geometry", "text": "GMAT geometry note: Some hybrid problems feature content areas that are fairly closely related. Other hybrid problems feature content areas that share little in common.", "topic": "geometry"}
{"id": "book 2_p390_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 390, "page_end": 390, "topic_guess": "geometry", "text": "GMAT geometry note: These problems must be solved in separate steps. The difficulty of a hybrid problem is related to the following questions:.", "topic": "geometry"}
{"id": "book 2_p390_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 390, "page_end": 390, "topic_guess": "geometry", "text": "GMAT geometry note: Problems with minor additional content areas are generally easier to solve than hybrids that blend topics together in an unusual, fundamental, and How closely related are the subjects being tested? Are the content areas covered in the same section of the Manhattan Prep All the Quant guide?", "topic": "geometry"}
{"id": "book 2_p390_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 390, "page_end": 390, "topic_guess": "geometry", "text": "GMAT geometry note: The more closely related the subjects, the easier it will be to navigate the problem. How important is each of the subject areas?", "topic": "geometry"}
{"id": "book 2_p390_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 390, "page_end": 390, "topic_guess": "geometry", "text": "GMAT geometry note: Are each of the subject areas fundamental to the problem or is one of them just a low-level disguise that can be quickly disposed of? The more important each topic is in solving the problem, the more difficult the problem will usually be.", "topic": "geometry"}
{"id": "book 2_p391_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 391, "page_end": 391, "topic_guess": "general", "text": "GMAT quant note: Focus on equations, expressions, variables, and step-by-step manipulation. The best hybrids are one of a kind.", "topic": "algebra"}
{"id": "book 2_p391_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 391, "page_end": 391, "topic_guess": "general", "text": "GMAT quant note: You will have to bring your A game to solve them. That’s why you’re reading this chapter!", "topic": "algebra"}
{"id": "book 2_p392_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 392, "page_end": 392, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Identify and Sequence the Parts When you encounter a hybrid problem, first pick out all of the topics tested.", "topic": "number_properties"}
{"id": "book 2_p392_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 392, "page_end": 392, "topic_guess": "number_theory", "text": "GMAT number properties note: Take a look at the first problem from the pop quiz:. Try-It #8-1 Set A consists of four consecutive positive integers.", "topic": "number_properties"}
{"id": "book 2_p392_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 392, "page_end": 392, "topic_guess": "number_theory", "text": "GMAT number properties note: Set B is constructed as follows: each integer in set A is randomly either increased by 10% or decreased by 10%, and the four resulting values compose set B. This problem refers to consecutive integers, but it involves percents and also asks for the range of a set of numbers, which is a statistics concept.", "topic": "number_properties"}
{"id": "book 2_p392_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 392, "page_end": 392, "topic_guess": "number_theory", "text": "GMAT number properties note: Which aspect of the problem should you tackle first? In this problem, a set begins as four consecutive integers and is then changed.", "topic": "number_properties"}
{"id": "book 2_p392_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 392, "page_end": 392, "topic_guess": "number_theory", "text": "GMAT number properties note: So you might start your approach by jotting down some sets of consecutive integers—either as numbers or in terms of a variable. Then, The greatest integer in set A was increased by 10% when set B was constructed.", "topic": "number_properties"}
{"id": "book 2_p392_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 392, "page_end": 392, "topic_guess": "number_theory", "text": "GMAT number properties note: The smallest integer in set A was increased by 10% when set B was constructed.", "topic": "number_properties"}
{"id": "book 2_p393_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 393, "page_end": 393, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Apply the percent change described in the problem and, finally, check the range of the resulting set.", "topic": "percent"}
{"id": "book 2_p393_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 393, "page_end": 393, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: In general, start with wherever you feel the logical beginning is or whichever part you feel is an easier, cleaner starting point.", "topic": "percent"}
{"id": "book 2_p394_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 394, "page_end": 394, "topic_guess": "number_theory", "text": "GMAT number properties note: Percent problems usually relate part, whole, rate, and percent change. Where to Start STARTING AT THE BEGINNING As you contemplate the logical order of steps, you might feel less confident with the second stage than with the first.", "topic": "percent"}
{"id": "book 2_p394_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 394, "page_end": 394, "topic_guess": "number_theory", "text": "GMAT number properties note: If so, go ahead and start at the beginning. Just articulate very clearly, “What intermediate result will I get once I’m finished with the first part of the problem?” Begin by defining set A.", "topic": "percent"}
{"id": "book 2_p394_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 394, "page_end": 394, "topic_guess": "number_theory", "text": "GMAT number properties note: Set A consists of four consecutive integers, such as 1, 2, 3, 4, or 10, 11, 12, 13. Next, choose a statement to work with first.", "topic": "percent"}
{"id": "book 2_p394_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 394, "page_end": 394, "topic_guess": "number_theory", "text": "GMAT number properties note: The other integers might have been either increased or decreased. Using one of your cases, calculate what might happen to the set:.", "topic": "percent"}
{"id": "book 2_p394_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 394, "page_end": 394, "topic_guess": "number_theory", "text": "GMAT number properties note: Set A Scenarios Set B 1, 2, 3, 4 All values increased by 10% 1.1, 2.2, 3.3, 4.4 1, 2, 3, 4 Smallest value increased by 10%, other values decreased by 10% 1.1, 1.8, 2.7, 3.6 Finally, calculate the range of the resulting set. The range of a set is the difference between the greatest number and the smallest number in the set.", "topic": "percent"}
{"id": "book 2_p395_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 395, "page_end": 395, "topic_guess": "statistics", "text": "GMAT statistics note: Percent problems usually relate part, whole, rate, and percent change. Set A Scenarios Set B Range All values increased by 10% 1.1, 2.2, 3.3, Smallest value increased by 10%, other values decreased by 10% 1.1, 1.8, 2.7, Since different ranges are possible, the statement is insufficient.", "topic": "percent"}
{"id": "book 2_p395_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 395, "page_end": 395, "topic_guess": "statistics", "text": "GMAT statistics note: STARTING AT THE END You might decide that it is easier to start at the end and work backwards. Just ask yourself, “What information do I need to have as a last step before arriving at a solution to the question?” In this problem, you might first consider what would cause set B to have a different range.", "topic": "percent"}
{"id": "book 2_p395_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 395, "page_end": 395, "topic_guess": "statistics", "text": "GMAT statistics note: The range of a set is the difference between the greatest and smallest numbers in the set. If the difference between these two numbers can vary, the range can vary as well.", "topic": "percent"}
{"id": "book 2_p395_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 395, "page_end": 395, "topic_guess": "statistics", "text": "GMAT statistics note: However, the smallest integer might have increased by 10%, bringing it closer to the greatest integer, or it might have decreased by 10%, bringing it farther from the greatest integer.", "topic": "percent"}
{"id": "book 2_p396_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 396, "page_end": 396, "topic_guess": "statistics", "text": "GMAT statistics note: Percent problems usually relate part, whole, rate, and percent change. Now, put the two statements together.", "topic": "percent"}
{"id": "book 2_p396_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 396, "page_end": 396, "topic_guess": "statistics", "text": "GMAT statistics note: One option is to try a few different cases, making sure to increase the smallest integer in set A and decrease the greatest integer in set A. Set A Scenario Set B Range 1, 2, 3, 4 All values increased by 10% 1.1, 2.2, 3.3, 4.4 4.4—1.1 = 3.3 Try another case, using different numbers:.", "topic": "percent"}
{"id": "book 2_p396_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 396, "page_end": 396, "topic_guess": "statistics", "text": "GMAT statistics note: Set A Scenario Set B Range 10, 11, 12, 13 All values increased by 10% 11, 12.1, 13.2, 14.3 14.3—11 = 3.3 The range is the same, but this doesn’t mean that the statements are sufficient together. To see why, use a different approach.", "topic": "percent"}
{"id": "book 2_p396_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 396, "page_end": 396, "topic_guess": "statistics", "text": "GMAT statistics note: Set A always contains the numbers x, x + 1, x + 2, and x + 3. When the smallest value in set A is increased by 10%, it goes from x to 1.1x.", "topic": "percent"}
{"id": "book 2_p396_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 396, "page_end": 396, "topic_guess": "statistics", "text": "GMAT statistics note: When the greatest value is increased by 10%, it goes from x + 3 to 1.1x + 3.3. So these two values differ by 3.3.", "topic": "percent"}
{"id": "book 2_p396_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 396, "page_end": 396, "topic_guess": "statistics", "text": "GMAT statistics note: However, you can’t assume that the smallest and greatest values in set A always become the smallest and greatest values in set B. If two of the values switch places, the range might change.", "topic": "percent"}
{"id": "book 2_p397_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 397, "page_end": 397, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Set A Scenario Set B Range Smallest and largest values increased by 10%, other values decreased by 10% The range of the set might vary, even when both statements are used.", "topic": "statistics"}
{"id": "book 2_p397_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 397, "page_end": 397, "topic_guess": "statistics", "text": "GMAT statistics note: Try-It #8-2 Set S contains 100 consecutive integers. If the range of the negative elements of set S equals 80, what is the average (arithmetic mean) of the positive numbers in the set?", "topic": "statistics"}
{"id": "book 2_p397_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 397, "page_end": 397, "topic_guess": "statistics", "text": "GMAT statistics note: In this problem, information is given about the range of numbers in a set. Thus, knowing how to work with statistics techniques will be important in solving the problem.", "topic": "statistics"}
{"id": "book 2_p397_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 397, "page_end": 397, "topic_guess": "statistics", "text": "GMAT statistics note: Solving this problem also requires using consecutive integers techniques— namely, counting consecutive integers and computing their average. These two topics are closely related but still cover different ideas.", "topic": "statistics"}
{"id": "book 2_p397_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 397, "page_end": 397, "topic_guess": "statistics", "text": "GMAT statistics note: If you start with the statistics piece of the problem, then you’ll be able to find the highest and lowest negative numbers in the set of consecutive integers. This will act as an input to the formulas for computing the largest and smallest positive integers in the set, and subsequently the average of the positive integers in the set.", "topic": "statistics"}
{"id": "book 2_p398_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 398, "page_end": 398, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. First, determine what set of consecutive negative integers will result in a range of 80.", "topic": "number_properties"}
{"id": "book 2_p398_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 398, "page_end": 398, "topic_guess": "number_theory", "text": "GMAT number properties note: Range is defined as the difference between the highest and lowest numbers in a set:. The high number among the negative terms is the largest negative integer, Therefore, the lowest number in set S is −81.", "topic": "number_properties"}
{"id": "book 2_p398_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 398, "page_end": 398, "topic_guess": "number_theory", "text": "GMAT number properties note: Use this result to jump to the consecutive integers portion of the solution. In this case, −81 is the smallest or “first” element in the set:.", "topic": "number_properties"}
{"id": "book 2_p398_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 398, "page_end": 398, "topic_guess": "number_theory", "text": "GMAT number properties note: Therefore, the highest or “last” number in set S is 18. Finally, calculate the average of the positive terms in the set using 1 as the smallest positive integer (“first pos”) and 18 as the largest positive integer (“last pos”).", "topic": "number_properties"}
{"id": "book 2_p399_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 399, "page_end": 399, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The average of the positive numbers in the set is 9.5.", "topic": "number_properties"}
{"id": "book 2_p399_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 399, "page_end": 399, "topic_guess": "number_theory", "text": "GMAT number properties note: If instead you decided to start with the second step, you would write the formula for the average of consecutive integers first, focusing on the positive integers:. The two unknowns you need to solve for are the greatest positive integer and the least positive integer because the question asks only about the positive integers in the set.", "topic": "number_properties"}
{"id": "book 2_p399_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 399, "page_end": 399, "topic_guess": "number_theory", "text": "GMAT number properties note: Some of the integers in set S are negative and some are positive, so clearly the least positive integer will be 1. Therefore, you only need to figure out what the greatest integer in the set will be.", "topic": "number_properties"}
{"id": "book 2_p399_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 399, "page_end": 399, "topic_guess": "number_theory", "text": "GMAT number properties note: This is the information needed in the last step of solving the problem. In order to calculate this number, you would now need to apply the definition of range:.", "topic": "number_properties"}
{"id": "book 2_p399_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 399, "page_end": 399, "topic_guess": "number_theory", "text": "GMAT number properties note: The greatest negative integer (the “high” number) is −1.", "topic": "number_properties"}
{"id": "book 2_p400_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 400, "page_end": 400, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Therefore, the least number in the set is −81.", "topic": "number_properties"}
{"id": "book 2_p400_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 400, "page_end": 400, "topic_guess": "number_theory", "text": "GMAT number properties note: Plug this into the formula for counting consecutive integers:. Therefore, the highest number in the set is 18.", "topic": "number_properties"}
{"id": "book 2_p400_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 400, "page_end": 400, "topic_guess": "number_theory", "text": "GMAT number properties note: Finally, plug 18 into the average formula:. Whether you start solving this problem from the beginning or from the end, study how the steps hook together: the output of one step becomes the input to another.", "topic": "number_properties"}
{"id": "book 2_p400_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 400, "page_end": 400, "topic_guess": "number_theory", "text": "GMAT number properties note: These are the “turns” you have to make in solving any hybrid problem. O en, a number that plays one role in a particular formula or calculation plays a completely different role in the next step.", "topic": "number_properties"}
{"id": "book 2_p401_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 401, "page_end": 401, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Minor Hybrids In minor hybrid problems, one of the following conditions applies:.", "topic": "number_properties"}
{"id": "book 2_p401_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 401, "page_end": 401, "topic_guess": "number_theory", "text": "GMAT number properties note: These problems can be easier to solve than major hybrids. However, you will still benefit greatly from paying close attention to the turns as you move through the stages of the solution.", "topic": "number_properties"}
{"id": "book 2_p401_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 401, "page_end": 401, "topic_guess": "number_theory", "text": "GMAT number properties note: Try-It #8-3 If a and b are consecutive positive integers, is ab divisible by 30? This problem tests your skill with both Divisibility & Primes and Consecutive Integers.", "topic": "number_properties"}
{"id": "book 2_p401_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 401, "page_end": 401, "topic_guess": "number_theory", "text": "GMAT number properties note: Consecutive integer concepts o en lend themselves well to questions about divisibility. The content areas in the problem are closely related.", "topic": "number_properties"}
{"id": "book 2_p401_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 401, "page_end": 401, "topic_guess": "number_theory", "text": "GMAT number properties note: For instance, they are covered in the same section of the Manhattan Prep All the Quant guide. One of the content areas is a low-level disguise or some other minor feature.", "topic": "number_properties"}
{"id": "book 2_p401_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 401, "page_end": 401, "topic_guess": "number_theory", "text": "GMAT number properties note: A2 is divisible by 25.(1) 63 is a factor of b2.(2).", "topic": "number_properties"}
{"id": "book 2_p402_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 402, "page_end": 402, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. To be divisible by 30, a number must have 2, 3, and 5 as prime factors.", "topic": "number_properties"}
{"id": "book 2_p402_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 402, "page_end": 402, "topic_guess": "number_theory", "text": "GMAT number properties note: Thus, the question becomes: “Does ab have 2, 3, and 5 as prime factors?” Next, use a concept from consecutive integers: a and b are consecutive positive integers. Thus, either a or b is an even number, which means that the product ab is automatically divisible by 2.", "topic": "number_properties"}
{"id": "book 2_p402_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 402, "page_end": 402, "topic_guess": "number_theory", "text": "GMAT number properties note: The question can be further simplified: “Does the product ab have 3 and 5 as prime factors?” (1) INSUFFICIENT:. Knowing that 5 is a prime factor of a indicates that 5 is a factor of ab, but does not indicate whether 3 is a factor.", "topic": "number_properties"}
{"id": "book 2_p402_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 402, "page_end": 402, "topic_guess": "number_theory", "text": "GMAT number properties note: Knowing that 3 is a prime factor of b indicates that 3 is a factor of ab, but does not indicate whether 5 is a factor. According to the combined statements, a is divisible by 5 and b is divisible by 3.", "topic": "number_properties"}
{"id": "book 2_p402_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 402, "page_end": 402, "topic_guess": "number_theory", "text": "GMAT number properties note: This is sufficient information to answer the rephrased question. Notice that the solution never indicated exactly what consecutive integers a and b are—this isn’t necessary.", "topic": "number_properties"}
{"id": "book 2_p402_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 402, "page_end": 402, "topic_guess": "number_theory", "text": "GMAT number properties note: Also notice a potential trap in this problem—assuming that a < b bec.", "topic": "number_properties"}
{"id": "book 2_p403_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 403, "page_end": 403, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Try-It #8-4 A carnival card game gives the contestant a one in three probability of successfully choosing the correct card and thereby winning the game.", "topic": "probability"}
{"id": "book 2_p403_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 403, "page_end": 403, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: If a contestant plays the game repeatedly, what is the minimum number of times that he must play the game so that the probability that he never loses is less than 1%? This problem primarily tests Probability theory.", "topic": "probability"}
{"id": "book 2_p403_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 403, "page_end": 403, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: In addition, Exponents are needed to represent the impact of playing multiple games on the probability of the outcomes. The probabilities are given in fractions, yet the question is asked in terms of percents, so Fraction, Decimal, and Percent (FDP) connections are relevant.", "topic": "probability"}
{"id": "book 2_p403_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 403, "page_end": 403, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Finally, the question is phrased in terms of an inequality, so Inequalities come into play. “The probability that he never loses” can be rephrased as “the probability that he always wins.” This probability can be expressed as , where is the chance of winning on a single play and n is the number of times the contestant plays.", "topic": "probability"}
{"id": "book 2_p403_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 403, "page_end": 403, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Number of Plays P(All Wins) Approx.", "topic": "probability"}
{"id": "book 2_p404_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 404, "page_end": 404, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Percent problems usually relate part, whole, rate, and percent change. Number of Plays P(All Wins) Approx.", "topic": "percent"}
{"id": "book 2_p404_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 404, "page_end": 404, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Equivalent You might also have noticed that only the denominator in the probabilities mattered, as the numerator was always 1. To have a probability of less than 1%, the fractional probability must be “1 over something greater than 100.” In order for 3n >100, n must be at least 5.", "topic": "percent"}
{"id": "book 2_p405_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 405, "page_end": 405, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Problem Set Solve the following problems and identify the topics being tested.", "topic": "number_properties"}
{"id": "book 2_p405_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 405, "page_end": 405, "topic_guess": "number_theory", "text": "GMAT number properties note: The average (arithmetic mean) of a list of six numbers is equal to 0. What is the positive difference between the number of positive numbers in the list and the number of negative numbers in the list?", "topic": "number_properties"}
{"id": "book 2_p405_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 405, "page_end": 405, "topic_guess": "number_theory", "text": "GMAT number properties note: Each of the positive numbers in the list equals 10.(1) Each of the negative numbers in the list equals −5.(2) 2. If a, b, and c are positive, is a > b?", "topic": "number_properties"}
{"id": "book 2_p405_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 405, "page_end": 405, "topic_guess": "number_theory", "text": "GMAT number properties note: If c is randomly chosen from the integers 20 to 99, inclusive, what is the probability that c3 − c is divisible by 12?", "topic": "number_properties"}
{"id": "book 2_p406_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 406, "page_end": 406, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If x and y are positive integers greater than 1 such that x − y and are both even integers, which of the following numbers must be non-prime integers?", "topic": "number_properties"}
{"id": "book 2_p406_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 406, "page_end": 406, "topic_guess": "number_theory", "text": "GMAT number properties note: A number cube with faces numbered 1 through 6 has an equal chance of landing on any face when rolled. If the number cube is rolled twice, what is the probability that the sum of the two rolls is a prime number?", "topic": "number_properties"}
{"id": "book 2_p406_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 406, "page_end": 406, "topic_guess": "number_theory", "text": "GMAT number properties note: The function {x} is defined as the area of a square with diagonal of length x. If x > 0 and {x2} = x2, what is the value of x?", "topic": "number_properties"}
{"id": "book 2_p407_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 407, "page_end": 407, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. (1 centimeter = 10 millimeters) 9.", "topic": "number_properties"}
{"id": "book 2_p407_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 407, "page_end": 407, "topic_guess": "number_theory", "text": "GMAT number properties note: Three consecutive integers are selected from the integers 1 to 50, inclusive. What is the sum of the remainders that result when each of the three integers is divided by x?", "topic": "number_properties"}
{"id": "book 2_p407_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 407, "page_end": 407, "topic_guess": "number_theory", "text": "GMAT number properties note: When the greatest of the consecutive integers is divided by x, the remainder is 0. When the least of the consecutive integers is divided by x, the remainder is 1.", "topic": "number_properties"}
{"id": "book 2_p408_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 408, "page_end": 408, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If x, y, and z are all distinct positive integers, and the percent increase from x to y is equal to the percent increase from y to z, what is x?", "topic": "number_properties"}
{"id": "book 2_p409_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 409, "page_end": 409, "topic_guess": "general", "text": "GMAT quant note: Track mean, median, range, distribution, and how operations change spread. Since the average of the six numbers in the list is 0, the sum of the six numbers is 0.", "topic": "statistics"}
{"id": "book 2_p409_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 409, "page_end": 409, "topic_guess": "general", "text": "GMAT quant note: There could be positive numbers and negative numbers in the set. Zero is not mentioned, but this does not rule it out.", "topic": "statistics"}
{"id": "book 2_p409_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 409, "page_end": 409, "topic_guess": "general", "text": "GMAT quant note: In order for the sum of the numbers in the set to be 0, either all the terms are 0 or there are some positives and some negatives. The list could be {−2, −2, −2, −2, −2, 10}, and the positive difference between the number of positive terms and the number of negative terms would be 4.", "topic": "statistics"}
{"id": "book 2_p409_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 409, "page_end": 409, "topic_guess": "general", "text": "GMAT quant note: Alternatively, the list could be {−20, −20, 10, 10, 10, 10}, and the positive difference would be 2. The set could be {−5, 1, 1, 1, 1, 1}, and the positive difference between the number of positive terms and the number of negative terms would be 4.", "topic": "statistics"}
{"id": "book 2_p409_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 409, "page_end": 409, "topic_guess": "general", "text": "GMAT quant note: The set could be {−5, −5, −5, 5, 5, 5}, and the positive difference would be 0. The statements together suggest that the set has twice as many −5 terms as 10 terms, in order to maintain a sum of 0.", "topic": "statistics"}
{"id": "book 2_p409_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 409, "page_end": 409, "topic_guess": "general", "text": "GMAT quant note: If every term is negative or positive, then the set would have.", "topic": "statistics"}
{"id": "book 2_p410_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 410, "page_end": 410, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. Terms are possible, so the set could be {−5, −5, 0, 0, 0, 10}, and the positive difference would be 1.", "topic": "number_properties"}
{"id": "book 2_p410_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 410, "page_end": 410, "topic_guess": "general", "text": "GMAT quant note: You could simplify the numerator arithmetically (multiply out the terms and then add). Alternatively, factor a 22 out of the numerator.", "topic": "number_properties"}
{"id": "book 2_p410_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 410, "page_end": 410, "topic_guess": "general", "text": "GMAT quant note: Then, distribute the denominator (which becomes the difference of squares). If a, b, and c are all positive, then a > b + positive.", "topic": "number_properties"}
{"id": "book 2_p410_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 410, "page_end": 410, "topic_guess": "general", "text": "GMAT quant note: Therefore, a must be greater than b. You could also prove this fact by Testing Cases.", "topic": "number_properties"}
{"id": "book 2_p410_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 410, "page_end": 410, "topic_guess": "general", "text": "GMAT quant note: Case b + c < a Is this case possible according to (2)? A > b b + positive < number greater than b Possible a = b b + c < b Impossible, since c is positive a < b b + c < number less than b Impossible, since c is positive.", "topic": "number_properties"}
{"id": "book 2_p411_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 411, "page_end": 411, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Only the a > b case is possible, so the answer is a definite Yes.", "topic": "algebra"}
{"id": "book 2_p411_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 411, "page_end": 411, "topic_guess": "algebra", "text": "GMAT algebra note: A(a + c) > b(b + c) a2 + ac > b2 + bc a2 + ac − b2 − bc > 0 a2 − b2 + ac − bc > 0 Group similar terms to simplify. [a2 − b2] + [ac − bc] > 0 Note the Quadratic Template.", "topic": "algebra"}
{"id": "book 2_p411_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 411, "page_end": 411, "topic_guess": "algebra", "text": "GMAT algebra note: (a − b)(a + b) + c(a − b) > 0 (a − b)[(a + b) + c] > 0 Factor out a − b. Therefore, a + b + c is positive, because all three additive terms are positive.", "topic": "algebra"}
{"id": "book 2_p411_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 411, "page_end": 411, "topic_guess": "algebra", "text": "GMAT algebra note: By Number Properties sign rules, (a − b) must also be positive in order for the product to be greater than 0. It is hard to see where to begin or what series of manipulations will be productive.", "topic": "algebra"}
{"id": "book 2_p411_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 411, "page_end": 411, "topic_guess": "algebra", "text": "GMAT algebra note: LT = less than and GT = greater than. GTb means “a number greater than b.” Case Is this case possible according to (1)?", "topic": "algebra"}
{"id": "book 2_p411_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 411, "page_end": 411, "topic_guess": "algebra", "text": "GMAT algebra note: The le is greater than the right.", "topic": "algebra"}
{"id": "book 2_p412_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 412, "page_end": 412, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Case Is this case possible according to (1)?", "topic": "number_properties"}
{"id": "book 2_p412_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 412, "page_end": 412, "topic_guess": "number_theory", "text": "GMAT number properties note: The two expressions are equal. The le side is actually less than the right side, not greater than as (1) requires.", "topic": "number_properties"}
{"id": "book 2_p412_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 412, "page_end": 412, "topic_guess": "number_theory", "text": "GMAT number properties note: The words divisible and probability are used, so this question is about Divisibility & Primes and Probability. There are 99 − 20 + 1 = 80 possible values for c, so the unknown is how many of these c values yield a c3 − c that is divisible by 12.", "topic": "number_properties"}
{"id": "book 2_p412_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 412, "page_end": 412, "topic_guess": "number_theory", "text": "GMAT number properties note: The prime factorization of 12 is 2 × 2 × 3. There are at least two ways to think about this: numbers are divisible by 12 if they are divisible by 3 and by 2 twice, or if they are multiples of both 4 and 3.", "topic": "number_properties"}
{"id": "book 2_p412_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 412, "page_end": 412, "topic_guess": "number_theory", "text": "GMAT number properties note: The expression involving c can be factored. These are consecutive integers.", "topic": "number_properties"}
{"id": "book 2_p412_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 412, "page_end": 412, "topic_guess": "number_theory", "text": "GMAT number properties note: It may help to put them in increasing order: (c − 1)c(c + 1). Thus, this question has a lot to do with Consecutive Integers, and not only because the integers 20 to 99 themselves are consecutive.", "topic": "number_properties"}
{"id": "book 2_p413_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 413, "page_end": 413, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. In any set of three consecutive integers, a multiple of 3 will be included.", "topic": "number_properties"}
{"id": "book 2_p413_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 413, "page_end": 413, "topic_guess": "number_theory", "text": "GMAT number properties note: Thus, (c − 1)c(c + 1) is always divisible by 3 for any integer c. This takes care of part of the 12.", "topic": "number_properties"}
{"id": "book 2_p413_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 413, "page_end": 413, "topic_guess": "number_theory", "text": "GMAT number properties note: So the question becomes: “How many of the possible (c − 1)c(c + 1) values are divisible by 4?” Since the prime factors of 4 are 2’s, it makes sense to think in terms of Odds and Evens. Thus, c3 − c is divisible by 12 whenever c is odd, possibilities.", "topic": "number_properties"}
{"id": "book 2_p413_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 413, "page_end": 413, "topic_guess": "number_theory", "text": "GMAT number properties note: Thus, c3 − c is also divisible by 12 whenever c is a multiple of 4, which are the cases c = possibilities. X and y are both positive and is an integer, so x > y.", "topic": "number_properties"}
{"id": "book 2_p413_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 413, "page_end": 413, "topic_guess": "number_theory", "text": "GMAT number properties note: Since x − y is even, either x and y are both even or they are both odd. Since = an even integer, x = y × (even integer).", "topic": "number_properties"}
{"id": "book 2_p414_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 414, "page_end": 414, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Therefore, x is an even integer, as is y.", "topic": "number_properties"}
{"id": "book 2_p414_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 414, "page_end": 414, "topic_guess": "number_theory", "text": "GMAT number properties note: X and y are both positive even integers and x > y. No even number greater than 2 is prime, so x can’t be prime.", "topic": "number_properties"}
{"id": "book 2_p414_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 414, "page_end": 414, "topic_guess": "number_theory", "text": "GMAT number properties note: X and y are each positive even integers and x > y. Thus, x + y is even and the least possible value of x + y = 4 + 2 = 6.", "topic": "number_properties"}
{"id": "book 2_p414_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 414, "page_end": 414, "topic_guess": "number_theory", "text": "GMAT number properties note: All even numbers greater than or equal to 6 are non-prime. It could be that x = 4 and y = 2, so , which is not prime, but is also not an integer.", "topic": "number_properties"}
{"id": "book 2_p414_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 414, "page_end": 414, "topic_guess": "number_theory", "text": "GMAT number properties note: In fact, if an even integer, then III. First, think about the prime numbers less than 12, the maximum sum of the numbers.", "topic": "number_properties"}
{"id": "book 2_p414_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 414, "page_end": 414, "topic_guess": "number_theory", "text": "GMAT number properties note: These primes are 2, 3, 5, 7, 11. The probability of rolling 2, 3, 5, 7, or 11 is equal to the number of ways to roll any of these sums divided by the total number of possible outcomes.", "topic": "number_properties"}
{"id": "book 2_p414_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 414, "page_end": 414, "topic_guess": "number_theory", "text": "GMAT number properties note: The total number of possible outcomes is 6 × 6 = 36. Sum of 2 can happen 1 way: 1 + 1 Sum of 3 can happen 2 ways: 1 + 2, 2 + 1 Sum of 5 can happen 4 ways: 1 + 4, 2 + 3, 3 + 2, 4 + 1 Sum of 7 can happen 6 ways: 1 + 6, 2 + 5, 3 + 4, 4 + 3, 5 + 2, 6 + 1.", "topic": "number_properties"}
{"id": "book 2_p415_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 415, "page_end": 415, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Sum of 11 can happen 2 ways: 5 + 6, 6 + 5 That’s a total of 1 + 2 + 4 + 6 + 2 = 15 ways to roll a prime sum. The problem defines the function in words.", "topic": "geometry"}
{"id": "book 2_p415_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 415, "page_end": 415, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: You’ll need to translate into math. “The function {x} is defined as the area of a square with diagonal of length x.” {x} = s2, where s is the side of the square.", "topic": "geometry"}
{"id": "book 2_p415_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 415, "page_end": 415, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: If the diagonal equals x, then the side of the square equals The question stem also indicates that {x2} = x2. In other words, applying the defined function to x2 will result in an answer of x2.", "topic": "geometry"}
{"id": "book 2_p416_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 416, "page_end": 416, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Now, set that equal to x2 and solve. It is acceptable to divide by x2 because the question stem indicates that x ≠ 0.) Therefore, x is equal to or .", "topic": "geometry"}
{"id": "book 2_p416_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 416, "page_end": 416, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Since the question stem indicates that x > 0, only is a valid solution. Alternatively, you could work backwards from the answers.", "topic": "geometry"}
{"id": "book 2_p416_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 416, "page_end": 416, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Next, the question specifies that that becomes {2}, so what does this function return? “The function {x} is defined as the area of a square with diagonal of length x.” The function {2} is defined as the area of a square with diagonal length 2.", "topic": "geometry"}
{"id": "book 2_p416_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 416, "page_end": 416, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: The side length of this square is , so the area of this square is . Therefore, {2} = 2, which matches the question stem specification that {x2} = x2.", "topic": "geometry"}
{"id": "book 2_p417_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 417, "page_end": 417, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Microchip radius = (2.5 cm)(10 mm/cm) = 25 mm 9.", "topic": "number_properties"}
{"id": "book 2_p417_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 417, "page_end": 417, "topic_guess": "number_theory", "text": "GMAT number properties note: When consecutive integers are divided by the same number, the remainders follow a repeating pattern. For instance, when consecutive integers are divided by 4, the remainders form a repeating pattern of [1, 2, 3, 0], with every fourth integer being divisible by 4.", "topic": "number_properties"}
{"id": "book 2_p417_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 417, "page_end": 417, "topic_guess": "number_theory", "text": "GMAT number properties note: This statement indicates that the greatest of the three integers is divisible by x. However, the sum of the remainders depends on the value of x.", "topic": "number_properties"}
{"id": "book 2_p417_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 417, "page_end": 417, "topic_guess": "number_theory", "text": "GMAT number properties note: For instance, if x = 10, the three remainders would be 8, 9, and 0, respectively, and their sum would be 17. If x = 5, the three remainders would be 3, 4, and 0, with a sum of 7.", "topic": "number_properties"}
{"id": "book 2_p418_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 418, "page_end": 418, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. However, the sum of the remainders again depends on the value of x.", "topic": "number_properties"}
{"id": "book 2_p418_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 418, "page_end": 418, "topic_guess": "number_theory", "text": "GMAT number properties note: If x = 10, the three remainders would be 1, 2, and 3, with a sum of 6. If x = 2, the three remainders would be 1, 0, and 1, with a sum of 2.", "topic": "number_properties"}
{"id": "book 2_p418_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 418, "page_end": 418, "topic_guess": "number_theory", "text": "GMAT number properties note: Call the three consecutive integers y, y + 1, and y + 2. The remainder when y is divided by x is 1 and the remainder when y + 2 is divided by x is 0.", "topic": "number_properties"}
{"id": "book 2_p418_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 418, "page_end": 418, "topic_guess": "number_theory", "text": "GMAT number properties note: Write out the known remainders and write out enough other terms to see where the pattern repeats. In this case, the term y – 1 also has a remainder of 0, so the pattern repeats every three terms.", "topic": "number_properties"}
{"id": "book 2_p418_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 418, "page_end": 418, "topic_guess": "number_theory", "text": "GMAT number properties note: Y − 1 y y + 1 y + 2 The only way for the remainders to have a consistent repeating pattern is for the missing remainder to equal 2 (in which case x must equal 3). Therefore, the sum of the three remainders is 1 + 2 + 0 = 3.", "topic": "number_properties"}
{"id": "book 2_p418_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 418, "page_end": 418, "topic_guess": "number_theory", "text": "GMAT number properties note: The question asks for the value of x. Translate the equation given in the question stem.", "topic": "number_properties"}
{"id": "book 2_p419_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 419, "page_end": 419, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The problem stem specifies that x is an integer, so the right-hand side of the equation must also be an integer.", "topic": "number_properties"}
{"id": "book 2_p419_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 419, "page_end": 419, "topic_guess": "number_theory", "text": "GMAT number properties note: Therefore, z must be a factor of y2. The question asks for the value of x.", "topic": "number_properties"}
{"id": "book 2_p419_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 419, "page_end": 419, "topic_guess": "number_theory", "text": "GMAT number properties note: If y is prime, then there are two possible scenarios. This isn’t allowed, though, because the question stem indicates that the variables represent three different positive integers.", "topic": "number_properties"}
{"id": "book 2_p419_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 419, "page_end": 419, "topic_guess": "number_theory", "text": "GMAT number properties note: Case 2: y2 = z, in which case x must equal 1. If z = 9, then y2 is a multiple of 9, so z must be a multiple of 3.", "topic": "number_properties"}
{"id": "book 2_p419_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 419, "page_end": 419, "topic_guess": "number_theory", "text": "GMAT number properties note: It’s possible that z = 9, y = 3, and x = 1. It’s also possible that z = 9, y = 36, and x = 4.", "topic": "number_properties"}
{"id": "book 2_p420_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 420, "page_end": 420, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "book 2_p421_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 421, "page_end": 421, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "book 2_p422_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 422, "page_end": 422, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating. Workout Set 1 Workout Set 2 Workout Set 3 Workout Set 4 Workout Set 5 Workout Set 6 Workout Set 7 Workout Set 8 Workout Set 9 Workout Set 10 Workout Set 11 Workout Set 12 Workout Set 13 Workout Set 14 Workout Set 15 Workout Set 16.", "topic": "word_problems"}
{"id": "book 2_p423_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 423, "page_end": 423, "topic_guess": "geometry", "text": "GMAT geometry note: The circle with center O has a circumference of . If AC is a diameter of the circle, what is the length of BC?", "topic": "geometry"}
{"id": "book 2_p424_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 424, "page_end": 424, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. A batch of widgets costs p + 15 dollars for a company to produce and each batch sells for p(9 − p) dollars.", "topic": "number_properties"}
{"id": "book 2_p424_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 424, "page_end": 424, "topic_guess": "number_theory", "text": "GMAT number properties note: For which of the following values of p, does the company make a profit? If K is the sum of the reciprocals of the consecutive integers from 41 to 60 inclusive, which of the following is less than K?", "topic": "number_properties"}
{"id": "book 2_p425_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 425, "page_end": 425, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Triplets Adam, Bruce, and Charlie enter a triathlon.", "topic": "probability"}
{"id": "book 2_p425_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 425, "page_end": 425, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: There are nine competitors in the triathlon. If every competitor has an equal chance of winning, and three medals will be awarded, what is the probability that at least two of the triplets will win a medal?", "topic": "probability"}
{"id": "book 2_p426_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 426, "page_end": 426, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Floyd started at one end of a trail at 8:07 a.m.", "topic": "word_problems"}
{"id": "book 2_p426_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 426, "page_end": 426, "topic_guess": "word_problems", "text": "GMAT word problem note: And hiked the entire 5 miles to the other end of the trail at an average rate of 3 miles per hour. Connie started at the same end of the trail at 8:30 a.m.", "topic": "word_problems"}
{"id": "book 2_p426_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 426, "page_end": 426, "topic_guess": "word_problems", "text": "GMAT word problem note: And hiked the entire length of the trail at an average rate of 2.5 miles per hour. Finally, Karen started her hike in the same place at 10:00 a.m.", "topic": "word_problems"}
{"id": "book 2_p426_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 426, "page_end": 426, "topic_guess": "word_problems", "text": "GMAT word problem note: And hiked the entire length of the trail at an average rate of 4 miles per hour. During approximately what percent of the 4-hour period from 8:00 a.m.", "topic": "word_problems"}
{"id": "book 2_p426_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 426, "page_end": 426, "topic_guess": "word_problems", "text": "GMAT word problem note: To noon were at least two of the three hikers on the trail?", "topic": "word_problems"}
{"id": "book 2_p427_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 427, "page_end": 427, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Half an hour a er Car A started traveling from Newtown to Oldtown, a distance of 62 miles, Car B started traveling along the same road from Oldtown to Newtown.", "topic": "word_problems"}
{"id": "book 2_p427_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 427, "page_end": 427, "topic_guess": "word_problems", "text": "GMAT word problem note: The cars met each other on the road 15 minutes a er Car B started its trip. If Car A traveled at a constant rate that was 8 miles per hour greater than Car B’s constant rate, how many miles had Car B driven when they met?", "topic": "word_problems"}
{"id": "book 2_p427_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 427, "page_end": 427, "topic_guess": "word_problems", "text": "GMAT word problem note: If x = 2b − (88 + 86), for which of the following b values is x closest to zero?", "topic": "word_problems"}
{"id": "book 2_p428_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 428, "page_end": 428, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. If y is 20% less than 90% of x and z is 25% more than 130% of y, then z is what percent of x?", "topic": "percent"}
{"id": "book 2_p428_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 428, "page_end": 428, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Set A consists of three consecutive positive multiples of 3, and set B consists of five consecutive positive multiples of 5.", "topic": "percent"}
{"id": "book 2_p429_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 429, "page_end": 429, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Integers in set A is equal to the sum of the integers in set B, what is the least number that could be a member of set A?", "topic": "number_properties"}
{"id": "book 2_p430_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 430, "page_end": 430, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "book 2_p431_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 431, "page_end": 431, "topic_guess": "geometry", "text": "GMAT geometry note: If AC is a diameter of the circle, then triangle ABC is a right triangle, with angle ABC = 90 degrees. The shortest side of a triangle is across from its least angle and the longest side of a triangle is across from its greatest angle.", "topic": "geometry"}
{"id": "book 2_p431_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 431, "page_end": 431, "topic_guess": "geometry", "text": "GMAT geometry note: The circumference of the circle = , so . Thus, AC ≈ 10.2 and BC < 10.2.", "topic": "geometry"}
{"id": "book 2_p431_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 431, "page_end": 431, "topic_guess": "geometry", "text": "GMAT geometry note: The value should be less than 9. Since angle ACB is 30 degrees, angle CAB is 60 degrees.", "topic": "geometry"}
{"id": "book 2_p431_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 431, "page_end": 431, "topic_guess": "geometry", "text": "GMAT geometry note: The sides in a 30–60–90 triangle have the ratio , so use the ratio to compute the desired side, BC.", "topic": "geometry"}
{"id": "book 2_p432_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 432, "page_end": 432, "topic_guess": "strategy", "text": "GMAT strategy note: Watch divisibility, factors, remainders, parity, and prime structure. You can work backwards from the answers.", "topic": "number_properties"}
{"id": "book 2_p432_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 432, "page_end": 432, "topic_guess": "strategy", "text": "GMAT strategy note: The company makes a profit when p = 4. Alternatively, profit equals revenue minus cost.", "topic": "number_properties"}
{"id": "book 2_p433_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 433, "page_end": 433, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. For p > 5, both (p − 5) and (p − 3) are positive.", "topic": "percent"}
{"id": "book 2_p433_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 433, "page_end": 433, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: In that case, the profit is negative (i.e., the company loses money). The profit is only positive if (p − 5) and (p − 3) have opposite signs, which occurs when 3 < p < 5.", "topic": "percent"}
{"id": "book 2_p433_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 433, "page_end": 433, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: The sum has 20 fractional terms. It is impossible to compute this by hand in two minutes.", "topic": "percent"}
{"id": "book 2_p433_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 433, "page_end": 433, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Instead, look at the maximum and minimum possible values for the sum. The greatest fraction in the sum is .", "topic": "percent"}
{"id": "book 2_p433_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 433, "page_end": 433, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: K is definitely less than 20 × , or , which is less than Minimum:. The least fraction in the sum is .", "topic": "percent"}
{"id": "book 2_p433_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 433, "page_end": 433, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: K is definitely greater than 20 × Therefore, < K < I.", "topic": "percent"}
{"id": "book 2_p434_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 434, "page_end": 434, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. With nine competitors and only three medals awarded, only of the competitors will win overall.", "topic": "probability"}
{"id": "book 2_p434_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 434, "page_end": 434, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Although a simplification, it is reasonable for each competitor to see his or her chance of winning a medal as , or to expect to win of a medal (pretending for a moment that medals can be shared). The question asks for the probability that at least two of the triplets will win a medal.", "topic": "probability"}
{"id": "book 2_p434_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 434, "page_end": 434, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: In other words, you want to of the triplets to win medals or for each triplet to win to of a medal. Since and are both greater than , you are looking for the probability that the triplets will win medals at a rate greater than that expected for competitors overall.", "topic": "probability"}
{"id": "book 2_p434_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 434, "page_end": 434, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: In other words, this would be an unusual outcome. Thus, the probability should be less than .", "topic": "probability"}
{"id": "book 2_p434_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 434, "page_end": 434, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: You could then guess from among the remaining answers with a 1 in 3 chance of guessing correctly. To solve, use the probability formula and combinatorics.", "topic": "probability"}
{"id": "book 2_p435_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 435, "page_end": 435, "topic_guess": "general", "text": "GMAT quant note: Separate total outcomes from desired outcomes and count carefully. First, find the total number of outcomes for the triathlon.", "topic": "probability"}
{"id": "book 2_p435_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 435, "page_end": 435, "topic_guess": "general", "text": "GMAT quant note: Three will win medals and six will not. Set up an anagram grid where Y represents a medal, N no medal.", "topic": "probability"}
{"id": "book 2_p435_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 435, "page_end": 435, "topic_guess": "general", "text": "GMAT quant note: Competitor C1 C2 C3 C4 C5 C6 C7 C8 C9 Medal Y Y Y N N N N N N Therefore, the number of ways three medals can be awarded is Now, determine the number of instances when at least two brothers win a medal. Practically speaking, this could happen when 1) exactly three brothers win or 2) exactly two brothers win.", "topic": "probability"}
{"id": "book 2_p435_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 435, "page_end": 435, "topic_guess": "general", "text": "GMAT quant note: Start with all three triplets winning medals, where Y represents a medal. Triplet A B C Non-triplet C1 C2 C3 C4 C5 C6 Medal Y Y Y Medal N N N N N N.", "topic": "probability"}
{"id": "book 2_p436_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 436, "page_end": 436, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. The number of ways this could happen is .", "topic": "probability"}
{"id": "book 2_p436_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 436, "page_end": 436, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: This makes sense, as there is only one instance in which all three triplets would win medals and all of the other competitors would not. (If you recognize this immediately, no need to write out the math.) Next, calculate the instances when exactly two of the triplets win medals.", "topic": "probability"}
{"id": "book 2_p436_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 436, "page_end": 436, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Triplet A B C Non-triplet C1 C2 C3 C4 C5 C6 Medal Y Y N Medal Y N N N N N Since both triplets and non-triplets win medals in this scenario, consider the possibilities for both sides of the grid.", "topic": "probability"}
{"id": "book 2_p436_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 436, "page_end": 436, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: For the triplets, the number of ways that two could win medals is For the non-triplet competitors, the number of ways that one could win the remaining medal is Multiply these two numbers to get the total number of instances: 3 × 6 = The brothers win at least two medals in 18 + 1 = 19 cases. The total number of cases is 84, so the probability is.", "topic": "probability"}
{"id": "book 2_p437_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 437, "page_end": 437, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. To answer the question, you’ll need to identify the periods of time during which different numbers of hikers are on the trail.", "topic": "word_problems"}
{"id": "book 2_p437_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 437, "page_end": 437, "topic_guess": "word_problems", "text": "GMAT word problem note: Next, identify the time period during which each hiker was on the trail. Floyd traveled the 5 miles at a rate of 3 miles per hour (mph), so he spent = 1 hour and 40 minutes on the trail.", "topic": "word_problems"}
{"id": "book 2_p437_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 437, "page_end": 437, "topic_guess": "word_problems", "text": "GMAT word problem note: Therefore, Floyd was on the trail from 8:07 to 9:47. Connie traveled the 5 miles at a rate of 2.5 mph, so she spent = 2 hours on the trail.", "topic": "word_problems"}
{"id": "book 2_p437_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 437, "page_end": 437, "topic_guess": "word_problems", "text": "GMAT word problem note: Connie was on the trail from 8:30 to Finally, Karen traveled the 5 miles at a rate of 4 mph, so she spent = 1 hour 15 minutes on the trail. Karen was on the trail from 10:00 to 11:15.", "topic": "word_problems"}
{"id": "book 2_p437_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 437, "page_end": 437, "topic_guess": "word_problems", "text": "GMAT word problem note: Plot each of these time periods on your drawing.", "topic": "word_problems"}
{"id": "book 2_p438_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 438, "page_end": 438, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. There were no periods of time during which all three hikers were on the trail.", "topic": "word_problems"}
{"id": "book 2_p438_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 438, "page_end": 438, "topic_guess": "word_problems", "text": "GMAT word problem note: However, there were two periods during which exactly two of the three hikers were on the trail. Floyd and Connie were both on the trail from 8:30 to 9:47, and Connie and Karen were both on the trail from 10:00 to 10:30.", "topic": "word_problems"}
{"id": "book 2_p438_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 438, "page_end": 438, "topic_guess": "word_problems", "text": "GMAT word problem note: This represents a total of 1 hour and 47 minutes, or approximately 1.75 hours. 75 hours is more than a third of 4 hours, but less than half of 4 hours.", "topic": "word_problems"}
{"id": "book 2_p438_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 438, "page_end": 438, "topic_guess": "word_problems", "text": "GMAT word problem note: It is also possible to simplify the fraction more precisely.", "topic": "word_problems"}
{"id": "book 2_p439_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 439, "page_end": 439, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Note that you must convert 15 minutes to 0.25 hours.", "topic": "word_problems"}
{"id": "book 2_p440_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 440, "page_end": 440, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Rate Time Distance Car A (r + 8) mph 0.75 hours (0.75) (r + 8) miles Car B r mph 0.25 hours 0.25r miles Total 62 miles Set up and solve an equation for the total distance.", "topic": "word_problems"}
{"id": "book 2_p440_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 440, "page_end": 440, "topic_guess": "word_problems", "text": "GMAT word problem note: Therefore, in 15 minutes, Car B traveled a distance of 0.25r = (0.25)(56) = 14 miles. Testing the choices would be a natural way to solve this problem, since the question doesn’t ask you to solve for b in general, but rather “for which of the following is x closest to zero?” However, numbers between 220 and 242 are too great to plug and compute.", "topic": "word_problems"}
{"id": "book 2_p440_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 440, "page_end": 440, "topic_guess": "word_problems", "text": "GMAT word problem note: Instead, manipulate the terms with base 8 to see how they might balance with 2b.", "topic": "word_problems"}
{"id": "book 2_p441_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 441, "page_end": 441, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. If k = 2, then which is less than 1.", "topic": "algebra"}
{"id": "book 2_p441_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 441, "page_end": 441, "topic_guess": "algebra", "text": "GMAT algebra note: Stop if you can tell that an answer won’t equal the target. Alternatively, solve the problem algebraically.", "topic": "algebra"}
{"id": "book 2_p441_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 441, "page_end": 441, "topic_guess": "algebra", "text": "GMAT algebra note: The expression given is of the form , where and.", "topic": "algebra"}
{"id": "book 2_p442_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 442, "page_end": 442, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. First, try to eliminate the radical signs entirely, leaving only a2 and b2 in the denominator.", "topic": "algebra"}
{"id": "book 2_p442_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 442, "page_end": 442, "topic_guess": "algebra", "text": "GMAT algebra note: To do so, multiply by a fraction that is a convenient form of 1. Notice the “difference of two squares” special product created in the denominator.", "topic": "algebra"}
{"id": "book 2_p442_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 442, "page_end": 442, "topic_guess": "algebra", "text": "GMAT algebra note: An algebraic setup would be fairly ugly on this problem, but a combination of two other techniques work quite nicely: choose smart numbers and estimate. There are no real values for the variables, so Choosing Smart Numbers is a valid approach.", "topic": "algebra"}
{"id": "book 2_p443_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 443, "page_end": 443, "topic_guess": "algebra", "text": "GMAT algebra note: Percent problems usually relate part, whole, rate, and percent change. That you can estimate if the Smart Numbers get a little messy.", "topic": "percent"}
{"id": "book 2_p443_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 443, "page_end": 443, "topic_guess": "algebra", "text": "GMAT algebra note: Of the three variables, it is easiest to pick for x. If you were to pick for y or z, you would have to do “reverse” calculations to find the other variables.", "topic": "percent"}
{"id": "book 2_p443_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 443, "page_end": 443, "topic_guess": "algebra", "text": "GMAT algebra note: Since this is a percent problem, try 100 first. E.g., Since x = 100, 90% of x is equal to 90.", "topic": "percent"}
{"id": "book 2_p443_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 443, "page_end": 443, "topic_guess": "algebra", "text": "GMAT algebra note: Therefore, you is equal to 20% less than 90 = 90 − 18 = 72 This number is a little ugly, so round down to 70. Therefore, y = 70, so 130% of y is equal to 70 + 21 = 91.", "topic": "percent"}
{"id": "book 2_p443_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 443, "page_end": 443, "topic_guess": "algebra", "text": "GMAT algebra note: The next step requires you to take 25% of that number, so 91 is going to get messy. Estimate again, but this time round up a little bit (to offset the error you introduced when you rounded down earlier).", "topic": "percent"}
{"id": "book 2_p443_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 443, "page_end": 443, "topic_guess": "algebra", "text": "GMAT algebra note: Round to the nearest number that is divisible by 4, which is 92. Next, find 25% more than 92.", "topic": "percent"}
{"id": "book 2_p443_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 443, "page_end": 443, "topic_guess": "algebra", "text": "GMAT algebra note: Z = 92 + 23 = 115 If x = 100 and z ≈ 115, then the final calculation is which is closest to 117%. The values in set A and set B are unknown.", "topic": "percent"}
{"id": "book 2_p443_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 443, "page_end": 443, "topic_guess": "algebra", "text": "GMAT algebra note: Jot them down in terms of variables. If x is a positive integer, then x, x + 1, and x + 2 are consecutive integers, and 3x, 3(x + 1), and 3(x + 2) are three consecutive.", "topic": "percent"}
{"id": "book 2_p444_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 444, "page_end": 444, "topic_guess": "number_theory", "text": "GMAT number properties note: Track mean, median, range, distribution, and how operations change spread. Similarly, set B contains 5y, 5(y + 1), 5(y + 2), 5(y + 3), and 5(y + 4), where y is a positive integer.", "topic": "statistics"}
{"id": "book 2_p444_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 444, "page_end": 444, "topic_guess": "number_theory", "text": "GMAT number properties note: Do not use the same variable for both sets, since it is unclear whether or how they are related. Sum the numbers in each set.", "topic": "statistics"}
{"id": "book 2_p444_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 444, "page_end": 444, "topic_guess": "number_theory", "text": "GMAT number properties note: Set A: 3x + 3(x + 1) + 3(x + 2) = 3(3x + 3) Set B: 5y + 5(y + 1) + 5(y + 2) + 5(y + 3) + 5(y + 4) = 5(5y + 10) The sums of the sets are equal. Set them equal to each other and simplify as much as possible.", "topic": "statistics"}
{"id": "book 2_p444_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 444, "page_end": 444, "topic_guess": "number_theory", "text": "GMAT number properties note: 3(3x + 3) = 5(5y + 10) 9(x + 1) = 25(y + 2) Because x and y are positive integers, the right side of the equation is a multiple of 25 and the le side is a multiple of 9. In other words, each side of the equation must be a multiple of both 9 and 25.", "topic": "statistics"}
{"id": "book 2_p444_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 444, "page_end": 444, "topic_guess": "number_theory", "text": "GMAT number properties note: Because 9 and 25 have no common factors, x + 1 must be a multiple of 25 and y + 2 must be a multiple of 9. The least possible value for x is x = 24, and the least number that could be a member of set A is 3x = 3(24) = 72.", "topic": "statistics"}
{"id": "book 2_p444_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 444, "page_end": 444, "topic_guess": "number_theory", "text": "GMAT number properties note: Manually add them up or note that for an evenly spaced set the sum is (# of terms) × (the median term). The integers in set B are multiples of 5, so the sum of set B is a multiple of 5.", "topic": "statistics"}
{"id": "book 2_p444_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 444, "page_end": 444, "topic_guess": "number_theory", "text": "GMAT number properties note: By extension, the (equivalent) sum of set A must be a multiple of 5.", "topic": "statistics"}
{"id": "book 2_p446_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 446, "page_end": 446, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. If m = 4n + 9, where n is a positive integer, what is the greatest common factor of m and n?", "topic": "number_properties"}
{"id": "book 2_p446_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 446, "page_end": 446, "topic_guess": "general", "text": "GMAT quant note: M = 9s, where s is a positive integer(1) n = 4t, where t is a positive integer(2) 13. A museum sold 30 tickets on Saturday.", "topic": "number_properties"}
{"id": "book 2_p446_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 446, "page_end": 446, "topic_guess": "general", "text": "GMAT quant note: Some of the tickets sold were $10 general exhibit tickets and the rest were $70 special exhibit tickets. How many general exhibit tickets did the museum sell on Saturday?", "topic": "number_properties"}
{"id": "book 2_p446_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 446, "page_end": 446, "topic_guess": "general", "text": "GMAT quant note: The museum’s total revenue from ticket sales on Saturday was greater than $1,570 and less than $1,670. The museum sold more than 20, but fewer than 25, special exhibit tickets on Saturday.", "topic": "number_properties"}
{"id": "book 2_p447_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 447, "page_end": 447, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If x and y are integers, is xy = yx?", "topic": "number_properties"}
{"id": "book 2_p447_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 447, "page_end": 447, "topic_guess": "number_theory", "text": "GMAT number properties note: What is the perimeter of isosceles triangle ABC? The length of side AB is 9.(1) The length of side BC is 4.(2) 17.", "topic": "number_properties"}
{"id": "book 2_p447_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 447, "page_end": 447, "topic_guess": "number_theory", "text": "GMAT number properties note: If x, y, and z are integers and 2x 5y z = 6.4 × 106, what is the value of xy?", "topic": "number_properties"}
{"id": "book 2_p448_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 448, "page_end": 448, "topic_guess": "number_theory", "text": "GMAT number properties note: Track mean, median, range, distribution, and how operations change spread. A school has a students and b teachers.", "topic": "statistics"}
{"id": "book 2_p448_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 448, "page_end": 448, "topic_guess": "number_theory", "text": "GMAT number properties note: If a < 150, b < 25, and classes have a maximum of 15 students, can the a students be distributed among the b teachers so that each class has the same number of students? (Assume that any student can be taught by any teacher.) It is possible to divide the students evenly into groups of 2, 3, 5, 6, 9, 10, or 15.", "topic": "statistics"}
{"id": "book 2_p448_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 448, "page_end": 448, "topic_guess": "number_theory", "text": "GMAT number properties note: The greatest common factor of a and b is 10.(2) 20. A set of five distinct positive integers has a median of 3 and a range of 12.", "topic": "statistics"}
{"id": "book 2_p448_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 448, "page_end": 448, "topic_guess": "number_theory", "text": "GMAT number properties note: What is the mean of the set of integers? The product of the integers in the set is a multiple of 14.(1) The sum of the integers in the set is a multiple of 13.(2).", "topic": "statistics"}
{"id": "book 2_p449_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 449, "page_end": 449, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "book 2_p450_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 450, "page_end": 450, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating. You can cross-multiply the inequality, as long as you consider both cases.", "topic": "word_problems"}
{"id": "book 2_p450_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 450, "page_end": 450, "topic_guess": "general", "text": "GMAT quant note: This is a conditional rephrased question—importantly, one with completely opposite questions as possibilities. Any statement that doesn’t at least answer the question of whether a − b is positive or negative is unlikely to be sufficient.", "topic": "word_problems"}
{"id": "book 2_p450_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 450, "page_end": 450, "topic_guess": "general", "text": "GMAT quant note: You would also have to carry each statement through both questions, so this path is not very efficient. It’s better to use the original inequality to test cases in order to eliminate the incorrect answers.", "topic": "word_problems"}
{"id": "book 2_p450_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 450, "page_end": 450, "topic_guess": "general", "text": "GMAT quant note: Test cases to determine whether this statement is sufficient. A b Is Case 1 2 1 ✓ No Case 2 2 Yes If a − b is negative, the question becomes “Is 1 < ab(a − b)?” (FLIPPED inequality sign) If a − b is positive, the question becomes “Is 1 > ab(a − b)?” (ORIGINAL inequality sign).", "topic": "word_problems"}
{"id": "book 2_p451_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 451, "page_end": 451, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. A b a < b Is Case 1 Yes Case 2 ?", "topic": "number_properties"}
{"id": "book 2_p451_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 451, "page_end": 451, "topic_guess": "general", "text": "GMAT quant note: No (1) AND (2) INSUFFICIENT:. Whenever possible, reuse cases you’ve already tested (you can only do this when the case makes both statements true).", "topic": "number_properties"}
{"id": "book 2_p451_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 451, "page_end": 451, "topic_guess": "general", "text": "GMAT quant note: No Once again, two valid sets of numbers return two different answers, so even together, the statements are insufficient.", "topic": "number_properties"}
{"id": "book 2_p452_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 452, "page_end": 452, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The greatest common factor (GCF) of any two numbers is given by the product of the shared or overlapping primes in the two numbers.", "topic": "number_properties"}
{"id": "book 2_p452_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 452, "page_end": 452, "topic_guess": "number_theory", "text": "GMAT number properties note: Any such question can be rephrased accordingly: “What exactly are the overlapping factors of m and n?” (1) SUFFICIENT:. This statement indicates that m must be a multiple of 9.", "topic": "number_properties"}
{"id": "book 2_p452_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 452, "page_end": 452, "topic_guess": "number_theory", "text": "GMAT number properties note: If m is a multiple of 9 and m = 4n + 9, then this equation is really saying. Multiple of 9 = 4n + 9 In order for this to be true, 4n must also be a multiple of 9.", "topic": "number_properties"}
{"id": "book 2_p452_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 452, "page_end": 452, "topic_guess": "number_theory", "text": "GMAT number properties note: The number 4 does not contain any factors of 9, so n itself must be a multiple of 9. If both m and n are multiples of 9, and m is exactly 9 units away from 4n, then the greatest possible common factor is 9, so this statement is sufficient.", "topic": "number_properties"}
{"id": "book 2_p452_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 452, "page_end": 452, "topic_guess": "number_theory", "text": "GMAT number properties note: (If you’re not sure about the logic of that last part, test out a couple of real numbers. n must be a multiple of 9.", "topic": "number_properties"}
{"id": "book 2_p452_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 452, "page_end": 452, "topic_guess": "number_theory", "text": "GMAT number properties note: If n = 9, then m = 4(9) + 9 = 45. The GCF of the numbers 9 and 45 is 9.", "topic": "number_properties"}
{"id": "book 2_p452_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 452, "page_end": 452, "topic_guess": "number_theory", "text": "GMAT number properties note: If n = 18, then m = 4(18) + 9 = 81. The GCF of the numbers 18 and 81 is still 9.) Alternatively, methodically list possible values of s, m, and n.", "topic": "number_properties"}
{"id": "book 2_p452_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 452, "page_end": 452, "topic_guess": "number_theory", "text": "GMAT number properties note: Ignore any case for which the statement is not true. For the remaining cases, determine the answer to the question.", "topic": "number_properties"}
{"id": "book 2_p452_c1_r9", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 452, "page_end": 452, "topic_guess": "number_theory", "text": "GMAT number properties note: S = pos int m = 9s n = pos int GCF of m and n?", "topic": "number_properties"}
{"id": "book 2_p453_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 453, "page_end": 453, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. S = pos int m = 9s n = pos int GCF of m and n?", "topic": "percent"}
{"id": "book 2_p453_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 453, "page_end": 453, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Zero ✗ Fraction ✗ Fraction ✗ Fraction ✗ Fraction ✗ Fraction ✗ Fraction ✗ Fraction ✗.", "topic": "percent"}
{"id": "book 2_p454_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 454, "page_end": 454, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. In this case, n is only a positive integer when (s − 1) is a multiple of 4.", "topic": "number_properties"}
{"id": "book 2_p454_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 454, "page_end": 454, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Also, m and n are both multiples of 9. Try the next number in the pattern, 9 + 4 = 13, to be confident that the apparent GCF pattern will continue.", "topic": "number_properties"}
{"id": "book 2_p454_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 454, "page_end": 454, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: S = pos int m = 9s n = pos int GCF of m and n? 5 (9)(5) = 45 9 (9)(9) = 81 13 (9)(13) = 117 The variables m and n always share an overlapping factor of 9, but there is never any overlap between their remaining factors.", "topic": "number_properties"}
{"id": "book 2_p454_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 454, "page_end": 454, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The GCF is always (2) INSUFFICIENT: n = 4t, where t is a positive integer. In this case, n must be a multiple of 4.", "topic": "number_properties"}
{"id": "book 2_p455_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 455, "page_end": 455, "topic_guess": "statistics", "text": "GMAT statistics note: Watch divisibility, factors, remainders, parity, and prime structure. T = pos int n = 4t m = 4n+ 9 GCF of m and n?", "topic": "number_properties"}
{"id": "book 2_p455_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 455, "page_end": 455, "topic_guess": "statistics", "text": "GMAT statistics note: 1 4 16 + 9 = 25 1 2 8 32 + 9 = 41 1 3 12 48 + 9 = 57 3 There is more than one possible GCF, so the statement is not sufficient. Use the integer constraint to test possible cases.", "topic": "number_properties"}
{"id": "book 2_p455_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 455, "page_end": 455, "topic_guess": "statistics", "text": "GMAT statistics note: Because the special tickets are so much more expensive, begin by choosing numbers for which (70)(# of special tickets) is approximately $1,600. If g = 7, total revenue = 10(7) + 70(23) = 70 + 1,610 = 1,680.", "topic": "number_properties"}
{"id": "book 2_p455_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 455, "page_end": 455, "topic_guess": "statistics", "text": "GMAT statistics note: If g = 8, total revenue = 10(8) + 70(22) = 80 + 1,540 = 1,620. If g = 9, total revenue = 10(9) + 70(21) = 90 + 1,470 = 1,560.", "topic": "number_properties"}
{"id": "book 2_p455_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 455, "page_end": 455, "topic_guess": "statistics", "text": "GMAT statistics note: Only g = 8 gives a total revenue in the given range. Don’t assume that having a range of values automatically means a statement is insufficient to answer a Value question.", "topic": "number_properties"}
{"id": "book 2_p456_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 456, "page_end": 456, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Constraint (such as an integer constraint) may limit the number of cases to just one possibility.", "topic": "algebra"}
{"id": "book 2_p456_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 456, "page_end": 456, "topic_guess": "algebra", "text": "GMAT algebra note: Alternatively, you can use algebra, though that path is a bit long on this problem. Then (30 − g) represents the number of special exhibit tickets sold.", "topic": "algebra"}
{"id": "book 2_p456_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 456, "page_end": 456, "topic_guess": "algebra", "text": "GMAT algebra note: Set up and solve the following inequality:. The only integer between 7.2 and 8.8 is 8, so g must be 8.", "topic": "algebra"}
{"id": "book 2_p456_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 456, "page_end": 456, "topic_guess": "algebra", "text": "GMAT algebra note: The museum sold 8 general exhibit tickets. This statement indicates that the museum sold 21, 22, 23, or 24 special exhibit tickets.", "topic": "algebra"}
{"id": "book 2_p456_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 456, "page_end": 456, "topic_guess": "algebra", "text": "GMAT algebra note: Since the museum sold a total of 30 tickets, this means that it sold 9, 8, 7, or 6 general exhibit tickets. Test cases on this theory problem.", "topic": "algebra"}
{"id": "book 2_p456_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 456, "page_end": 456, "topic_guess": "algebra", "text": "GMAT algebra note: If you can generate both Yes and No cases, you will prove insufficiency.", "topic": "algebra"}
{"id": "book 2_p457_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 457, "page_end": 457, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Case 1 2 0 20 = 1 02 = 0 No Case 2 3 1 31 = 3 13 = 1 No Case 3 4 2 42 = 16 24 = 16 Yes (2) INSUFFICIENT:.", "topic": "number_properties"}
{"id": "book 2_p457_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 457, "page_end": 457, "topic_guess": "number_theory", "text": "GMAT number properties note: Case 1 8 1 81 = 8 18 = 1 No Case 2 4 2 42 = 16 24 = 16 Yes (1) AND (2) SUFFICIENT:. Test whether any cases work for both statements.", "topic": "number_properties"}
{"id": "book 2_p457_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 457, "page_end": 457, "topic_guess": "number_theory", "text": "GMAT number properties note: The case x = 4 and y = 2 was already used for both statements and returned a Yes answer. Can any of the No cases work for both statements?", "topic": "number_properties"}
{"id": "book 2_p457_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 457, "page_end": 457, "topic_guess": "number_theory", "text": "GMAT number properties note: If you are good with numbers, you can try to find another pair of integers that will work with both equations. Since x and y have to be integers, there are only a limited number of possibilities.", "topic": "number_properties"}
{"id": "book 2_p457_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 457, "page_end": 457, "topic_guess": "number_theory", "text": "GMAT number properties note: The two numbers need to multiply to 4 and.", "topic": "number_properties"}
{"id": "book 2_p458_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 458, "page_end": 458, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. In addition to 4 and 2, the pair −4 and −2 fit the bill, as long as x = −2 and y = −4.", "topic": "algebra"}
{"id": "book 2_p458_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 458, "page_end": 458, "topic_guess": "algebra", "text": "GMAT algebra note: Alternatively, solve algebraically to determine the possible values for x and y. You have already tested the case where x = 4 (and y = 2), so test the other case.", "topic": "algebra"}
{"id": "book 2_p458_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 458, "page_end": 458, "topic_guess": "algebra", "text": "GMAT algebra note: Yes In either case, xy = yx. Odd exponents do not “hide the sign” of the base.", "topic": "algebra"}
{"id": "book 2_p458_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 458, "page_end": 458, "topic_guess": "algebra", "text": "GMAT algebra note: If ab3c4 > 0, then a and b must have the same sign so that their product is positive.", "topic": "algebra"}
{"id": "book 2_p459_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 459, "page_end": 459, "topic_guess": "geometry", "text": "GMAT geometry note: Case, a3b must also be positive. As a result, in order for the inequality a3bc5 to be positive, c5 must be positive.", "topic": "geometry"}
{"id": "book 2_p459_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 459, "page_end": 459, "topic_guess": "geometry", "text": "GMAT geometry note: The rephrased question is, “Is c (1) INSUFFICIENT:. No information about the sign of c.", "topic": "geometry"}
{"id": "book 2_p459_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 459, "page_end": 459, "topic_guess": "geometry", "text": "GMAT geometry note: Answers the rephrased question directly. The perimeter of a triangle is equal to the sum of the three sides.", "topic": "geometry"}
{"id": "book 2_p459_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 459, "page_end": 459, "topic_guess": "geometry", "text": "GMAT geometry note: Knowing the length of one side of the triangle is not enough to find the sum of all three sides. Triangle ABC is an isosceles triangle, which means that two of the sides are equal in length.", "topic": "geometry"}
{"id": "book 2_p459_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 459, "page_end": 459, "topic_guess": "geometry", "text": "GMAT geometry note: The statements provide two of the side lengths, so the third side, AC, must equal one of the given sides. There is a hidden constraint in this problem: the triangle must be valid.", "topic": "geometry"}
{"id": "book 2_p459_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 459, "page_end": 459, "topic_guess": "geometry", "text": "GMAT geometry note: Recall that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. AB BC AC Perimeter Valid triangle?", "topic": "geometry"}
{"id": "book 2_p459_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 459, "page_end": 459, "topic_guess": "geometry", "text": "GMAT geometry note: 9 4 4 9 + 4 + 4 = 17 No: 4 + 4 < 9.", "topic": "geometry"}
{"id": "book 2_p460_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 460, "page_end": 460, "topic_guess": "geometry", "text": "GMAT geometry note: AB BC AC Perimeter Valid triangle? 9 4 9 9 + 4 + 9 = 22 Yes: 4 + 9 > 9 A “triangle” with three sides of 4, 4, and 9 is not really a triangle, as it cannot be drawn with those dimensions.", "topic": "geometry"}
{"id": "book 2_p460_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 460, "page_end": 460, "topic_guess": "geometry", "text": "GMAT geometry note: Therefore, the actual sides of the triangle must be AB = 9, BC = 4, and AC = 9. Express both sides of the equation in terms of prime numbers.", "topic": "geometry"}
{"id": "book 2_p460_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 460, "page_end": 460, "topic_guess": "geometry", "text": "GMAT geometry note: The right side of the equation is composed of only 2’s and 5’s. The le side of the equation has x number of 2’s and y number of 5’s along with some factor z.", "topic": "geometry"}
{"id": "book 2_p460_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 460, "page_end": 460, "topic_guess": "geometry", "text": "GMAT geometry note: This unknown factor z must be composed of only 2’s and/or 5’s, or it must be 1 (i.e., with no prime factors). If z = 1, then x = 11 and y = 5.", "topic": "geometry"}
{"id": "book 2_p461_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 461, "page_end": 461, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If z = 2?5?, where the exponents are not 0, then x and y will depend on the value of those exponents.", "topic": "number_properties"}
{"id": "book 2_p461_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 461, "page_end": 461, "topic_guess": "number_theory", "text": "GMAT number properties note: The rephrased question is thus, “How many factors of 2 and 5 are in z?” (1) SUFFICIENT:. If z = 20 = 2251, then this answers the rephrased question.", "topic": "number_properties"}
{"id": "book 2_p461_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 461, "page_end": 461, "topic_guess": "number_theory", "text": "GMAT number properties note: Incidentally, this implies that 2x5y(2251) = 21155, so x = 9 and y = 4, making xy = 36 (though you do not have to solve for x and y). (2) INSUFFICIENT: x = 9, but the statement doesn’t indicate anything about the value of y.", "topic": "number_properties"}
{"id": "book 2_p461_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 461, "page_end": 461, "topic_guess": "number_theory", "text": "GMAT number properties note: This problem deals with both squares and inequalities. negative numbers become positive when squared and that squaring a number less than 1 results in a value that is even closer to 0.", "topic": "number_properties"}
{"id": "book 2_p461_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 461, "page_end": 461, "topic_guess": "number_theory", "text": "GMAT number properties note: Without any more information about z, this statement only tells you that x2 is greater than y2. However, x could be either greater than y or less than y, depending on whether x is positive or negative.", "topic": "number_properties"}
{"id": "book 2_p461_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 461, "page_end": 461, "topic_guess": "number_theory", "text": "GMAT number properties note: Since all squares are positive, x2 is much greater than y2. In this case, x2 is greater than y2 and x is greater than y.", "topic": "number_properties"}
{"id": "book 2_p462_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 462, "page_end": 462, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. This statement does not provide any information about y, so y could be greater than, less than, or equal to x.", "topic": "word_problems"}
{"id": "book 2_p462_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 462, "page_end": 462, "topic_guess": "word_problems", "text": "GMAT word problem note: This implies that |x| is greater than |z|. There are only two situations in which x is greater than z and the absolute value of x is greater than the absolute value of z.", "topic": "word_problems"}
{"id": "book 2_p462_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 462, "page_end": 462, "topic_guess": "word_problems", "text": "GMAT word problem note: One possibility is that x is positive and z is a lesser positive number. Another possibility is that x is positive and z is a negative number with an absolute value that is less than x.", "topic": "word_problems"}
{"id": "book 2_p462_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 462, "page_end": 462, "topic_guess": "word_problems", "text": "GMAT word problem note: These scenarios are illustrated below. Therefore, x is definitely positive.", "topic": "word_problems"}
{"id": "book 2_p462_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 462, "page_end": 462, "topic_guess": "word_problems", "text": "GMAT word problem note: Since x is positive, |x| = x, so x > |y|. If x is greater than the absolute value of y, x must also be greater than y itself, so the answer to the question is definitely Yes.", "topic": "word_problems"}
{"id": "book 2_p463_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 463, "page_end": 463, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. In order for the a students to be distributed evenly among the b teachers, a must be divisible by b.", "topic": "number_properties"}
{"id": "book 2_p463_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 463, "page_end": 463, "topic_guess": "number_theory", "text": "GMAT number properties note: This is a factor question in disguise: “Is a divisible by b?” (1) INSUFFICIENT:. The given numbers represent factors of a.", "topic": "number_properties"}
{"id": "book 2_p463_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 463, "page_end": 463, "topic_guess": "number_theory", "text": "GMAT number properties note: If the a students can be divided evenly into a group of 2, then 2 is a factor of a. Likewise, 3 and 5 must be factors of a.", "topic": "number_properties"}
{"id": "book 2_p463_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 463, "page_end": 463, "topic_guess": "number_theory", "text": "GMAT number properties note: If 2 and 3 are already factors of a, then 6 must also be a factor (since 2 × 3 = 6). Likewise, you’ve already counted one factor of 3, so you need only one more factor of 3 to make 9.", "topic": "number_properties"}
{"id": "book 2_p463_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 463, "page_end": 463, "topic_guess": "number_theory", "text": "GMAT number properties note: You already have the necessary factors to make both 10 and 15, so ignore those numbers as well. The final list of factors is: 2, 3, 5, and 3.", "topic": "number_properties"}
{"id": "book 2_p463_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 463, "page_end": 463, "topic_guess": "number_theory", "text": "GMAT number properties note: As a result, a must equal (2)(3)(5)(3) = 90 or a multiple of 90. Since the question stem indicated that a < 150, a must actually be 90.", "topic": "number_properties"}
{"id": "book 2_p463_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 463, "page_end": 463, "topic_guess": "number_theory", "text": "GMAT number properties note: This statement provides no information about the teachers, however, so it is insufficient. If the greatest common factor (GCF) of a and b is 10, then a and b must both be multiples of 10.", "topic": "number_properties"}
{"id": "book 2_p463_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 463, "page_end": 463, "topic_guess": "number_theory", "text": "GMAT number properties note: Since the question stem indicated that b < 25, b must be 10 or 20. The value of a is any multiple of 10 up to 140, inclusive.", "topic": "number_properties"}
{"id": "book 2_p463_c1_r9", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 463, "page_end": 463, "topic_guess": "number_theory", "text": "GMAT number properties note: This is not enough to determine whether a is divisible by b.", "topic": "number_properties"}
{"id": "book 2_p464_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 464, "page_end": 464, "topic_guess": "number_theory", "text": "GMAT number properties note: Track mean, median, range, distribution, and how operations change spread. All five of the integers in the set are positive, and all of the integers are distinct, or different from each other.", "topic": "statistics"}
{"id": "book 2_p464_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 464, "page_end": 464, "topic_guess": "number_theory", "text": "GMAT number properties note: Jot down blanks to represent the values in the set. The least two integers in the set must be less than the median, different from each other, and positive: the first two numbers must be 1 and 2.", "topic": "statistics"}
{"id": "book 2_p464_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 464, "page_end": 464, "topic_guess": "number_theory", "text": "GMAT number properties note: Finally, the range of the set, or the difference between the greatest and least integers, is 12. Therefore, the greatest integer in the set is 13.", "topic": "statistics"}
{"id": "book 2_p464_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 464, "page_end": 464, "topic_guess": "number_theory", "text": "GMAT number properties note: The only unknown value in the set is the second-greatest integer, which must be between 3 and 13, exclusive. Finding this integer will give you enough information to find the mean of the set.", "topic": "statistics"}
{"id": "book 2_p464_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 464, "page_end": 464, "topic_guess": "number_theory", "text": "GMAT number properties note: For a number to be a multiple of 14, it must be a multiple of both 2 and 7. The product of the four known values is already a multiple of 2, since the set already contains the number 2.", "topic": "statistics"}
{"id": "book 2_p464_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 464, "page_end": 464, "topic_guess": "number_theory", "text": "GMAT number properties note: However, there are no multiples of 7 among the known values.", "topic": "statistics"}
{"id": "book 2_p465_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 465, "page_end": 465, "topic_guess": "statistics", "text": "GMAT statistics note: Watch divisibility, factors, remainders, parity, and prime structure. Missing fourth value must be a multiple of 7.", "topic": "number_properties"}
{"id": "book 2_p465_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 465, "page_end": 465, "topic_guess": "statistics", "text": "GMAT statistics note: The only multiple of 7 between 3 and 13 is 7 itself, so this is the missing integer and the mean can now be calculated. The four known integers sum to 19.", "topic": "number_properties"}
{"id": "book 2_p465_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 465, "page_end": 465, "topic_guess": "statistics", "text": "GMAT statistics note: Since the unknown integer must be positive, the sum of the entire set must be greater than 19. The next multiples of 13 are 13(2) = 26 and 13(3) = 39, which would imply that the missing integer is 26 – 19 = 7 or 39 – 19 = 20, respectively.", "topic": "number_properties"}
{"id": "book 2_p465_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 465, "page_end": 465, "topic_guess": "statistics", "text": "GMAT statistics note: (For greater multiples of 13, the missing integer would have to be even greater than 20.) However, the missing integer must be between 3 and 13, so 20 is too great. The missing integer must be 7, and the mean can be calculated.", "topic": "number_properties"}
{"id": "book 2_p466_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 466, "page_end": 466, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If , which of the following is true of g?", "topic": "number_properties"}
{"id": "book 2_p466_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 466, "page_end": 466, "topic_guess": "number_theory", "text": "GMAT number properties note: If x and y are positive integers, what is the remainder when xyis divided by 10? For all positive integers n, the sequence An is defined by the following relationship.", "topic": "number_properties"}
{"id": "book 2_p467_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 467, "page_end": 467, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. What is the sum of all the terms in the sequence from A1 through A10, inclusive?", "topic": "number_properties"}
{"id": "book 2_p467_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 467, "page_end": 467, "topic_guess": "number_theory", "text": "GMAT number properties note: If x and y are positive integers and n = 5x + 7y + 3, what is the units digit of n? Y = 2x − 16(1) y is divisible by 4.(2).", "topic": "number_properties"}
{"id": "book 2_p468_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 468, "page_end": 468, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. At a certain store, all notebooks have the same price and all pencils have the same price.", "topic": "algebra"}
{"id": "book 2_p468_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 468, "page_end": 468, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The price of four notebooks and three pencils is more than $12 and less than $13. The price of two notebooks and five pencils is more than $8 and less than $9.", "topic": "algebra"}
{"id": "book 2_p468_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 468, "page_end": 468, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: If a notebook costs x more than a pencil, which of the following could be the value of x? A certain sequence is defined by the following rule:.", "topic": "algebra"}
{"id": "book 2_p468_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 468, "page_end": 468, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Sn = k(Sn − 1), where k is a constant. If S1 = 2 and S13 = 72, what is the value of S7?", "topic": "algebra"}
{"id": "book 2_p468_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 468, "page_end": 468, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: N is divisible by 3.(1) n is odd.(2).", "topic": "algebra"}
{"id": "book 2_p469_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 469, "page_end": 469, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. In a certain sequence, each term a er the first is twice the previous term.", "topic": "algebra"}
{"id": "book 2_p469_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 469, "page_end": 469, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: If the first term of the sequence is 3, what is the sum of the 14th, 15th, and 16th terms in the sequence? For n > 0, the sequence tn is defined as tn = tn − 1 + n.", "topic": "algebra"}
{"id": "book 2_p469_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 469, "page_end": 469, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: N + 1 is divisible by 3.(1) n − 1 is divisible by 4.(2) 30. To complete a 120-mile trip, a train first travels at a constant rate of x miles per hour (mph) for 80 miles.", "topic": "algebra"}
{"id": "book 2_p469_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 469, "page_end": 469, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Then it travels the remaining 40 miles at a constant rate of y mph. If the train had instead completed the entire trip at a constant rate of z mph, the trip would have taken 1 hour longer.", "topic": "algebra"}
{"id": "book 2_p469_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 469, "page_end": 469, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Which of the following is the value of z in terms of x and y?", "topic": "algebra"}
{"id": "book 2_p470_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 470, "page_end": 470, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "book 2_p471_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 471, "page_end": 471, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "book 2_p472_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 472, "page_end": 472, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Therefore, f(x) equals the average of the integers from 1 to x, which is , multiplied by the number of integers from 1 to x, which is x.", "topic": "number_properties"}
{"id": "book 2_p472_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 472, "page_end": 472, "topic_guess": "number_theory", "text": "GMAT number properties note: Therefore, f(x) = The question asks for the sum of the values of , and so on up to . Instead of summing these values by hand, look for a pattern, since the problem only asks for a range and not a specific answer.", "topic": "number_properties"}
{"id": "book 2_p472_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 472, "page_end": 472, "topic_guess": "number_theory", "text": "GMAT number properties note: One possibility is to use algebra. Can be simplified as follows.", "topic": "number_properties"}
{"id": "book 2_p473_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 473, "page_end": 473, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Using this formula, the first few terms of the sum are , and so on, until the tenth term, which is .", "topic": "algebra"}
{"id": "book 2_p473_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 473, "page_end": 473, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: All of these terms are greater than and no greater than 1. If all of the terms were 0.5, then the sum would be 10(0.5) = 5.", "topic": "algebra"}
{"id": "book 2_p473_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 473, "page_end": 473, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: However, if all of the terms were 1, then the sum would be 10. Therefore, the sum must be between 5 and 10.", "topic": "algebra"}
{"id": "book 2_p473_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 473, "page_end": 473, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: This problem can also be solved without using a formula for f(x). Manually calculate the first few values of f(x), and look for a pattern.", "topic": "algebra"}
{"id": "book 2_p474_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 474, "page_end": 474, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. The values appear to be decreasing and approaching .", "topic": "number_properties"}
{"id": "book 2_p474_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 474, "page_end": 474, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: If each term in the sum is greater than but no greater than 1, their sum will be greater than 5 but no greater than 10. The question asks for the remainder when xy is divided by 10.", "topic": "number_properties"}
{"id": "book 2_p474_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 474, "page_end": 474, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The remainder of any number when divided by 10 is equal to the units digit of that number, so the question is really asking for the units digit of xy. But 26 has a units digit of 6, and remember that 6any positive integer has a units digit of 6 (the pattern is a single-term repeat).", "topic": "number_properties"}
{"id": "book 2_p474_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 474, "page_end": 474, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Thus, 26 raised to ANY positive integer power will also have a units digit of 6 and therefore a remainder of 6 when divided by 10. Given that yx = 1, there are a few possible scenarios.", "topic": "number_properties"}
{"id": "book 2_p475_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 475, "page_end": 475, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. X y yx = 1 0 anything nonzero (anything nonzero)0 = 1 ✓ anything 1 1anything = 1 ✓ even −1 −1even = 1 ✓ However, the question stem indicates that x and y are POSITIVE integers, so eliminate the first and third scenarios.", "topic": "number_properties"}
{"id": "book 2_p475_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 475, "page_end": 475, "topic_guess": "number_theory", "text": "GMAT number properties note: The remaining scenario indicates that y = 1 and x = any positive integer. Without more information about x, you cannot determine the remainder when xy is divided by 10.", "topic": "number_properties"}
{"id": "book 2_p475_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 475, "page_end": 475, "topic_guess": "number_theory", "text": "GMAT number properties note: Set up a table to list the first few terms of the sequence and also the cumulative sum.", "topic": "number_properties"}
{"id": "book 2_p476_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 476, "page_end": 476, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. N Cumulative Sumn Cumulative Sum As you build the table, compare the input column values (n) with the output column values (cumulative sum), looking for a pattern.", "topic": "algebra"}
{"id": "book 2_p476_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 476, "page_end": 476, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The denominator of the cumulative sum is n! And the numerator is one less than n!.", "topic": "algebra"}
{"id": "book 2_p476_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 476, "page_end": 476, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Sum of terms through Substitute to find the sum through A10.", "topic": "algebra"}
{"id": "book 2_p477_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 477, "page_end": 477, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. The units digit of n is determined solely by the units digit of the expressions 5x and 7y + 3 because when two numbers are added together, the units digit of the sum is determined solely by the units digits of the added numbers.", "topic": "number_properties"}
{"id": "book 2_p477_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 477, "page_end": 477, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Since x is a positive integer, and 5any positive integer always has a units digit of 5, 5x always ends in a 5. However, the units digit of 7y + 3 is not certain, as the units digit pattern for the powers of 7 is a four-term repeat: [7, 9, The question can be rephrased as, “What is the units digit of 7y + 3?” Note:.", "topic": "number_properties"}
{"id": "book 2_p477_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 477, "page_end": 477, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Determining y would be one way of answering the question above, but don’t rephrase to, “What is y?” Because the units digits of the powers of 7 have a repeating pattern, you might get a single answer for the units digit of 7y + 3 despite having multiple values for y.", "topic": "number_properties"}
{"id": "book 2_p477_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 477, "page_end": 477, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: This statement indicates neither the value of y nor the units digit of 7y + 3, as y depends on the value of x, which could be any positive integer. By contrast, if x = 10, then y = 4, and 7y + 3 has a units digit of 3.", "topic": "number_properties"}
{"id": "book 2_p477_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 477, "page_end": 477, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The statement does indicate that y is an even number, so the exponent y + 3 must be odd. As a result, there are only two possible values for the units digit of the d.", "topic": "number_properties"}
{"id": "book 2_p478_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 478, "page_end": 478, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Y y + 3 Units Digit of 7y + 3y y + 3 Units Digit of 7y + 3 Ultimately, this means that n has a units digit of 5 + 3 = 8.", "topic": "algebra"}
{"id": "book 2_p478_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 478, "page_end": 478, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Instead, start by translating the information in the problem into math. The first piece of information in the problem translates to the following inequality:.", "topic": "algebra"}
{"id": "book 2_p478_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 478, "page_end": 478, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The second piece of information translates to this inequality.", "topic": "algebra"}
{"id": "book 2_p479_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 479, "page_end": 479, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. The question asks for the difference between the cost of a notebook and the cost of a pencil.", "topic": "statistics"}
{"id": "book 2_p479_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 479, "page_end": 479, "topic_guess": "statistics", "text": "GMAT statistics note: In other words, what is the value of n – p? One option is to use the previous two inequalities to create a third inequality, which includes the combo n – p.", "topic": "statistics"}
{"id": "book 2_p479_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 479, "page_end": 479, "topic_guess": "statistics", "text": "GMAT statistics note: One way to accomplish this is to multiply the second inequality by –1, taking care to flip the signs. Then, reorder the result so that the inequality signs point in the expected direction:.", "topic": "statistics"}
{"id": "book 2_p479_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 479, "page_end": 479, "topic_guess": "statistics", "text": "GMAT statistics note: Now, sum the two inequalities. You can add inequalities as long as the inequality signs are facing the same direction:.", "topic": "statistics"}
{"id": "book 2_p479_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 479, "page_end": 479, "topic_guess": "statistics", "text": "GMAT statistics note: This simplifies to 1.5 < n – p < 2.5. It is also possible to “solve” the two inequalities for n and for p, separately, then to use the resulting ranges to find the possible differences between n and p.", "topic": "statistics"}
{"id": "book 2_p480_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 480, "page_end": 480, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. To form each new term of the sequence, multiply the previous term by k.", "topic": "algebra"}
{"id": "book 2_p480_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 480, "page_end": 480, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: If S1 = 2, then S2 = 2k, and S3 = 2k2, and Sn = 2kn − 1. Since the problem indicates that S13 = 72, set up an equation to solve for k.", "topic": "algebra"}
{"id": "book 2_p480_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 480, "page_end": 480, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Plug the value of k into that expression to calculate S7. Remember the units digit pattern for 9x, where x is an integer.", "topic": "algebra"}
{"id": "book 2_p480_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 480, "page_end": 480, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The units digit of 9x is 9 if x is odd, but the units digit is 1 if x is even: a repeating pattern of [9, 1]. Now, consider the sums of the powers of 9 up to 9n.", "topic": "algebra"}
{"id": "book 2_p481_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 481, "page_end": 481, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. 1 1 + 9 = 10 0 2 1 + 9 + 81 = 91 1 3 [1 + 9 + 81] + 729 = [units digit of 1] + units digit of 9 0 4 Units digit of 0 + units digit of 1 1 Odd Units digit: pairs of (1 + 9) 0 Even Units digit: pairs of (1 + 9) + 1 1 The alternating 1’s and 9’s in the units digits pair to a sum of 10, or a units digit of 0.", "topic": "number_properties"}
{"id": "book 2_p481_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 481, "page_end": 481, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Thus, the units digit of the sum displays another two- term repeating pattern. The units digit of y is 0 if n is odd, but 1 if n is even.", "topic": "number_properties"}
{"id": "book 2_p481_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 481, "page_end": 481, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The remainder when y is divided by 5 depends only on the units digit and will be either 0 or 1 as well. The rephrased question is, “Is n odd or even?” (1) INSUFFICIENT:.", "topic": "number_properties"}
{"id": "book 2_p481_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 481, "page_end": 481, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: If n is a multiple of 3, it may be either odd or even. If n is odd, the units digit of y is 0, and the remainder is 0 when y is divided by 5.", "topic": "number_properties"}
{"id": "book 2_p482_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 482, "page_end": 482, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. The sequence is 3, 6, 12, 24, 48, and so on.", "topic": "algebra"}
{"id": "book 2_p482_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 482, "page_end": 482, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: You could write out the first 16 terms and add the 14th, 15th, and 16th together, but such an approach would be prone to error and time-consuming. Write the sequence in terms of the powers of 2: 3, 6, 12, 24, 48, and so on 3(20), 3(21), 3(22), 3(23), 3(24), and so on So Sn, the nth term of the sequence, equals 3(2n − 1).", "topic": "algebra"}
{"id": "book 2_p482_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 482, "page_end": 482, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Thus, the sum of the 14th, 15th, and 16th terms equals 3(213) + 3(214) + All of the terms share the common factors 3 and 213, so factor those terms out.", "topic": "algebra"}
{"id": "book 2_p483_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 483, "page_end": 483, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. Sequence problems are o en best approached by charting out the first several terms of the given sequence.", "topic": "number_properties"}
{"id": "book 2_p483_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 483, "page_end": 483, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: In this case, keep track of n, tn, and whether tn is even or odd. 0 3 Odd 1 3 + 1 = 4 Even 2 4 + 2 = 6 Even 3 6 + 3 = 9 Odd 4 9 + 4 = 13 Odd 5 13 + 5 = 18 Even 6 18 + 6 = 24 Even 7 24 + 7 = 31 Odd 8 31 + 8 = 39 Odd Notice that beginning with n = 1, a four-term repeating cycle of [even, even, odd, odd] emerges for tn.", "topic": "number_properties"}
{"id": "book 2_p483_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 483, "page_end": 483, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Thus, a statement will be sufficient only.", "topic": "number_properties"}
{"id": "book 2_p484_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 484, "page_end": 484, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. If it indicates how n relates to a multiple of 4 (i.e., n = a multiple of 4 ± known constant).", "topic": "algebra"}
{"id": "book 2_p484_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 484, "page_end": 484, "topic_guess": "algebra", "text": "GMAT algebra note: This statement does not indicate how n relates to a multiple of 4. If n + 1 is a multiple of 3, then n + 1 could be 3, 6, 9, 12, 15, etc.", "topic": "algebra"}
{"id": "book 2_p484_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 484, "page_end": 484, "topic_guess": "algebra", "text": "GMAT algebra note: This means that n could be 2, 5, 8, 11, 14, etc. From the chart, if n = 2 or n = 5, then tn is even.", "topic": "algebra"}
{"id": "book 2_p484_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 484, "page_end": 484, "topic_guess": "algebra", "text": "GMAT algebra note: However, if n = 8 or n = 11, then tn is odd. This statement indicates exactly how n relates to a multiple of 4.", "topic": "algebra"}
{"id": "book 2_p484_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 484, "page_end": 484, "topic_guess": "algebra", "text": "GMAT algebra note: If n − 1 is a multiple of 4, then n − 1 could be 4, 8, 12, 16, 20, etc., and n could be 5, 9, 13, 17, 21, etc. From the chart (and the continuation of the four-term pattern), tn must be even.", "topic": "algebra"}
{"id": "book 2_p484_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 484, "page_end": 484, "topic_guess": "algebra", "text": "GMAT algebra note: If you choose to use Smart Numbers, select your numbers cautiously to simplify the arithmetic. Choose x = y = 40, so that the train completes the entire trip at 40 mph, taking = 3 hours.", "topic": "algebra"}
{"id": "book 2_p484_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 484, "page_end": 484, "topic_guess": "algebra", "text": "GMAT algebra note: If the train had instead traveled at z mph, it would have taken one more hour, or 4 hours in total.", "topic": "algebra"}
{"id": "book 2_p485_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 485, "page_end": 485, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. The correct answer will yield a value of z = 30 mph when x = 40 and y = 40 are plugged in.", "topic": "word_problems"}
{"id": "book 2_p485_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 485, "page_end": 485, "topic_guess": "word_problems", "text": "GMAT word problem note: Alternatively, solve algebraically by creating an RTD chart. Rate Time Distance x mph hours 80 miles.", "topic": "word_problems"}
{"id": "book 2_p486_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 486, "page_end": 486, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Rate Time Distance y mph hours 40 miles (Original speed) hours 120 miles z mph hours 120 miles Use the last row of the chart to create an equation, which can then be solved for z.", "topic": "word_problems"}
{"id": "book 2_p487_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 487, "page_end": 487, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure.", "topic": "number_properties"}
{"id": "book 2_p488_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 488, "page_end": 488, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. If y ≠ x, then is the equivalent of which of the following?", "topic": "algebra"}
{"id": "book 2_p488_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 488, "page_end": 488, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: In the sequence gn defined for all positive integer values of n, g1 = g2 = 1 and, for n ≥ 3, gn = gn − 1 + 2n − 3. If the function equals the sum of the terms g1, g2...", "topic": "algebra"}
{"id": "book 2_p489_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 489, "page_end": 489, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If 3k + 3k = (39)39 − 3k, then what is the value of k?", "topic": "number_properties"}
{"id": "book 2_p489_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 489, "page_end": 489, "topic_guess": "number_theory", "text": "GMAT number properties note: If x and y are positive integers, what is the value of xy?", "topic": "number_properties"}
{"id": "book 2_p490_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 490, "page_end": 490, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If k is an integer and is divisible by 6k, what is the maximum possible value of k?", "topic": "number_properties"}
{"id": "book 2_p490_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 490, "page_end": 490, "topic_guess": "number_theory", "text": "GMAT number properties note: In a certain sequence, the term Sn is given by the formula Sn = (n + 1)! Which of the following is equivalent to the difference between S100 and S99?", "topic": "number_properties"}
{"id": "book 2_p492_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 492, "page_end": 492, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "book 2_p493_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 493, "page_end": 493, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Simplify the right side of the equation.", "topic": "algebra"}
{"id": "book 2_p493_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 493, "page_end": 493, "topic_guess": "algebra", "text": "GMAT algebra note: This still needs to be manipulated in order to match the le side of the equation, since the bases need to match their exponents. The number 33 is okay, but 28 is not.", "topic": "algebra"}
{"id": "book 2_p493_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 493, "page_end": 493, "topic_guess": "algebra", "text": "GMAT algebra note: Try turning it into a base of 4 instead. That’s it: x = 3 and y = 4, since the problem states that y > x.", "topic": "algebra"}
{"id": "book 2_p493_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 493, "page_end": 493, "topic_guess": "algebra", "text": "GMAT algebra note: Note that the expression can be rewritten as follows by pairing up consecutive terms in the sequence. In each of these pairs, the 94 term cancels out.", "topic": "algebra"}
{"id": "book 2_p494_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 494, "page_end": 494, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Therefore, the value of K is as follows.", "topic": "algebra"}
{"id": "book 2_p494_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 494, "page_end": 494, "topic_guess": "algebra", "text": "GMAT algebra note: First, rephrase the question stem by subtracting xy from both sides: “Is xy < 0?” The question is whether xy is negative, or “Do x and y have opposite signs?” Be careful! Dividing by variables is the mistake:.", "topic": "algebra"}
{"id": "book 2_p494_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 494, "page_end": 494, "topic_guess": "algebra", "text": "GMAT algebra note: Incorrect as a result of the mistake in the previous step.", "topic": "algebra"}
{"id": "book 2_p495_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 495, "page_end": 495, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Not only does this rephrase make the statements moot (2 is definitely not less than 1, no matter what the statements say), but it also ignores some special cases.", "topic": "percent"}
{"id": "book 2_p495_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 495, "page_end": 495, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: If xy = 0, then dividing by xy yields an undefined value. If xy < 0, you should have flipped the sign of the inequality.", "topic": "percent"}
{"id": "book 2_p495_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 495, "page_end": 495, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Instead, use the correct rephrasing: “Do x and y have opposite signs?” (1) INSUFFICIENT:. If , then x2 and y must have opposite signs.", "topic": "percent"}
{"id": "book 2_p495_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 495, "page_end": 495, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Since x2 must be positive, y must be negative. However, x could be either positive or negative.", "topic": "percent"}
{"id": "book 2_p495_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 495, "page_end": 495, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: If x3y3 < (xy)2, then you can divide both sides by (xy)2 (since that quantity is positive). The simplified inequality is xy < 1, which is not sufficient to answer the question.", "topic": "percent"}
{"id": "book 2_p495_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 495, "page_end": 495, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Each statement indicates that xy could be either positive or negative. The statements are equally insufficient and neither provides any additional information to the other.", "topic": "percent"}
{"id": "book 2_p495_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 495, "page_end": 495, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: First, distribute the numerator. Group the numerator terms with x and −y in mind.", "topic": "percent"}
{"id": "book 2_p496_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 496, "page_end": 496, "topic_guess": "strategy", "text": "GMAT strategy note: Alternatively, you could choose smart numbers. Plug the selected values into the choices.", "topic": "general"}
{"id": "book 2_p496_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 496, "page_end": 496, "topic_guess": "strategy", "text": "GMAT strategy note: The choice that equals 7 is the correct answer. Begin by listing some values of gn to get a sense for how gn progresses.", "topic": "general"}
{"id": "book 2_p497_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 497, "page_end": 497, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Now, look for a pattern in the sums defined by Each value is double the previous value:.", "topic": "algebra"}
{"id": "book 2_p497_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 497, "page_end": 497, "topic_guess": "algebra", "text": "GMAT algebra note: You have already discovered that g3 = 2, so you can select g3 as the answer. The common term in this problem is the recurring base of 3.", "topic": "algebra"}
{"id": "book 2_p497_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 497, "page_end": 497, "topic_guess": "algebra", "text": "GMAT algebra note: Group like terms (i.e., all the terms with k on the le side, all the other.", "topic": "algebra"}
{"id": "book 2_p498_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 498, "page_end": 498, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Powers of 3 on the right side), then simplify each power of 3 using exponent rules.", "topic": "number_properties"}
{"id": "book 2_p498_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 498, "page_end": 498, "topic_guess": "number_theory", "text": "GMAT number properties note: Note the constraints that x and y are positive integers. If y = 5 and x = 6, then this statement is true: 6!", "topic": "number_properties"}
{"id": "book 2_p498_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 498, "page_end": 498, "topic_guess": "number_theory", "text": "GMAT number properties note: Are these the only possible positive integer values for x and y? The constant 6 is equal to 3!, so there’s one more: when y = 1 and x = 3, the statement is also true: 3!", "topic": "number_properties"}
{"id": "book 2_p498_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 498, "page_end": 498, "topic_guess": "number_theory", "text": "GMAT number properties note: You can multiply both sides by y to eliminate that variable entirely. In other words, this statement doesn’t indicate anything about y.", "topic": "number_properties"}
{"id": "book 2_p499_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 499, "page_end": 499, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The quadratic produces two solutions, but only x = 6 is valid, since x must be a positive integer.", "topic": "number_properties"}
{"id": "book 2_p499_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 499, "page_end": 499, "topic_guess": "number_theory", "text": "GMAT number properties note: This is sufficient to calculate a value for xy. The question asks for the greatest value of k such that is divisible by 6k.", "topic": "number_properties"}
{"id": "book 2_p499_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 499, "page_end": 499, "topic_guess": "number_theory", "text": "GMAT number properties note: Since 6 = 3 × 2, the greatest value of k will equal the number of 3 × 2 pairs among the prime factors of.", "topic": "number_properties"}
{"id": "book 2_p500_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 500, "page_end": 500, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. To count the number of times 3 appears as a factor of , rewrite the expression, pulling out any factor(s) of 3 from each term.", "topic": "number_properties"}
{"id": "book 2_p500_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 500, "page_end": 500, "topic_guess": "number_theory", "text": "GMAT number properties note: There are six factors of 3 in To count the number of times 2 appears as a factor of , rewrite the expression, pulling out any factor(s) of 2 from each term. = (33)(25)(31)(2 × 15)(29)(22 × 7)(27)(2 × 13)(25)(23 ×3)(23) There are twelve factors of 2 in Since there are twelve 2’s but only six 3’s, there are only six 3 × 2 pairs among the prime factors of .", "topic": "number_properties"}
{"id": "book 2_p500_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 500, "page_end": 500, "topic_guess": "number_theory", "text": "GMAT number properties note: In general, focus on the greatest prime in the divisor (in this case, the six maximum possible factors of 3 in 6k), as it will be the limiting factor.", "topic": "number_properties"}
{"id": "book 2_p501_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 501, "page_end": 501, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. This is one of the incorrect answers.", "topic": "number_properties"}
{"id": "book 2_p503_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 503, "page_end": 503, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If x and y are positive integers, what is the value of x2 = 2xy − y2(1) 2xy = 8(2) 42.", "topic": "number_properties"}
{"id": "book 2_p503_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 503, "page_end": 503, "topic_guess": "number_theory", "text": "GMAT number properties note: What is the remainder when (47)(49) is divided by 8? If a = 4x2 + 4xy and b = 4y2 + 4xy, which of the following is equivalent to x + y?", "topic": "number_properties"}
{"id": "book 2_p504_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 504, "page_end": 504, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "book 2_p505_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 505, "page_end": 505, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If x is the square of an integer, is y the square of an integer?", "topic": "number_properties"}
{"id": "book 2_p505_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 505, "page_end": 505, "topic_guess": "number_theory", "text": "GMAT number properties note: Xy is the square of an integer.(1) is the square of an integer.(2) 47. If x and y are positive integers such that x > y and , which of the following is equivalent to 2b?", "topic": "number_properties"}
{"id": "book 2_p506_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 506, "page_end": 506, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "book 2_p507_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 507, "page_end": 507, "topic_guess": "general", "text": "GMAT quant note: If , what is the value of xy?", "topic": "general"}
{"id": "book 2_p508_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 508, "page_end": 508, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "book 2_p509_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 509, "page_end": 509, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. Factor this “square of a difference” special product and simplify.", "topic": "number_properties"}
{"id": "book 2_p509_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 509, "page_end": 509, "topic_guess": "general", "text": "GMAT quant note: Since x = y, the value of is 1. Test different values for x and y.", "topic": "number_properties"}
{"id": "book 2_p509_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 509, "page_end": 509, "topic_guess": "general", "text": "GMAT quant note: X (positive) y (positive) xy = 4 2 2 2 × 2 = 4 ✓ 1.", "topic": "number_properties"}
{"id": "book 2_p510_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 510, "page_end": 510, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. X (positive) y (positive) xy = 4 1 4 1 × 4 = 4 ✓ Alternatively, manipulate the equation xy = 4 to get on one side.", "topic": "algebra"}
{"id": "book 2_p510_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 510, "page_end": 510, "topic_guess": "algebra", "text": "GMAT algebra note: The expression does not equal a constant, so could take on many different values. One way to solve would be to multiply (47)(49), then either divide the result by 8 or repeatedly subtract known multiples of 8 from the result until you are le with a remainder less than 8.", "topic": "algebra"}
{"id": "book 2_p510_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 510, "page_end": 510, "topic_guess": "algebra", "text": "GMAT algebra note: An alternative is to rewrite the given product as an equivalent yet easier- to-manipulate product. Note that 47 and 49 are equidistant from 48, a multiple of 8.", "topic": "algebra"}
{"id": "book 2_p510_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 510, "page_end": 510, "topic_guess": "algebra", "text": "GMAT algebra note: Write each of the original factors as terms in the form (a + b) or (a − b).", "topic": "algebra"}
{"id": "book 2_p511_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 511, "page_end": 511, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. (47)(49) = (48 + 1)(48 − 1) This form is the difference of two squares special product, (a + b)(a − b) = a2 − b2.", "topic": "number_properties"}
{"id": "book 2_p511_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 511, "page_end": 511, "topic_guess": "general", "text": "GMAT quant note: (48 + 1)(48 − 1) = (482 − 12) Forty-eight is a multiple of 8, and therefore so is 482. Thus, (482 − 12) is 1 less than a multiple of 8.", "topic": "number_properties"}
{"id": "book 2_p511_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 511, "page_end": 511, "topic_guess": "general", "text": "GMAT quant note: All such numbers (e.g., 7, 15, 23, 31) have a remainder of 7 when divided by 8. Add a and b to get a + b = 4x2 + 8xy + 4y2 = 4(x2 + 2xy + y2).", "topic": "number_properties"}
{"id": "book 2_p511_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 511, "page_end": 511, "topic_guess": "general", "text": "GMAT quant note: The right side is the square of a sum, so factor and solve. Note that you could safely take the square root of both sides because any square is non-negative.", "topic": "number_properties"}
{"id": "book 2_p512_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 512, "page_end": 512, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Alternatively, choose smart numbers.", "topic": "algebra"}
{"id": "book 2_p512_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 512, "page_end": 512, "topic_guess": "algebra", "text": "GMAT algebra note: For values to plug into the choices, first compute a and b. The one that equals 5 is the correct answer.", "topic": "algebra"}
{"id": "book 2_p512_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 512, "page_end": 512, "topic_guess": "algebra", "text": "GMAT algebra note: Manipulate the question expression, noting the special products in the exponents.", "topic": "algebra"}
{"id": "book 2_p513_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 513, "page_end": 513, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. The rephrased question is, “What is the value of ab?” Knowing the values of a and b individually would be sufficient, of course, but the individual values are not required as long as you can determine ab.", "topic": "number_properties"}
{"id": "book 2_p513_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 513, "page_end": 513, "topic_guess": "general", "text": "GMAT quant note: It is impossible to manipulate a + b = 7 to get ab, nor can you solve for a and b individually. This statement answers the rephrased question directly.", "topic": "number_properties"}
{"id": "book 2_p513_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 513, "page_end": 513, "topic_guess": "general", "text": "GMAT quant note: Notice the special product of the form (a + b)(a − b) = a2 − b2 under the square root symbol. Jot down the information provided in the question stem.", "topic": "number_properties"}
{"id": "book 2_p513_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 513, "page_end": 513, "topic_guess": "general", "text": "GMAT quant note: If x is the square of an integer, then x = i2, where i is an unknown integer.", "topic": "number_properties"}
{"id": "book 2_p514_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 514, "page_end": 514, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. (1) INSUFFICIENT: x is a perfect square, and when multiplied by the unknown value y, it turns into a (possibly different) perfect square.", "topic": "number_properties"}
{"id": "book 2_p514_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 514, "page_end": 514, "topic_guess": "general", "text": "GMAT quant note: This situation can occur if y is also a perfect square. Since y is the square of an integer, the answer is Yes.", "topic": "number_properties"}
{"id": "book 2_p514_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 514, "page_end": 514, "topic_guess": "general", "text": "GMAT quant note: If you only consider integer values for y, it seems as though y must be a square. For instance, if x = 22, y cannot equal 2, 3, 5, 6, or any non-square, since 2(22), 3(22), 5(22), etc., are not squares.", "topic": "number_properties"}
{"id": "book 2_p514_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 514, "page_end": 514, "topic_guess": "general", "text": "GMAT quant note: However, the question does not restrict y to integer values. To find a non- integer value of y that fits the known information, start by choosing square values for both x and xy.", "topic": "number_properties"}
{"id": "book 2_p514_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 514, "page_end": 514, "topic_guess": "general", "text": "GMAT quant note: In this case, y = , which is not a perfect square. Therefore, the answer is No.", "topic": "number_properties"}
{"id": "book 2_p514_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 514, "page_end": 514, "topic_guess": "general", "text": "GMAT quant note: (2) SUFFICIENT: is the square of an integer, so = j2, for some unknown integer j. From the question stem, x = i2, for some unknown integer i.", "topic": "number_properties"}
{"id": "book 2_p515_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 515, "page_end": 515, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Therefore, y must be the square of an integer.", "topic": "algebra"}
{"id": "book 2_p515_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 515, "page_end": 515, "topic_guess": "algebra", "text": "GMAT algebra note: The answer is definitely Yes, and this statement is sufficient. To solve this problem, isolate b in the equation.", "topic": "algebra"}
{"id": "book 2_p515_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 515, "page_end": 515, "topic_guess": "algebra", "text": "GMAT algebra note: Most of the choices are not fractions, so try to cancel the denominator. Since x − y is a well-disguised difference of two squares, factor the numerator and denominator.", "topic": "algebra"}
{"id": "book 2_p516_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 516, "page_end": 516, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Each of the expressions given is equal to 16, so set them equal to each other.", "topic": "algebra"}
{"id": "book 2_p516_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 516, "page_end": 516, "topic_guess": "algebra", "text": "GMAT algebra note: Note that you have the square of a sum and the difference of two squares special products. Put them both in distributed form, then simplify.", "topic": "algebra"}
{"id": "book 2_p516_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 516, "page_end": 516, "topic_guess": "algebra", "text": "GMAT algebra note: Since 2y(x + y) = 0, it must be true that either 2y or (x + y) is equal to 0. However, (x + y) cannot equal 0, since the problem indicates that (x + y)2 = 16.", "topic": "algebra"}
{"id": "book 2_p516_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 516, "page_end": 516, "topic_guess": "algebra", "text": "GMAT algebra note: So it must be that 2y = 0, and therefore y = 0. Plug in 0 for y in x2 – y2 = 16 to get x2 = 16.", "topic": "algebra"}
{"id": "book 2_p516_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 516, "page_end": 516, "topic_guess": "algebra", "text": "GMAT algebra note: Use this to solve for y in the second equation.", "topic": "algebra"}
{"id": "book 2_p517_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 517, "page_end": 517, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Do x = ±4 and y = 0 also work in the first equation?", "topic": "algebra"}
{"id": "book 2_p517_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 517, "page_end": 517, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: There are several ways to approach this problem without extensive calculation. The units digit of 5,9952 must equal the units digit of 52, which is 5.", "topic": "algebra"}
{"id": "book 2_p517_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 517, "page_end": 517, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Similarly, the units digit of 4,7962 must equal the units digit of 62, which is 6. Subtracting a number with a units digit of 6 from a number with a units digit of 5 results in a units digit of 9.", "topic": "algebra"}
{"id": "book 2_p517_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 517, "page_end": 517, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Estimation It is also possible to estimate the answer: 5,995 is approximately equal to 6,000, and 4,796 is approximately equal to 5,000.", "topic": "algebra"}
{"id": "book 2_p518_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 518, "page_end": 518, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. 6,0002 – 5,0002 = 36,000,000 – 25,000,000 = 11,000,000 Since 3,0002 = 9,000,000, and 4,0002 = 16,000,000, the correct answer must be between 3,0002 and 4,0002.", "topic": "algebra"}
{"id": "book 2_p518_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 518, "page_end": 518, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Factoring The number 5,995 is a multiple of 5 just under 6,000, which equals 5(1,200). The number 4,796 is a multiple of 4 just under 4,800, which equals 4(1,200).", "topic": "algebra"}
{"id": "book 2_p518_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 518, "page_end": 518, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: This “coincidence” suggests factoring. 5,995 = 6,000 – 5 = 5(1,200 – 1) = 5(1,199) 4,796 = 4,800 – 4 = 4(1,200 – 1) = 4(1,199) The shared factor of 1,199 means that simplification is possible.", "topic": "algebra"}
{"id": "book 2_p519_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 519, "page_end": 519, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Begin by simplifying the equation given in the question.", "topic": "algebra"}
{"id": "book 2_p519_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 519, "page_end": 519, "topic_guess": "algebra", "text": "GMAT algebra note: At this stage, you might be tempted to divide both sides of the equation by xy, in order to arrive at the simpler equation y2 + x2 = 2xy. However, that would be a mistake—never divide both sides of an equation by an unknown (in this case, xy), unless you are certain that the unknown cannot equal zero.", "topic": "algebra"}
{"id": "book 2_p519_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 519, "page_end": 519, "topic_guess": "algebra", "text": "GMAT algebra note: (Division by zero is undefined, and can lead to nonsensical results.) So rather than divide by xy, subtract 2x2y2 from both sides to group all terms on one side of the equals sign. This last line implies that either xy = 0 or y – x = 0.", "topic": "algebra"}
{"id": "book 2_p519_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 519, "page_end": 519, "topic_guess": "algebra", "text": "GMAT algebra note: In other words, either xy = 0 or y = x. If y > x, then it is impossible that y = x.", "topic": "algebra"}
{"id": "book 2_p519_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 519, "page_end": 519, "topic_guess": "algebra", "text": "GMAT algebra note: If x < 0, then it is possible that xy = 0 (i.e., if y = 0) or that y = x (i.e., y is negative, too). If y = x = any negative number, then there are infinitely many solutions for xy.", "topic": "algebra"}
{"id": "book 2_p520_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 520, "page_end": 520, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "book 2_p521_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 521, "page_end": 521, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Is it possible for a wooden block in the shape of a rectangular solid measuring l by w by h centimeter to pass through a square hole with sides of length 4 centimeter? The volume of the wooden block is 16 cm3.(1) L > w > h > 1(2) 52.", "topic": "geometry"}
{"id": "book 2_p521_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 521, "page_end": 521, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Three of the four vertices of a rectangle in the xy-coordinate plane are (–5, 1), (–4, 4), and (8, 0). In the coordinate plane, circle C has radius r and its center is at point (x, y).", "topic": "geometry"}
{"id": "book 2_p521_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 521, "page_end": 521, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Is at least 50% of the area of circle C contained in a single quadrant of the coordinate plane?", "topic": "geometry"}
{"id": "book 2_p522_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 522, "page_end": 522, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. A group of men and women gathered to compete in a marathon.", "topic": "statistics"}
{"id": "book 2_p522_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 522, "page_end": 522, "topic_guess": "statistics", "text": "GMAT statistics note: Each competitor was weighed before the competition. The average weight of the women was 120 pounds, and the average weight of the men was greater than 120 pounds.", "topic": "statistics"}
{"id": "book 2_p522_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 522, "page_end": 522, "topic_guess": "statistics", "text": "GMAT statistics note: If the average weight of the entire group was w pounds, what percent of the competitors were women? The average weight of the men was 150 pounds.(1) The difference between w and the average weight of the women was twice the difference between w and the average weight of the men.", "topic": "statistics"}
{"id": "book 2_p522_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 522, "page_end": 522, "topic_guess": "statistics", "text": "GMAT statistics note: What is the average (arithmetic mean) of these numbers: 12. A set of 5 numbers has an average (arithmetic mean) of 50.", "topic": "statistics"}
{"id": "book 2_p522_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 522, "page_end": 522, "topic_guess": "statistics", "text": "GMAT statistics note: The greatest element in the set is 5 greater than 3 times the least.", "topic": "statistics"}
{"id": "book 2_p523_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 523, "page_end": 523, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. If the median of the set equals the mean, what is the greatest possible value in the set?", "topic": "statistics"}
{"id": "book 2_p523_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 523, "page_end": 523, "topic_guess": "statistics", "text": "GMAT statistics note: 5, 2, 4, m, 9, 5 For the list of numbers above, what is the median? The median is an integer.(1).", "topic": "statistics"}
{"id": "book 2_p524_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 524, "page_end": 524, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Set M contains seven consecutive integers, and set N contains three values chosen from set M.", "topic": "statistics"}
{"id": "book 2_p524_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 524, "page_end": 524, "topic_guess": "statistics", "text": "GMAT statistics note: Is the standard deviation of set N greater than the standard deviation of set M? Set N contains the median of set M.(1) The range of set M and set N are equal.(2) 60.", "topic": "statistics"}
{"id": "book 2_p524_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 524, "page_end": 524, "topic_guess": "statistics", "text": "GMAT statistics note: Four different children have jelly beans:. Aaron has 5, Bianca has 7, Callie has 8, and Dante has 11.", "topic": "statistics"}
{"id": "book 2_p524_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 524, "page_end": 524, "topic_guess": "statistics", "text": "GMAT statistics note: How many jelly beans must Dante give to Aaron to ensure that each child has within 1 jelly bean of all the other children?", "topic": "statistics"}
{"id": "book 2_p525_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 525, "page_end": 525, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "book 2_p526_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 526, "page_end": 526, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Start by considering the different types of blocks that could or could not fit through the hole.", "topic": "word_problems"}
{"id": "book 2_p526_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 526, "page_end": 526, "topic_guess": "word_problems", "text": "GMAT word problem note: If all dimensions of the block are less than 4 centimeters (cm), the block can definitely fit through the hole. If exactly one of the three dimensions is greater than 4 cm, the block can still definitely fit through the hole.", "topic": "word_problems"}
{"id": "book 2_p526_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 526, "page_end": 526, "topic_guess": "word_problems", "text": "GMAT word problem note: To do so, rotate the block so that the two shorter dimensions are aligned with the sides of the hole.", "topic": "word_problems"}
{"id": "book 2_p527_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 527, "page_end": 527, "topic_guess": "general", "text": "GMAT quant note: However, if at least two of the dimensions of the block are greater than 4 cm, further investigation is required. It may be possible to fit the block through the hole by rotating it to take advantage of the longest dimension of the hole, which is the diagonal of but if the dimensions of the block are too great, this may be impossible.", "topic": "general"}
{"id": "book 2_p527_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 527, "page_end": 527, "topic_guess": "general", "text": "GMAT quant note: The block can have any dimensions such that Lwh = 16. However, a block with dimensions 8, 8, and 0.25 cannot fit through the hole.", "topic": "general"}
{"id": "book 2_p527_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 527, "page_end": 527, "topic_guess": "general", "text": "GMAT quant note: Since the answer can be Yes or No, this statement is insufficient. A block with dimensions 2, 3, and 4 can fit through the hole, but a block with dimensions 100, 101, and 102 cannot.", "topic": "general"}
{"id": "book 2_p527_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 527, "page_end": 527, "topic_guess": "general", "text": "GMAT quant note: Lwh = 16 and h is greater than 1, so Lw is less than 16. Therefore, L and w cannot both be greater than 4.", "topic": "general"}
{"id": "book 2_p527_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 527, "page_end": 527, "topic_guess": "general", "text": "GMAT quant note: Since h is the least dimension, it must also be less than 4. Your GMAT scratch pad has a grid.", "topic": "general"}
{"id": "book 2_p528_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 528, "page_end": 528, "topic_guess": "geometry", "text": "GMAT geometry note: Complete the rectangle with the dashed lines shown. The fourth point must be located approximately where the bigger dot is drawn.", "topic": "geometry"}
{"id": "book 2_p528_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 528, "page_end": 528, "topic_guess": "geometry", "text": "GMAT geometry note: Alternatively, compute the location of the fourth point, using the fact that the short sides have the same slope. The known short side connects the points (−5, 1) and (−4, 4).", "topic": "geometry"}
{"id": "book 2_p528_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 528, "page_end": 528, "topic_guess": "geometry", "text": "GMAT geometry note: In other words, the bottom le corner is 1 to the le and 3 down from the top le corner. The question asks whether at least half of the circle is contained in one quadrant.", "topic": "geometry"}
{"id": "book 2_p528_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 528, "page_end": 528, "topic_guess": "geometry", "text": "GMAT geometry note: However, it does not specify a specific quadrant. If at least half of the circle is contained in any one of the four quadrants, the answer to the question will be Yes.", "topic": "geometry"}
{"id": "book 2_p528_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 528, "page_end": 528, "topic_guess": "geometry", "text": "GMAT geometry note: Note that if circle C lies in exactly one or two quadrants, then the answer to the question must be Yes, because at least half of the circle must be in.", "topic": "geometry"}
{"id": "book 2_p529_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 529, "page_end": 529, "topic_guess": "geometry", "text": "GMAT geometry note: If x = 2, y = 1, and r = 0.5, then the answer to the question is Yes, since the entire circle is contained in the first quadrant. However, if x = 2, y = 1, and r is much greater than 2, then approximately a quarter of the circle is contained in each quadrant and the answer to the question is No.", "topic": "geometry"}
{"id": "book 2_p529_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 529, "page_end": 529, "topic_guess": "geometry", "text": "GMAT geometry note: Since the x coordinate is greater than the radius of the circle, no part of the circle can be in the second or third quadrants.", "topic": "geometry"}
{"id": "book 2_p530_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 530, "page_end": 530, "topic_guess": "geometry", "text": "GMAT geometry note: Track mean, median, range, distribution, and how operations change spread. Therefore, at least half of the circle must be in the first quadrant or at least half of the circle must be in the fourth quadrant.", "topic": "statistics"}
{"id": "book 2_p530_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 530, "page_end": 530, "topic_guess": "geometry", "text": "GMAT geometry note: The answer to the question is definitely Yes, and this statement is sufficient. The question stem indicates that the average weight of the women was 120 and the average weight of the men was greater than 120.", "topic": "statistics"}
{"id": "book 2_p530_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 530, "page_end": 530, "topic_guess": "geometry", "text": "GMAT geometry note: In order to determine what percent of the competitors were women, you would need to know more about the weight of the men and you’d also need to know something about the relative number of women vs. If there were not an equal number of each, then this is a weighted average question.", "topic": "statistics"}
{"id": "book 2_p530_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 530, "page_end": 530, "topic_guess": "geometry", "text": "GMAT geometry note: This does not provide the weight of the men or even the value of w, but it does indicate where the weighted average falls between the two values.", "topic": "statistics"}
{"id": "book 2_p531_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 531, "page_end": 531, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. If the weighted average is twice as far from the women’s end of the line, then the men are responsible for of the total weight and the women are responsible for of the total weight.", "topic": "statistics"}
{"id": "book 2_p531_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 531, "page_end": 531, "topic_guess": "statistics", "text": "GMAT statistics note: Therefore, , or approximately of the competitors were women. You do not need to calculate this figure.", "topic": "statistics"}
{"id": "book 2_p531_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 531, "page_end": 531, "topic_guess": "statistics", "text": "GMAT statistics note: The simple average formula applies to this problem.", "topic": "statistics"}
{"id": "book 2_p532_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 532, "page_end": 532, "topic_guess": "word_problems", "text": "GMAT word problem note: Track mean, median, range, distribution, and how operations change spread. However, the chance of computational error is high on a problem with this many terms of such great value.", "topic": "statistics"}
{"id": "book 2_p532_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 532, "page_end": 532, "topic_guess": "word_problems", "text": "GMAT word problem note: Try grouping the similar terms. Group A: 12, 13, 14 (equidistant terms with an average of 13, the middle term) Group B: 510, 520, 530 (equidistant terms with an average of 520, the middle term) Group C: 1,115.", "topic": "statistics"}
{"id": "book 2_p532_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 532, "page_end": 532, "topic_guess": "word_problems", "text": "GMAT word problem note: 1,125 (equidistant terms with an average of 1,120, the middle term) Since each group of terms consists of three values (and are therefore equally weighted in the set of nine terms), the average of all nine original terms is the average of the respective averages of Groups A, B, and C.", "topic": "statistics"}
{"id": "book 2_p532_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 532, "page_end": 532, "topic_guess": "word_problems", "text": "GMAT word problem note: Average = You can make that division easier by breaking 1,653 into lesser numbers that are divisible by 3: 1,653 = 1,500 + 150 + 3. Divide each separately by 3 to get 500 + 50 + 1 = 551.", "topic": "statistics"}
{"id": "book 2_p533_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 533, "page_end": 533, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Two techniques will help you efficiently interpret the information given in the question.", "topic": "statistics"}
{"id": "book 2_p533_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 533, "page_end": 533, "topic_guess": "statistics", "text": "GMAT statistics note: First, draw a number line with five dots representing the five numbers in the set. Second, label these numbers A, B, C, D, and E, with the understanding that A ≤ B ≤ C ≤ D ≤ E.", "topic": "statistics"}
{"id": "book 2_p533_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 533, "page_end": 533, "topic_guess": "statistics", "text": "GMAT statistics note: A + B + C + D + E = 250 (The set of five numbers has an average of 50.) E = 5 + 3A (The greatest element is five greater than three times the least element in the set.) C = 50 (The median of the set equals the mean.) You’re asked to maximize E. Arrange the dots on the number line such that you obey the constraints, yet also note where you have some flexibility.", "topic": "statistics"}
{"id": "book 2_p533_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 533, "page_end": 533, "topic_guess": "statistics", "text": "GMAT statistics note: Point D can be anywhere on the line from point C to point E. Given the information from the question stem, you can maximize E by minimizing D.", "topic": "statistics"}
{"id": "book 2_p534_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 534, "page_end": 534, "topic_guess": "geometry", "text": "GMAT geometry note: Similarly, point B can be anywhere on the line from point A to point C. Maximize E by minimizing B, so make B = A.", "topic": "geometry"}
{"id": "book 2_p534_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 534, "page_end": 534, "topic_guess": "geometry", "text": "GMAT geometry note: Therefore, E = 5 + 3A = 5 + 3(29) = 5 + 87 = 92. Many elements in this triangle could vary.", "topic": "geometry"}
{"id": "book 2_p534_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 534, "page_end": 534, "topic_guess": "geometry", "text": "GMAT geometry note: You don’t even know the placement of B relative to AC, so the triangle itself might stretch. Even for a fixed triangle, DE could slide up or down, so various lengths are possible for DE, as shown here.", "topic": "geometry"}
{"id": "book 2_p535_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 535, "page_end": 535, "topic_guess": "geometry", "text": "GMAT geometry note: You don’t know the lengths of any sides of the triangle. The side that most affects the length of DE is AC, so stretch that side.", "topic": "geometry"}
{"id": "book 2_p535_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 535, "page_end": 535, "topic_guess": "geometry", "text": "GMAT geometry note: Stretching the triangle out to the right stretches DE. AC must be 14 and DE must be parallel to AC and halfway between AC and B, in order to maintain BE = EC.", "topic": "geometry"}
{"id": "book 2_p535_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 535, "page_end": 535, "topic_guess": "geometry", "text": "GMAT geometry note: Even though vertex B is free to move, DE will always be the average of the width of the triangle at AC (14) and the width at B (0). Thus, DE must be 7, no matter how the picture shi s.", "topic": "geometry"}
{"id": "book 2_p536_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 536, "page_end": 536, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. To find the median of a set of numbers, line them up in order of value.", "topic": "statistics"}
{"id": "book 2_p536_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 536, "page_end": 536, "topic_guess": "statistics", "text": "GMAT statistics note: The question of interest is, “Where is m relative to the other values?” This set has six values, an even number of terms, so the median is the average of the two middle terms. These are the three scenarios shown above.", "topic": "statistics"}
{"id": "book 2_p536_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 536, "page_end": 536, "topic_guess": "statistics", "text": "GMAT statistics note: If m ≤ 4, then the two middle terms are 4 and 5, and the median is If 4 < m < 5, then the two middle terms are m and 5, and the median is If m ≥ 5, then the two middle terms are 5 and 5, and the median is 5. For the case where 4 < m < 5, the median is Thus, if the median is an integer, it must be 5.", "topic": "statistics"}
{"id": "book 2_p537_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 537, "page_end": 537, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Standard deviation is a measure of the “spread” of a group of numbers.", "topic": "statistics"}
{"id": "book 2_p537_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 537, "page_end": 537, "topic_guess": "statistics", "text": "GMAT statistics note: If set M contains the numbers {1, 2, 3, 4, 5, 6, 7}, then 4 is the median. Set N could be {3, 4, 5}, which has a lesser standard deviation than set M because the two sets have the same average but N is not as spread out as M.", "topic": "statistics"}
{"id": "book 2_p537_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 537, "page_end": 537, "topic_guess": "statistics", "text": "GMAT statistics note: Alternatively, N could be {1, 4, 7}, which has a greater standard deviation than set M because the two sets still have the same average, but N is now more spread out than M. (It doesn’t have additional values that are closer to the average.) (2) SUFFICIENT:.", "topic": "statistics"}
{"id": "book 2_p537_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 537, "page_end": 537, "topic_guess": "statistics", "text": "GMAT statistics note: If set M contains the numbers {1, 2, 3, 4, 5, 6, 7}, and set N has the same range, then N must contain 1 and 7. If set N contains {1, 4, 7}, then N has a greater standard deviation than M.", "topic": "statistics"}
{"id": "book 2_p537_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 537, "page_end": 537, "topic_guess": "statistics", "text": "GMAT statistics note: If set N contains {1, 2, 7}, the standard deviation actually increases even more (because the numbers are no longer evenly distributed}. No matter what combination you try, the standard deviation of N has to be greater than the standard deviation of M.", "topic": "statistics"}
{"id": "book 2_p537_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 537, "page_end": 537, "topic_guess": "statistics", "text": "GMAT statistics note: Conceptually, the transfer of jelly beans from Dante to Aaron reduces the range of the number of jelly beans held by individual children. The constraint is that the final distribution represents a range of just 1 jelly bean, a condition Bian.", "topic": "statistics"}
{"id": "book 2_p538_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 538, "page_end": 538, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Aaron and Dante must end up with a number of jelly beans that is either 7 or 8.", "topic": "algebra"}
{"id": "book 2_p538_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 538, "page_end": 538, "topic_guess": "algebra", "text": "GMAT algebra note: If either Aaron or Dante has a number of jelly beans other than 7 or 8, he will differ too much from either Bianca’s or Callie’s number. You can count out the necessary change on the number line above or you can write out the algebra.", "topic": "algebra"}
{"id": "book 2_p538_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 538, "page_end": 538, "topic_guess": "algebra", "text": "GMAT algebra note: The solution to both equations is x = 3:. Add 3 to Aaron to get 8, and subtract 3 from Dante to get 8.", "topic": "algebra"}
{"id": "book 2_p538_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 538, "page_end": 538, "topic_guess": "algebra", "text": "GMAT algebra note: The resulting number of jelly beans is A = 8, B = 7, C = 8, and D = 8.", "topic": "algebra"}
{"id": "book 2_p539_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 539, "page_end": 539, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. At least one side of rectangle R has an integer length.(1) The diagonals of rectangle R have integer lengths.(2) 62.", "topic": "percent"}
{"id": "book 2_p539_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 539, "page_end": 539, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: A group of friends charters a boat for $540 and each person contributes equally to the cost. They determine that if they can get three more of their friends to join them, every person in the group will pay $9 less.", "topic": "percent"}
{"id": "book 2_p539_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 539, "page_end": 539, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: If they find three more friends to join them, what is the total number of people renting the boat? In Smithtown, the ratio of right-handed people to le -handed people is 3 to 1 and the ratio of men to women is 3 to 2.", "topic": "percent"}
{"id": "book 2_p539_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 539, "page_end": 539, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: If the number of right-handed men is maximized, then what percent of all the people in Smithtown are le -handed women?", "topic": "percent"}
{"id": "book 2_p540_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 540, "page_end": 540, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Track mean, median, range, distribution, and how operations change spread. The sum of the interior angle measures for any n-sided polygon equals 180(n − 2).", "topic": "statistics"}
{"id": "book 2_p540_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 540, "page_end": 540, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: If Polygon A has interior angle measures that correspond to a set of consecutive integers, and if the median angle measure for Polygon A is 140°, what is the least angle measure in the polygon? When the positive integer x is divided by the positive integer y, the quotient is 2 and the remainder is z.", "topic": "statistics"}
{"id": "book 2_p540_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 540, "page_end": 540, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: When x is divided by the positive integer a, the quotient is 3 and the remainder is b. The ratio of y to a is less than 3 to 2.(1) The ratio of y to a is greater than 2 to 3.(2).", "topic": "statistics"}
{"id": "book 2_p541_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 541, "page_end": 541, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If a and b are odd integers, a ∆ b represents the product of all odd integers between a and b,inclusive.", "topic": "number_properties"}
{"id": "book 2_p541_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 541, "page_end": 541, "topic_guess": "number_theory", "text": "GMAT number properties note: If y is the least prime factor of (3 ∆ 47) + 2, which of the following must be true? Set S is the set of all prime integers between 0 and 20.", "topic": "number_properties"}
{"id": "book 2_p541_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 541, "page_end": 541, "topic_guess": "number_theory", "text": "GMAT number properties note: If three numbers are chosen randomly from set S, and no number is chosen more than once, what is the probability that the sum of all three numbers is odd?", "topic": "number_properties"}
{"id": "book 2_p542_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 542, "page_end": 542, "topic_guess": "geometry", "text": "GMAT geometry note: Sets A and B each consist of three terms selected from the first five prime integers. No term appears more than once within a set, but any integer may be a term in both sets.", "topic": "geometry"}
{"id": "book 2_p542_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 542, "page_end": 542, "topic_guess": "geometry", "text": "GMAT geometry note: If the average of the terms in set A is 4 and the product of the terms in set B is divisible by 22, how many terms are shared by both sets? If Q is the sum of the areas of squares SAND and SURF and C is the area of the circle, then the fraction is.", "topic": "geometry"}
{"id": "book 2_p543_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 543, "page_end": 543, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Set S consists of n consecutive positive integers.", "topic": "number_properties"}
{"id": "book 2_p543_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 543, "page_end": 543, "topic_guess": "number_theory", "text": "GMAT number properties note: If n > 3, what is the value of n? The number of multiples of 2 contained in set S is equal to the number of multiples of 3 contained in set S.", "topic": "number_properties"}
{"id": "book 2_p544_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 544, "page_end": 544, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "book 2_p545_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 545, "page_end": 545, "topic_guess": "geometry", "text": "GMAT geometry note: A rectangle is a square if and only if its length equals its width. If the length is 1 and the width is 2, the rectangle is not a square.", "topic": "geometry"}
{"id": "book 2_p545_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 545, "page_end": 545, "topic_guess": "geometry", "text": "GMAT geometry note: If the length and width are both 2, the rectangle is a square. Since R is a rectangle, both of its diagonals have the same length.", "topic": "geometry"}
{"id": "book 2_p545_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 545, "page_end": 545, "topic_guess": "geometry", "text": "GMAT geometry note: It is only necessary to examine one of the diagonals. Since the diagonal and two of the sides form a right triangle, the length of the diagonal is always equal to The diagonal of the rectangle below has an integer length, but the rectangle is not a square.", "topic": "geometry"}
{"id": "book 2_p546_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 546, "page_end": 546, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: However, in the rectangle below, the diagonal has an integer length and the rectangle is a square. The answer could be Yes or No, so this statement is insufficient.", "topic": "geometry"}
{"id": "book 2_p546_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 546, "page_end": 546, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: According to the statements, the rectangle has at least one integer side, and it also has integer diagonals. In a square, the legs and hypotenuse must be in the ratio .", "topic": "geometry"}
{"id": "book 2_p546_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 546, "page_end": 546, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: To be a square, a rectangle must have a diagonal that is times as long as its sides. However, is an irrational number, so if x is an integer, then is not an integer (and vice versa).", "topic": "geometry"}
{"id": "book 2_p546_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 546, "page_end": 546, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: It is impossible for the rectangle described by these statements to be a square, and the answer to the question is definitely No. The statements together are sufficient.", "topic": "geometry"}
{"id": "book 2_p547_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 547, "page_end": 547, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Call the initial number of friends in the group f and the initial cost c.", "topic": "algebra"}
{"id": "book 2_p547_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 547, "page_end": 547, "topic_guess": "algebra", "text": "GMAT algebra note: The problem allows you to write two equations. The question asks for f + 3.", "topic": "algebra"}
{"id": "book 2_p547_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 547, "page_end": 547, "topic_guess": "algebra", "text": "GMAT algebra note: These equations can be solved algebraically, but the math is going to result in having to solve a quadratic. Try plugging them into the problem to find the correct answer.", "topic": "algebra"}
{"id": "book 2_p547_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 547, "page_end": 547, "topic_guess": "algebra", "text": "GMAT algebra note: F + 3 f Orig Cost New Cost Are the costs $9 apart? More people need to join the group in order to bring the costs closer together.", "topic": "algebra"}
{"id": "book 2_p547_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 547, "page_end": 547, "topic_guess": "algebra", "text": "GMAT algebra note: The answer must be between 9 and 18.", "topic": "algebra"}
{"id": "book 2_p548_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 548, "page_end": 548, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Check the math if you’re not sure, but do practice this technique enough that you know when you can actually tell what the answer must be (without doing the math).", "topic": "number_properties"}
{"id": "book 2_p548_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 548, "page_end": 548, "topic_guess": "number_theory", "text": "GMAT number properties note: Use a double-set matrix to solve this problem. Thus, both x and y are integers.", "topic": "number_properties"}
{"id": "book 2_p548_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 548, "page_end": 548, "topic_guess": "number_theory", "text": "GMAT number properties note: Moreover, the total number of people must be a multiple of 4 and of 5 in order for the given ratios to be possible. From this constraint, there are two ways to solve.", "topic": "number_properties"}
{"id": "book 2_p548_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 548, "page_end": 548, "topic_guess": "number_theory", "text": "GMAT number properties note: Algebraic Solution Since the question specifies that the number of right-handed men be as great as possible, assume that all the men are right-handed. Of course, that means that none of the men are le -handed.", "topic": "number_properties"}
{"id": "book 2_p548_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 548, "page_end": 548, "topic_guess": "number_theory", "text": "GMAT number properties note: Because each column in a double-set matrix must total, you can also fill in the number of le - handed women (the group you want).", "topic": "number_properties"}
{"id": "book 2_p549_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 549, "page_end": 549, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Smart Number Solution Since the total number of people in Smithtown must be a multiple of 20, set the total to 20 and determine the subtotals of men, women, le - handed, and right-handed based on the ratios given in the problem.", "topic": "percent"}
{"id": "book 2_p549_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 549, "page_end": 549, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Right-Handed Le -Handed Total Men 12 Women 8 Total 15 5 20.", "topic": "percent"}
{"id": "book 2_p550_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 550, "page_end": 550, "topic_guess": "number_theory", "text": "GMAT number properties note: Track mean, median, range, distribution, and how operations change spread. To maximize the number of right-handed men, assign all the men to the “right-handed men” cell, and fill in the remaining cells.", "topic": "statistics"}
{"id": "book 2_p550_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 550, "page_end": 550, "topic_guess": "number_theory", "text": "GMAT number properties note: Right-Handed Le -Handed Total Men 12 0 12 Women 3 5 8 Total 15 5 20 Therefore, le -handed women represent of the population. If the median angle measure is 140° and the interior angle measures correspond to a set of consecutive integers, then the average angle measure must equal 140°.", "topic": "statistics"}
{"id": "book 2_p550_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 550, "page_end": 550, "topic_guess": "number_theory", "text": "GMAT number properties note: Since the sum of the angles must equal 180(n − 2), the average angle must equal.", "topic": "statistics"}
{"id": "book 2_p551_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 551, "page_end": 551, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Therefore, the polygon has nine sides and nine interior angles, and the measures of these angles are equal to a set of consecutive integers centered at 140.", "topic": "number_properties"}
{"id": "book 2_p551_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 551, "page_end": 551, "topic_guess": "number_theory", "text": "GMAT number properties note: If has a quotient of 2 and a remainder of z, then x is z more than 2y. If has a quotient of 3 and a remainder of b, then x is b more than 3a.", "topic": "number_properties"}
{"id": "book 2_p551_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 551, "page_end": 551, "topic_guess": "number_theory", "text": "GMAT number properties note: The question asks whether z > b, and the statements give information about . Simplify the question by replacing z and b with their equivalents and solving for the combination.", "topic": "number_properties"}
{"id": "book 2_p552_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 552, "page_end": 552, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Note that the inequality sign flipped in the last step because of the division by −2.", "topic": "number_properties"}
{"id": "book 2_p552_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 552, "page_end": 552, "topic_guess": "number_theory", "text": "GMAT number properties note: The variable a is a positive integer, so no additional flip is required for that manipulation. Rephrase the question as, “Is (1) SUFFICIENT:.", "topic": "number_properties"}
{"id": "book 2_p552_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 552, "page_end": 552, "topic_guess": "number_theory", "text": "GMAT number properties note: This directly answers the rephrased question: “Yes, .” Therefore, z > b. The answer might be Yes if .", "topic": "number_properties"}
{"id": "book 2_p552_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 552, "page_end": 552, "topic_guess": "number_theory", "text": "GMAT number properties note: This product is a very large odd number, as it is the product of only odd numbers and thus does not have 2 as a factor. Therefore, (3 ∆ 47) + 2 = Odd + Even = Odd, and (3 ∆ 47) + 2 does not have 2 as a factor either.", "topic": "number_properties"}
{"id": "book 2_p552_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 552, "page_end": 552, "topic_guess": "number_theory", "text": "GMAT number properties note: Every odd prime number between 3 and 47 inclusive is a factor of (3 ∆ 47), since each of these primes is a component of the product. The sum will only be divisible by 3 if k is also divisible by 3.", "topic": "number_properties"}
{"id": "book 2_p553_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 553, "page_end": 553, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Divide (3 ∆ 47) + k by 3, you are evaluating .", "topic": "number_properties"}
{"id": "book 2_p553_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 553, "page_end": 553, "topic_guess": "number_theory", "text": "GMAT number properties note: Because is an integer, must also be an integer to yield an integer sum. In this problem, k = 2, which is not divisible by any of the odd primes between 3 and 47.", "topic": "number_properties"}
{"id": "book 2_p553_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 553, "page_end": 553, "topic_guess": "number_theory", "text": "GMAT number properties note: Since (3 ∆ 47) IS divisible, but 2 is NOT divisible, the sum (3 ∆ 47) + 2 is NOT divisible by any of the odd primes between 3 and So (3 ∆ 47) + 2 is not divisible by any prime number less than or equal to 47. The least prime factor of (3 ∆ 47) + 2 must be greater than 47.", "topic": "number_properties"}
{"id": "book 2_p553_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 553, "page_end": 553, "topic_guess": "number_theory", "text": "GMAT number properties note: Thus, the minimum possible prime factor is 53, since that is the least prime greater than 47. If set S is the set of all prime integers between 0 and 20, then S = {2, 3, 5, 7, 11, 13, 17, 19}.", "topic": "number_properties"}
{"id": "book 2_p553_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 553, "page_end": 553, "topic_guess": "number_theory", "text": "GMAT number properties note: There are seven odd terms and one even term in set S. If the even term is among those selected, the sum will be even (E + O + O = E).", "topic": "number_properties"}
{"id": "book 2_p553_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 553, "page_end": 553, "topic_guess": "number_theory", "text": "GMAT number properties note: The sum will be odd if all three terms selected are odd (O + O + O = O).", "topic": "number_properties"}
{"id": "book 2_p554_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 554, "page_end": 554, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The first five prime integers are 2, 3, 5, 7, and 11.", "topic": "number_properties"}
{"id": "book 2_p554_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 554, "page_end": 554, "topic_guess": "number_theory", "text": "GMAT number properties note: These are the only terms that can appear in sets A and B. There are some other restrictions on the sets:.", "topic": "number_properties"}
{"id": "book 2_p554_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 554, "page_end": 554, "topic_guess": "number_theory", "text": "GMAT number properties note: The average of the terms in set A is 4, so the sum of the terms is (4) (3) = 12. There is only one way for three of the first five primes to sum to 12: 2 + 3 + 7.", "topic": "number_properties"}
{"id": "book 2_p554_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 554, "page_end": 554, "topic_guess": "number_theory", "text": "GMAT number properties note: The product of the terms in set B is divisible by 22, so 2 and 11 are terms in set B. Set B is {2, 11, x}, where x can be 3, 5, or 7, but not 2 or 11 (no duplicates).", "topic": "number_properties"}
{"id": "book 2_p554_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 554, "page_end": 554, "topic_guess": "number_theory", "text": "GMAT number properties note: Sets A and B share at least one term: the 2. If x is either 3 or 7, the sets will share two terms.", "topic": "number_properties"}
{"id": "book 2_p554_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 554, "page_end": 554, "topic_guess": "number_theory", "text": "GMAT number properties note: If x is 5, the sets will only share one term. The rephrased question is, “Is x = 5?” A definite Yes or No answer leads to a definite value answer for the number of shared terms (i.e., Yes = one shared term, No = two shared terms).", "topic": "number_properties"}
{"id": "book 2_p554_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 554, "page_end": 554, "topic_guess": "number_theory", "text": "GMAT number properties note: If the product of the terms in set B is not divisible by 5, x ≠ 5 and the answer to the rephrased question is a definite No. If the product of the terms in set B is divisible by 14, then 2 and 7 are terms in B.", "topic": "number_properties"}
{"id": "book 2_p554_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 554, "page_end": 554, "topic_guess": "number_theory", "text": "GMAT number properties note: Therefore, x = 7 and the answer to the rephrased question is a definite No.", "topic": "number_properties"}
{"id": "book 2_p555_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 555, "page_end": 555, "topic_guess": "geometry", "text": "GMAT geometry note: The area of a square is equal to the length of one side squared. The question asks for the fraction Inscribed angle USA cuts off a diameter (UA) of the circle, so angle USA must be a right angle.", "topic": "geometry"}
{"id": "book 2_p555_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 555, "page_end": 555, "topic_guess": "geometry", "text": "GMAT geometry note: Therefore, the triangle is a right triangle with hypotenuse UA. The Pythagorean theorem indicates that US2 + SA2 = UA2.", "topic": "geometry"}
{"id": "book 2_p555_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 555, "page_end": 555, "topic_guess": "geometry", "text": "GMAT geometry note: US2 also represents the area of the smaller square. SA2 also represents the area of the larger square.", "topic": "geometry"}
{"id": "book 2_p555_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 555, "page_end": 555, "topic_guess": "geometry", "text": "GMAT geometry note: The sum of the two equals the quantity Q mentioned in the question stem, so Q = UA2. UA is a diameter of the circle, so the quantity Q = (2r)2 = 4r2.", "topic": "geometry"}
{"id": "book 2_p556_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 556, "page_end": 556, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. The value of n must be 4 or greater.", "topic": "number_properties"}
{"id": "book 2_p556_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 556, "page_end": 556, "topic_guess": "general", "text": "GMAT quant note: If n is odd, it could equal 3, 5, 7, or so on. Test cases to figure out what is possible.", "topic": "number_properties"}
{"id": "book 2_p556_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 556, "page_end": 556, "topic_guess": "general", "text": "GMAT quant note: N Set S # Multiples 2 = # Multiples 3? 4 3, 4, 5, 6 Yes: 2 multiples of each 5 3, 4, 5, 6, 7 Yes: 2 multiples of each (1) AND (2) INSUFFICIENT:.", "topic": "number_properties"}
{"id": "book 2_p556_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 556, "page_end": 556, "topic_guess": "general", "text": "GMAT quant note: The case of n = 5 was already proven in the last step. Is there a way to have n = another odd number?", "topic": "number_properties"}
{"id": "book 2_p556_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 556, "page_end": 556, "topic_guess": "general", "text": "GMAT quant note: 5 3, 4, 5, 6, 7 Yes: 2 multiples of each 7 3, 4, 5, 6, 7, 8, 9 Yes: 3 multiples of each.", "topic": "number_properties"}
{"id": "book 2_p557_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 557, "page_end": 557, "topic_guess": "general", "text": "GMAT quant note: Since there are still at least two possible values for n, none of the information is sufficient to answer the question.", "topic": "general"}
{"id": "book 2_p558_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 558, "page_end": 558, "topic_guess": "word_problems", "text": "GMAT word problem note: Track mean, median, range, distribution, and how operations change spread. If the average (arithmetic mean) daily energy usage was greater than the median daily energy usage, what is the least possible value for n?", "topic": "statistics"}
{"id": "book 2_p558_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 558, "page_end": 558, "topic_guess": "word_problems", "text": "GMAT word problem note: A painting crew painted 80 houses. They painted the first y houses at a rate of x houses per week.", "topic": "statistics"}
{"id": "book 2_p558_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 558, "page_end": 558, "topic_guess": "word_problems", "text": "GMAT word problem note: Then, more painters arrived and everyone worked together to paint the remaining houses at a rate of 1.25x houses per week. How many weeks did it take to paint all 80 houses, in terms of x and y?", "topic": "statistics"}
{"id": "book 2_p559_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 559, "page_end": 559, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. If a2 = b2, what is the value of y?", "topic": "percent"}
{"id": "book 2_p559_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 559, "page_end": 559, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: A herd of 33 sheep is sheltered in a barn with 7 stalls, each of which is labeled with a unique letter from A to G, inclusive. Is there at least one sheep in every stall?", "topic": "percent"}
{"id": "book 2_p559_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 559, "page_end": 559, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: The ratio of the number of sheep in stall C to the number of sheep in stall E is 2 to 3.", "topic": "percent"}
{"id": "book 2_p560_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 560, "page_end": 560, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The ratio of the number of sheep in stall E to the number of sheep in stall F is 5 to 2.", "topic": "number_properties"}
{"id": "book 2_p560_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 560, "page_end": 560, "topic_guess": "number_theory", "text": "GMAT number properties note: If (s × 10q) − (t × 10r) = 10r, where q, r, s, and t are positive integers and q > r, then what is the units digit of t? What is the greatest prime factor of 21054 − 21352 + 214?", "topic": "number_properties"}
{"id": "book 2_p561_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 561, "page_end": 561, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Track mean, median, range, distribution, and how operations change spread. A decimal is called a “shrinking number” if its value is between 0 and 1 and each digit to the right of the decimal is not less than the digit to its immediate right.", "topic": "statistics"}
{"id": "book 2_p561_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 561, "page_end": 561, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: For instance, 0.86553221 is a shrinking number. If x is a shrinking number, which of the following must be true?", "topic": "statistics"}
{"id": "book 2_p561_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 561, "page_end": 561, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Each of the seven paintings in an art gallery has a different price. Is it possible to purchase at least three paintings for no more than $1,800 in total?", "topic": "statistics"}
{"id": "book 2_p561_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 561, "page_end": 561, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: The median price of the seven paintings is $550.(1) It is possible to purchase four paintings at the gallery for a total of $2,300.", "topic": "statistics"}
{"id": "book 2_p562_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 562, "page_end": 562, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: An (x, y) coordinate pair is to be chosen at random from the xy- plane.", "topic": "geometry"}
{"id": "book 2_p563_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 563, "page_end": 563, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "book 2_p564_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 564, "page_end": 564, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Here’s what you know about the arithmetic mean:.", "topic": "statistics"}
{"id": "book 2_p564_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 564, "page_end": 564, "topic_guess": "statistics", "text": "GMAT statistics note: In contrast, the median depends on n, but not in a linear way. To find the median, you order the terms and pick the middle one, so try various n values (i.e., vary the number of 8’s in the list).", "topic": "statistics"}
{"id": "book 2_p564_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 564, "page_end": 564, "topic_guess": "statistics", "text": "GMAT statistics note: This implies the eventual need to plug n values into the mean formula above, so draw a picture to help eliminate some answers first. The “low” and “high” grouping is a fast way to find the relationship between median and mean for this set.", "topic": "statistics"}
{"id": "book 2_p565_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 565, "page_end": 565, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. If the number of “lows” and “highs” are equal, the median is the average of the middle terms.", "topic": "statistics"}
{"id": "book 2_p565_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 565, "page_end": 565, "topic_guess": "statistics", "text": "GMAT statistics note: That is, if n = 6, then the median = 9. When n = 6, the mean must be greater than 9.", "topic": "statistics"}
{"id": "book 2_p565_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 565, "page_end": 565, "topic_guess": "statistics", "text": "GMAT statistics note: Pairs of 8 and 10 terms average to 9, but there is one “extra” 8. Pairs of 7 and 11 terms average to 9, but there is one “extra” 11.", "topic": "statistics"}
{"id": "book 2_p565_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 565, "page_end": 565, "topic_guess": "statistics", "text": "GMAT statistics note: These “extra” terms differ from 9 by −1 and +2 respectively, for a total difference of +1. That positive difference implies that (mean > 9), or (mean > median).", "topic": "statistics"}
{"id": "book 2_p565_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 565, "page_end": 565, "topic_guess": "statistics", "text": "GMAT statistics note: To prove that n = 5 is too low, you could take a more conventional approach. If n = 5, the number of terms is 3 + 5 + 5 + 4 = 17.", "topic": "statistics"}
{"id": "book 2_p565_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 565, "page_end": 565, "topic_guess": "statistics", "text": "GMAT statistics note: Thus, the median is 10, while the mean is closer to 9. That is, if n = 5, then median > mean.", "topic": "statistics"}
{"id": "book 2_p565_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 565, "page_end": 565, "topic_guess": "statistics", "text": "GMAT statistics note: If n = 6, then mean > median.", "topic": "statistics"}
{"id": "book 2_p566_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 566, "page_end": 566, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. This is a combined work problem, so use the work formula: rate × time = work.", "topic": "word_problems"}
{"id": "book 2_p566_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 566, "page_end": 566, "topic_guess": "word_problems", "text": "GMAT word problem note: The work and rates are given, but you need to calculate time, so manipulate the formula: . There are 80 houses, y houses are painted at a rate of x houses per week, and the rate increases to 1.25x houses per week for the remaining 80 − y houses.", "topic": "word_problems"}
{"id": "book 2_p566_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 566, "page_end": 566, "topic_guess": "word_problems", "text": "GMAT word problem note: To make the math easier, choose values such that x and 1.25x are integers (i.e., x is a multiple of 4) and y and 80 − y are divisible by x and 1.25x, respectively.", "topic": "word_problems"}
{"id": "book 2_p566_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 566, "page_end": 566, "topic_guess": "word_problems", "text": "GMAT word problem note: Variable Value Units Total Houses 80 80 Houses Houses Painted at Slower Rate y 20 Houses Houses Painted at Faster Rate 80 − y 60 Houses Initial Rate x 4 Houses/Week Increased Rate 1.25x 5 Houses/Week The total painting time is.", "topic": "word_problems"}
{"id": "book 2_p567_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 567, "page_end": 567, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. If a2 = b2, then (x + y)2 = (x − y)2.", "topic": "word_problems"}
{"id": "book 2_p567_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 567, "page_end": 567, "topic_guess": "word_problems", "text": "GMAT word problem note: Distribute both sides using the square of a sum and square of a difference special products, then simplify.", "topic": "word_problems"}
{"id": "book 2_p568_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 568, "page_end": 568, "topic_guess": "general", "text": "GMAT quant note: There are three basic scenarios. X y xy = 0 0 Any nonzero ✓ Any nonzero 0 ✓ (1) INSUFFICIENT:.", "topic": "general"}
{"id": "book 2_p568_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 568, "page_end": 568, "topic_guess": "general", "text": "GMAT quant note: This statement indicates that x and y must be non- negative for their square roots to be real values. The statement also eliminates the last scenario, in which x = y = 0.", "topic": "general"}
{"id": "book 2_p568_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 568, "page_end": 568, "topic_guess": "general", "text": "GMAT quant note: But y could still be 0 or any positive value. X y xy = 0 0 Any positive ✓ ✓ Any positive 0 ✓ ✓ (2) SUFFICIENT:.", "topic": "general"}
{"id": "book 2_p569_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 569, "page_end": 569, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Eliminates the last scenario.", "topic": "percent"}
{"id": "book 2_p569_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 569, "page_end": 569, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: If x and y were both 0, would equal 0. It also eliminates the first scenario.", "topic": "percent"}
{"id": "book 2_p569_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 569, "page_end": 569, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Thus, you can conclude that y = 0. Set up a table and assign sheep to stalls.", "topic": "percent"}
{"id": "book 2_p569_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 569, "page_end": 569, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Stall A B C D E F G # of Sheep 2x 3x Since a fractional sheep is not possible in this problem, x must be a positive integer. With 23 sheep remaining, it is possible for each of the other stalls to.", "topic": "percent"}
{"id": "book 2_p570_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 570, "page_end": 570, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Hold at least 1 sheep (a Yes answer).", "topic": "number_properties"}
{"id": "book 2_p570_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 570, "page_end": 570, "topic_guess": "number_theory", "text": "GMAT number properties note: However, the 23 other sheep might all be in stall B, leaving stalls A, D, F, and G empty (a No answer). Set up a table and assign sheep to stalls.", "topic": "number_properties"}
{"id": "book 2_p570_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 570, "page_end": 570, "topic_guess": "number_theory", "text": "GMAT number properties note: Stall A B C D E F G # of Sheep 5y 2y If y = 1, there are 5 sheep in E and 2 sheep in F. With 26 sheep remaining, it is possible for each of the other stalls to hold at least 1 sheep (a Yes answer).", "topic": "number_properties"}
{"id": "book 2_p570_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 570, "page_end": 570, "topic_guess": "number_theory", "text": "GMAT number properties note: However, 13 sheep might be in both stalls A and B, leaving stalls C, D, and G empty (a No answer). The only possibility is x = 5 and y = 3, since higher multiples would require more than 33 sheep total.", "topic": "number_properties"}
{"id": "book 2_p570_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 570, "page_end": 570, "topic_guess": "number_theory", "text": "GMAT number properties note: Thus, 31 sheep are allocated among these stalls as follows.", "topic": "number_properties"}
{"id": "book 2_p571_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 571, "page_end": 571, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Stall A B C D E F G # of Sheep 10 15 6 Thirty-one of the 33 sheep are in three of the 7 pens.", "topic": "algebra"}
{"id": "book 2_p571_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 571, "page_end": 571, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: With only 2 sheep unaccounted for, there is no way to place at least 1 sheep in each of the remaining four pens (a definite No answer). Group the 10r terms together.", "topic": "algebra"}
{"id": "book 2_p571_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 571, "page_end": 571, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Since q > r, the exponent on 10q − r is positive. Since s is a positive integer, s × 10q − r is a multiple of 10 and therefore ends in 0.", "topic": "algebra"}
{"id": "book 2_p571_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 571, "page_end": 571, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Any multiple of 10 minus 1 yields an integer with a units digit of 9.", "topic": "algebra"}
{"id": "book 2_p572_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 572, "page_end": 572, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. The complicated expression in the question stem leads to a disguised Positive/Negative problem.", "topic": "algebra"}
{"id": "book 2_p572_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 572, "page_end": 572, "topic_guess": "algebra", "text": "GMAT algebra note: In both cases (positive or negative 3), the end result is 3. Thus, in general, will always result in a positive value, or |x|.", "topic": "algebra"}
{"id": "book 2_p572_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 572, "page_end": 572, "topic_guess": "algebra", "text": "GMAT algebra note: Rephrase the original question using the absolute value symbol in place of the “square root of the square” symbols. Then try to make the right-hand side look more like the le.", "topic": "algebra"}
{"id": "book 2_p572_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 572, "page_end": 572, "topic_guess": "algebra", "text": "GMAT algebra note: Becomes Is |y − 4| = −(y − 4)? Since the absolute value of y − 4 must be positive or zero, you can rephrase the question further.", "topic": "algebra"}
{"id": "book 2_p572_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 572, "page_end": 572, "topic_guess": "algebra", "text": "GMAT algebra note: The absolute value |y − 3| can be interpreted as the distance between y and 3 on a number line.", "topic": "algebra"}
{"id": "book 2_p573_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 573, "page_end": 573, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If y × |y| > 0, then y × |y| is positive.", "topic": "number_properties"}
{"id": "book 2_p573_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 573, "page_end": 573, "topic_guess": "number_theory", "text": "GMAT number properties note: This means y and |y| must have the same sign. The term |y| is non-negative, so y must be positive.", "topic": "number_properties"}
{"id": "book 2_p573_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 573, "page_end": 573, "topic_guess": "number_theory", "text": "GMAT number properties note: A base of 2 is common to each term, and 10 is the smallest exponent appearing with that base. Factor 210 out from all of the terms in the expression.", "topic": "number_properties"}
{"id": "book 2_p573_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 573, "page_end": 573, "topic_guess": "number_theory", "text": "GMAT number properties note: Clearly, 2 is a prime factor, but is it the greatest? Examine (54 − 2352 + 24) to determine whether it has a larger prime factor.", "topic": "number_properties"}
{"id": "book 2_p573_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 573, "page_end": 573, "topic_guess": "number_theory", "text": "GMAT number properties note: The expression is of the form x2 − 2xy + y2, where:. Write the expression in factored form.", "topic": "number_properties"}
{"id": "book 2_p573_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 573, "page_end": 573, "topic_guess": "number_theory", "text": "GMAT number properties note: The prime factors of 21 are 3 and 7, so the largest prime factor of the original expression is 7.", "topic": "number_properties"}
{"id": "book 2_p574_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 574, "page_end": 574, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Alternatively, if you did not see the quadratic template in (54 − 2352 + 24), you could also perform the computation and factor the result.", "topic": "percent"}
{"id": "book 2_p574_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 574, "page_end": 574, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: According to the text, a “shrinking number” is a decimal between 0 and 1 in which each digit to the right of the decimal is no smaller than the digit to its immediate right. That is, in a shrinking number, each digit is either larger than the one to its right, or the same as the one to its right.", "topic": "percent"}
{"id": "book 2_p574_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 574, "page_end": 574, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: As you go through the digits from le to right, the digits will either stay the same or get smaller. For instance, 0.5331 is a shrinking number, but 0.37654 is not a shrinking number.", "topic": "percent"}
{"id": "book 2_p574_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 574, "page_end": 574, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: This Roman Numeral problem asks which of three options must be true. Test cases and eliminate any options that are not necessarily true.", "topic": "percent"}
{"id": "book 2_p574_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 574, "page_end": 574, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Because the digits do not get larger a er the decimal point.", "topic": "percent"}
{"id": "book 2_p575_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 575, "page_end": 575, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Number is relatively easy to do math with, start by using x = 0.1 to test cases.", "topic": "percent"}
{"id": "book 2_p575_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 575, "page_end": 575, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: However, this is not a shrinking number, since 9 is greater than 0. So statement I is not necessarily true.", "topic": "percent"}
{"id": "book 2_p575_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 575, "page_end": 575, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: , which is a shrinking number. In fact, statement II must always be true.", "topic": "percent"}
{"id": "book 2_p575_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 575, "page_end": 575, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Adding 9 and then dividing by 10 is equivalent to inserting the digit 9 to the immediate right of the decimal. Since 9 is the largest possible digit, it will never be lower than the digit to its immediate right, so the decimal will still be a shrinking number.", "topic": "percent"}
{"id": "book 2_p575_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 575, "page_end": 575, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: , which is not a shrinking number, since the digit two places to the right of the decimal is greater than the digit immediately to the right of the decimal. Statement III is not necessarily true.", "topic": "percent"}
{"id": "book 2_p576_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 576, "page_end": 576, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. A gallery has seven paintings with different prices.", "topic": "statistics"}
{"id": "book 2_p576_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 576, "page_end": 576, "topic_guess": "statistics", "text": "GMAT statistics note: Is it possible to purchase at least three of the paintings for no more than $1,800? This is possible only if the three least expensive paintings at the gallery have a total price of $1,800 or less.", "topic": "statistics"}
{"id": "book 2_p576_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 576, "page_end": 576, "topic_guess": "statistics", "text": "GMAT statistics note: The question can be rephrased as, “Is the total price of the three least expensive paintings $1,800 or less?” (1) SUFFICIENT:. The median price of the seven paintings is $550.", "topic": "statistics"}
{"id": "book 2_p576_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 576, "page_end": 576, "topic_guess": "statistics", "text": "GMAT statistics note: Since all of the paintings have different prices, the three least expensive paintings have prices below the median. Therefore, the total price of the three least expensive paintings is less than $550 + $550 + $550 = $1,650.", "topic": "statistics"}
{"id": "book 2_p576_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 576, "page_end": 576, "topic_guess": "statistics", "text": "GMAT statistics note: The answer is definitely Yes, so this statement is sufficient. This statement implies that the four least expensive paintings have a total price of $2,300 or less.", "topic": "statistics"}
{"id": "book 2_p576_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 576, "page_end": 576, "topic_guess": "statistics", "text": "GMAT statistics note: Jot down some possible cases. Case 1: $1, $2, $3, and $2,000.", "topic": "statistics"}
{"id": "book 2_p576_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 576, "page_end": 576, "topic_guess": "statistics", "text": "GMAT statistics note: The three least expensive paintings can be purchased for $1,800, and the answer to the question is Yes. To find a No answer, try to maximize the lowest three prices out of the four.", "topic": "statistics"}
{"id": "book 2_p576_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 576, "page_end": 576, "topic_guess": "statistics", "text": "GMAT statistics note: To do so, make the highest price as low as possible, or only slightly more than one-fourth of the total. Since one-fourth of $2,300 is $575, a good case to test is $573, $574, $576, and $577.", "topic": "statistics"}
{"id": "book 2_p577_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 577, "page_end": 577, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: The inequality y > |x| represents everything above the line (on either side of the y-axis)—that is, the shaded region. Since the equation y = |x| forms a 45-degree angle from the x-axis, there are 90 degrees above the line (on both sides of the y-axis).", "topic": "geometry"}
{"id": "book 2_p577_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 577, "page_end": 577, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: This represents one-fourth of the xy-plane. Therefore, if a random pair of (x, y) coordinates is chosen from the plane, the probability is that the point will fit the criterion y ≥ |x|.", "topic": "geometry"}
{"id": "book 2_p579_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 579, "page_end": 579, "topic_guess": "number_theory", "text": "GMAT number properties note: Track mean, median, range, distribution, and how operations change spread. If 3a + 3a − 2 = (90)(3b), what is b in terms of a?", "topic": "statistics"}
{"id": "book 2_p579_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 579, "page_end": 579, "topic_guess": "number_theory", "text": "GMAT number properties note: If xy ≠ 0, what is the value of y = 4 − x(1) x(x − 6y) = −9y2(2) 84. When one new integer is added to an existing list of six integers, does the median of the list change?", "topic": "statistics"}
{"id": "book 2_p579_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 579, "page_end": 579, "topic_guess": "number_theory", "text": "GMAT number properties note: The mean of the original six numbers is 50.(1) At least two of the numbers in the original list were 50.(2) 85. If abc + dcb = 598, then which of the following must be equivalent to a?", "topic": "statistics"}
{"id": "book 2_p580_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 580, "page_end": 580, "topic_guess": "number_theory", "text": "GMAT number properties note: Track mean, median, range, distribution, and how operations change spread. For nonzero integers a, b, c and d, is negative?", "topic": "statistics"}
{"id": "book 2_p580_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 580, "page_end": 580, "topic_guess": "number_theory", "text": "GMAT number properties note: Ad + bc = 0(1) abcd = −4(2) 87. Set A consists of 8 distinct prime numbers.", "topic": "statistics"}
{"id": "book 2_p580_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 580, "page_end": 580, "topic_guess": "number_theory", "text": "GMAT number properties note: If x is equal to the range of set A and y is equal to the median of set A, is the product xy even? The smallest integer in the set is 5.(1) The largest integer in the set is 101.(2) 88.", "topic": "statistics"}
{"id": "book 2_p580_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 580, "page_end": 580, "topic_guess": "number_theory", "text": "GMAT number properties note: If x3 < 16x, which of the following CANNOT be true?", "topic": "statistics"}
{"id": "book 2_p581_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 581, "page_end": 581, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. Is the two-digit positive integer n divisible by 3?", "topic": "number_properties"}
{"id": "book 2_p581_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 581, "page_end": 581, "topic_guess": "general", "text": "GMAT quant note: If the digits of n are reversed to produce the two-digit integer m, then m is divisible by 3. If the digits of n are reversed to produce the two-digit integer m, then m + n is divisible by 3.", "topic": "number_properties"}
{"id": "book 2_p581_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 581, "page_end": 581, "topic_guess": "general", "text": "GMAT quant note: Of all the sheets of paper in the tray of Printer A, 20% are removed and transferred to the tray of Printer B. A er the transfer, are there more sheets of paper in the tray of Printer B than in the tray of Printer A?", "topic": "number_properties"}
{"id": "book 2_p581_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 581, "page_end": 581, "topic_guess": "general", "text": "GMAT quant note: The transfer increases the number of sheets of paper in the tray of Printer B by more than 25%.", "topic": "number_properties"}
{"id": "book 2_p582_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 582, "page_end": 582, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "book 2_p583_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 583, "page_end": 583, "topic_guess": "geometry", "text": "GMAT geometry note: The circumference of the circle is equal to so The area of the circle is equal to The area of a square is side2. The diagonal of this square is the diameter of the circle, which is equal to .", "topic": "geometry"}
{"id": "book 2_p583_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 583, "page_end": 583, "topic_guess": "geometry", "text": "GMAT geometry note: The diagonal of a square is always , so the side of the square is equal to Therefore, side and the area of square ABCD is:. The shaded area is the area of the circle minus the area of the square = πx − 2x.", "topic": "geometry"}
{"id": "book 2_p584_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 584, "page_end": 584, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Because the problem never provides real values for a or b, you can choose smart numbers to solve.", "topic": "algebra"}
{"id": "book 2_p584_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 584, "page_end": 584, "topic_guess": "algebra", "text": "GMAT algebra note: Choose something for a that will make both exponents on the le side positive. Plug a = 3 into the answers and look for −1.", "topic": "algebra"}
{"id": "book 2_p584_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 584, "page_end": 584, "topic_guess": "algebra", "text": "GMAT algebra note: Alternatively, you can solve algebraically. Some manipulation is required to determine b in terms of a, but spend a minute thinking strategically about what manipulations to do.", "topic": "algebra"}
{"id": "book 2_p584_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 584, "page_end": 584, "topic_guess": "algebra", "text": "GMAT algebra note: It would help to isolate terms with an a by factoring 3a out on the le side. This will let you compare 3a to 3b, and thus compare a to b, once you clean up the constant terms that are le over.", "topic": "algebra"}
{"id": "book 2_p585_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 585, "page_end": 585, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Ninety is also the sum of 81 and 9, both powers of 3 themselves.", "topic": "algebra"}
{"id": "book 2_p585_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 585, "page_end": 585, "topic_guess": "algebra", "text": "GMAT algebra note: It is likely that some constant terms will cancel, as shown below. The constraint in the question stem indicates that neither x nor y equals zero.", "topic": "algebra"}
{"id": "book 2_p585_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 585, "page_end": 585, "topic_guess": "algebra", "text": "GMAT algebra note: You can prove insufficiency by Testing Cases. If y = 1, then x = 3, in which case is 3.", "topic": "algebra"}
{"id": "book 2_p585_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 585, "page_end": 585, "topic_guess": "algebra", "text": "GMAT algebra note: If y = 2, then x = 2, in which case is 1. Testing Cases might not be such a great idea for this statement because there are multiple instances of each variable.", "topic": "algebra"}
{"id": "book 2_p585_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 585, "page_end": 585, "topic_guess": "algebra", "text": "GMAT algebra note: Begin by distributing the le -hand side of the equation.", "topic": "algebra"}
{"id": "book 2_p586_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 586, "page_end": 586, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Now, you’ve got a quadratic, so solve that way.", "topic": "statistics"}
{"id": "book 2_p586_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 586, "page_end": 586, "topic_guess": "statistics", "text": "GMAT statistics note: You can test cases to try to prove or disprove the statements. If the original set is 50, 50, 50, 50, 50, 50, the mean and median were 50 originally.", "topic": "statistics"}
{"id": "book 2_p586_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 586, "page_end": 586, "topic_guess": "statistics", "text": "GMAT statistics note: The new term may be any value and the median will not change. If, on the other hand, the original set is 0, 0, 0, 100, 100, 100, the mean and median were 50 originally.", "topic": "statistics"}
{"id": "book 2_p586_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 586, "page_end": 586, "topic_guess": "statistics", "text": "GMAT statistics note: If the new term is 75, the median increases to 75. More generally for this set, if the new term is not equal to the original median of 50, the median will change.", "topic": "statistics"}
{"id": "book 2_p586_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 586, "page_end": 586, "topic_guess": "statistics", "text": "GMAT statistics note: If the original set is 50, 50, 50, 50, 50, 50, the median was 50 originally. If, on the other hand, the original set is 0, 0, 0, 50, 50, 50, the median was 25 originally.", "topic": "statistics"}
{"id": "book 2_p586_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 586, "page_end": 586, "topic_guess": "statistics", "text": "GMAT statistics note: If the new term is 40, the median increases to 40.", "topic": "statistics"}
{"id": "book 2_p587_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 587, "page_end": 587, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Generally for this set, if the new term is not equal to the original median of 25, the median will change.", "topic": "statistics"}
{"id": "book 2_p587_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 587, "page_end": 587, "topic_guess": "statistics", "text": "GMAT statistics note: The mean must be 50 and at least two terms in the original set must be 50. If the original set is 50, 50, 50, 50, 50, 50, the mean and median were 50 originally.", "topic": "statistics"}
{"id": "book 2_p587_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 587, "page_end": 587, "topic_guess": "statistics", "text": "GMAT statistics note: The new term may be any value and the median will not change. If the original set is 0, 0, 0, 50, 50, 200, the mean was 50 and the median was 25 originally.", "topic": "statistics"}
{"id": "book 2_p587_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 587, "page_end": 587, "topic_guess": "statistics", "text": "GMAT statistics note: If the new term is greater than 25, the median increases. If the new term is less than 25, the median decreases.", "topic": "statistics"}
{"id": "book 2_p587_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 587, "page_end": 587, "topic_guess": "statistics", "text": "GMAT statistics note: Set up the addition and extract several equations by summing each digit place individually. Note that both the ones digit and tens digit come from the sum c + b.", "topic": "statistics"}
{"id": "book 2_p587_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 587, "page_end": 587, "topic_guess": "statistics", "text": "GMAT statistics note: Since the result is different in the ones digit (8) and the tens digit (9), a 1 must be carried from the ones to the tens digit. If c + b = 18, then both c and b must be 9 (the largest digit).", "topic": "statistics"}
{"id": "book 2_p587_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 587, "page_end": 587, "topic_guess": "statistics", "text": "GMAT statistics note: Place an 8 in the ones digit of the sum and carry a 1 to the tens place.", "topic": "statistics"}
{"id": "book 2_p588_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 588, "page_end": 588, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Tens digit is thus 1 + b + c = 1 + 18 = 19.", "topic": "number_properties"}
{"id": "book 2_p588_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 588, "page_end": 588, "topic_guess": "number_theory", "text": "GMAT number properties note: Next, place a 9 in the tens digit of the sum and carry a 1 to the hundreds place. In the hundreds place, the sum is 1 + a + d = 5.", "topic": "number_properties"}
{"id": "book 2_p588_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 588, "page_end": 588, "topic_guess": "number_theory", "text": "GMAT number properties note: If an even number (0, 2, or 4) of the integers a, b, c, and d is negative, each pair of negatives will cancel, because (−1)(−1) = +1 and . This would yield a positive result for Thus, a way to rephrase the question is, “Among the integers a, b, c, and d, are an odd number (one or three) of them negative?” (1) SUFFICIENT:.", "topic": "number_properties"}
{"id": "book 2_p588_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 588, "page_end": 588, "topic_guess": "number_theory", "text": "GMAT number properties note: This statement can be rephrased as ad = −bc. A d b c ad = −bc Odd number of negatives?", "topic": "number_properties"}
{"id": "book 2_p589_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 589, "page_end": 589, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. A d b c ad = −bc Odd number of negatives?", "topic": "number_properties"}
{"id": "book 2_p589_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 589, "page_end": 589, "topic_guess": "number_theory", "text": "GMAT number properties note: − − − + ✓ Yes + − − − ✓ Yes Though the table doesn’t list all possibilities, it lists enough to realize that, in order for the signs of ad and bc to be opposite one another, either one or three of the four integers must be negative. You might recognize that the (−1)(−1) = +1 property implies that abcd is only negative when there are non-paired negatives among the integers.", "topic": "number_properties"}
{"id": "book 2_p589_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 589, "page_end": 589, "topic_guess": "number_theory", "text": "GMAT number properties note: That is, an odd number (one or three) of the integers a, b, c, and d must be negative. If not, you could list a few cases to see the pattern.", "topic": "number_properties"}
{"id": "book 2_p589_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 589, "page_end": 589, "topic_guess": "number_theory", "text": "GMAT number properties note: A b c d abcd = negative Odd number of negatives? + + − + ✓ Yes + − + + ✓ Yes − + − − ✓ Yes − − + − ✓ Yes.", "topic": "number_properties"}
{"id": "book 2_p590_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 590, "page_end": 590, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. The product xy will be even if x is even, y is even, or both are even.", "topic": "statistics"}
{"id": "book 2_p590_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 590, "page_end": 590, "topic_guess": "statistics", "text": "GMAT statistics note: The prime numbers include 2, 3, 5, 7, 11, 13, 17, 19, etc. The smallest possible term in set A is 2, which is the only even prime.", "topic": "statistics"}
{"id": "book 2_p590_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 590, "page_end": 590, "topic_guess": "statistics", "text": "GMAT statistics note: X = the range of set A = largest term − smallest term in set A. If the smallest term in set A is 2, then x = odd − even = odd.", "topic": "statistics"}
{"id": "book 2_p590_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 590, "page_end": 590, "topic_guess": "statistics", "text": "GMAT statistics note: If the smallest term in set A is odd (i.e., not 2), then x = odd − odd = even. The median of set A is the average of the two middle terms, since the number of terms in the set is even.", "topic": "statistics"}
{"id": "book 2_p590_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 590, "page_end": 590, "topic_guess": "statistics", "text": "GMAT statistics note: However, y could be either even (e.g., when the middle terms are 11 and 13) or odd (e.g., when the middle terms are 7 and 11). A useful rephrase of this question is, “Is either x or y even?” (1) SUFFICIENT:.", "topic": "statistics"}
{"id": "book 2_p590_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 590, "page_end": 590, "topic_guess": "statistics", "text": "GMAT statistics note: If the smallest prime in the set is 5, x = even, and therefore xy is even. If the largest integer in the set is 101, the range of the set can be odd or even (e.g., 101 − 3 = 98 or 101 − 2 = 99).", "topic": "statistics"}
{"id": "book 2_p590_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 590, "page_end": 590, "topic_guess": "statistics", "text": "GMAT statistics note: The median of the set can also be odd or even, as discussed. Therefore, xy can be either odd or even.", "topic": "statistics"}
{"id": "book 2_p591_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 591, "page_end": 591, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. It may be tempting to simplify this way:.", "topic": "algebra"}
{"id": "book 2_p591_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 591, "page_end": 591, "topic_guess": "algebra", "text": "GMAT algebra note: The first step of this solution is wrong because you can’t divide by x without knowing its sign. If x is negative, you would have to flip the sign.", "topic": "algebra"}
{"id": "book 2_p591_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 591, "page_end": 591, "topic_guess": "algebra", "text": "GMAT algebra note: Positive x2 < 16 (don’t flip) |x| < 4 0 < x < 4 Negative x2 > 16 (flip) |x| > 4 x < −4 There are two ranges of solutions for x. The question asks for something that is definitely NOT true.", "topic": "algebra"}
{"id": "book 2_p591_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 591, "page_end": 591, "topic_guess": "algebra", "text": "GMAT algebra note: Alternatively, there is a way to do algebra with inequalities containing variables. Instead of dividing by the variable, subtract it from both sides, then factor.", "topic": "algebra"}
{"id": "book 2_p592_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 592, "page_end": 592, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. In order for the product of three numbers to be negative, either all three need to be negative, or exactly one of the three needs to be negative.", "topic": "number_properties"}
{"id": "book 2_p592_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 592, "page_end": 592, "topic_guess": "number_theory", "text": "GMAT number properties note: This cannot occur if x is greater than 4, since all three values would be positive. Before diving into the statements, remind yourself of the divisibility rules.", "topic": "number_properties"}
{"id": "book 2_p592_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 592, "page_end": 592, "topic_guess": "number_theory", "text": "GMAT number properties note: One way to check for divisibility by 3 is to add the digits. If their sum is divisible by 3, then the number itself is divisible by 3.", "topic": "number_properties"}
{"id": "book 2_p592_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 592, "page_end": 592, "topic_guess": "number_theory", "text": "GMAT number properties note: If m is divisible by 3, then the sum of the digits of m must be divisible by 3 as well. Since m has the same digits as n, the sum of the digits of n must also be a multiple of 3.", "topic": "number_properties"}
{"id": "book 2_p592_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 592, "page_end": 592, "topic_guess": "number_theory", "text": "GMAT number properties note: Therefore, n itself is divisible by 3. This statement requires a bit more work because it’s not immediately clear what would happen when you add m and n.", "topic": "number_properties"}
{"id": "book 2_p592_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 592, "page_end": 592, "topic_guess": "number_theory", "text": "GMAT number properties note: Because the question asks you to move digits around to different places (the tens place, the units place), rewrite the information using place-value notation, where a = the tens digit of n and b = the units digit of n.", "topic": "number_properties"}
{"id": "book 2_p593_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 593, "page_end": 593, "topic_guess": "algebra", "text": "GMAT algebra note: Watch divisibility, factors, remainders, parity, and prime structure. The statement indicates that the sum m + n is divisible by 3, so 11(a + b) must also be divisible by 3.", "topic": "number_properties"}
{"id": "book 2_p593_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 593, "page_end": 593, "topic_guess": "algebra", "text": "GMAT algebra note: Eleven itself is not divisible by 3, so the sum a + b must be divisible by 3. This sum represents the sum of the two digits of the number m as well as the sum of the two digits of the number n.", "topic": "number_properties"}
{"id": "book 2_p593_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 593, "page_end": 593, "topic_guess": "algebra", "text": "GMAT algebra note: If the sum of n’s two digits is divisible by 3, then n is also divisible by 3. You can solve this problem algebraically or by Testing Cases.", "topic": "number_properties"}
{"id": "book 2_p593_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 593, "page_end": 593, "topic_guess": "algebra", "text": "GMAT algebra note: If Printer A starts out with 100 sheets, then 20 sheets are moved to Printer B, and Printer A ends up with 80 sheets. If B started out with 75 sheets, then an increase of 20 sheets would be more than 25% of the starting number, and B would end up with 95 sheets.", "topic": "number_properties"}
{"id": "book 2_p593_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 593, "page_end": 593, "topic_guess": "algebra", "text": "GMAT algebra note: In this case, B has more sheets than A a er the transfer. If, instead, B started out with 40 sheets, then an increase of 20 sheets would be more than 25% of the starting number, and B would end up with 60 sheets.", "topic": "number_properties"}
{"id": "book 2_p593_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 593, "page_end": 593, "topic_guess": "algebra", "text": "GMAT algebra note: In this case, B has fewer sheets than A a er the transfer. If Printer A starts out with 100 sheets and 20 are moved to Printer B, then A ends up with 80 sheets.", "topic": "number_properties"}
{"id": "book 2_p593_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 593, "page_end": 593, "topic_guess": "algebra", "text": "GMAT algebra note: If B started out with 2,000 sheets, then an increase of 20 sheets would be less than 30% of the starting number, and B would end up.", "topic": "number_properties"}
{"id": "book 2_p594_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 594, "page_end": 594, "topic_guess": "algebra", "text": "GMAT algebra note: Percent problems usually relate part, whole, rate, and percent change. If B started out with 80 sheets, then a 20-sheet increase would still be less than 30% of the starting number, and again B would end up with more sheets than A.", "topic": "percent"}
{"id": "book 2_p594_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 594, "page_end": 594, "topic_guess": "algebra", "text": "GMAT algebra note: If the 20 sheets represent a less-than-30% increase, then use 30% as the limiting figure. If 20 represented exactly 30% of B’s sheets, then B would have to start with sheets (ignore the fact that this isn’t an integer).", "topic": "percent"}
{"id": "book 2_p594_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 594, "page_end": 594, "topic_guess": "algebra", "text": "GMAT algebra note: In this case, B would end up with sheets, which is still greater than A’s 80 sheets. A er the transfer, B can’t drop below A.", "topic": "percent"}
{"id": "book 2_p594_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 594, "page_end": 594, "topic_guess": "algebra", "text": "GMAT algebra note: Algebraically, use a for the original number of sheets in Printer A and use b for the original number of sheets in Printer B. A er the transfer, A has a – 0.2a = 0.8a sheets.", "topic": "percent"}
{"id": "book 2_p594_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 594, "page_end": 594, "topic_guess": "algebra", "text": "GMAT algebra note: Rephrase the question: “Is b + 0.2a > 0.8a?” Or, “Is b > 0.6a?” (1) INSUFFICIENT:. Translate the statement into algebra.", "topic": "percent"}
{"id": "book 2_p594_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 594, "page_end": 594, "topic_guess": "algebra", "text": "GMAT algebra note: Flip the statement around to compare it more easily to the rephrased question: b < 0.8a.", "topic": "percent"}
{"id": "book 2_p595_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 595, "page_end": 595, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Therefore, or The decimal equivalent of is , so b is indeed always greater than 0.6a.", "topic": "percent"}
{"id": "book 2_p596_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 596, "page_end": 596, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. If |x| ≠ |y|, xy ≠ 0, , and , then is equal to which of the following?", "topic": "word_problems"}
{"id": "book 2_p596_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 596, "page_end": 596, "topic_guess": "word_problems", "text": "GMAT word problem note: If a certain culture of bacteria increases by a constant factor of x every y minutes, how long will it take for the culture to increase to 10,000 times its original size?", "topic": "word_problems"}
{"id": "book 2_p597_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 597, "page_end": 597, "topic_guess": "number_theory", "text": "GMAT number properties note: Percent problems usually relate part, whole, rate, and percent change. In two minutes, the culture will increase to 100 times its original size.", "topic": "percent"}
{"id": "book 2_p597_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 597, "page_end": 597, "topic_guess": "number_theory", "text": "GMAT number properties note: A cylinder of height h is full of water. When all of the water is poured into an empty cylinder whose radius is 25% larger than that of the original cylinder, the new cylinder is full.", "topic": "percent"}
{"id": "book 2_p597_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 597, "page_end": 597, "topic_guess": "number_theory", "text": "GMAT number properties note: The height of the new cylinder is what percent of h? If a and b are distinct positive integers, what is the units digit of 2a8b4a + b?", "topic": "percent"}
{"id": "book 2_p597_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 597, "page_end": 597, "topic_guess": "number_theory", "text": "GMAT number properties note: B = 24 and a < 24(1) The greatest common factor of a and b is 12.(2) 95. Employees of a certain company are each to receive a unique seven-digit identification code consisting of the digits 0, 1, 2, 3, 4, 5, and 6 such that no digit is used more than once in any given code.", "topic": "percent"}
{"id": "book 2_p598_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 598, "page_end": 598, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. In valid codes, the second digit in the code is exactly twice the first digit.", "topic": "probability"}
{"id": "book 2_p598_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 598, "page_end": 598, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: How many valid codes are there? An n-sided die has sides labeled with the numbers 1 through n, inclusive, and has an equal probability of landing on any side when rolled.", "topic": "probability"}
{"id": "book 2_p598_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 598, "page_end": 598, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: If the die is rolled twice, what is the probability of rolling 1 both times? If the die is rolled twice, the probability that the two rolls are different is 80%.", "topic": "probability"}
{"id": "book 2_p598_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 598, "page_end": 598, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: If the die is rolled twice, the probability that the two rolls are equal is 97. If w, x, y, and z are positive integers and , what is the proper order, increasing from le to right, of the following quantities.", "topic": "probability"}
{"id": "book 2_p599_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 599, "page_end": 599, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. What is the value of |a + b|?", "topic": "number_properties"}
{"id": "book 2_p599_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 599, "page_end": 599, "topic_guess": "number_theory", "text": "GMAT number properties note: (a + b + c + d) (a + b − c − d) = 16(1) c + d = 3(2) 99. Set A consists of all the integers between 10 and 21, inclusive.", "topic": "number_properties"}
{"id": "book 2_p599_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 599, "page_end": 599, "topic_guess": "number_theory", "text": "GMAT number properties note: Set B consists of all the integers between 10 and 50, inclusive. If x is a number chosen randomly from set A, y is a number chosen randomly from set B, and y has no factor z such that 1 < z < y, what is the probability that the product xy is divisible by 3?", "topic": "number_properties"}
{"id": "book 2_p600_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 600, "page_end": 600, "topic_guess": "general", "text": "GMAT quant note: At a birthday party, x children will be seated at two different tables. At the table with the birthday cake on it, exactly y children will be seated, including the birthday girl, Sally.", "topic": "general"}
{"id": "book 2_p600_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 600, "page_end": 600, "topic_guess": "general", "text": "GMAT quant note: How many different groups of children may be seated at the birthday cake table?", "topic": "general"}
{"id": "book 2_p601_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 601, "page_end": 601, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "book 2_p602_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 602, "page_end": 602, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "book 2_p603_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 603, "page_end": 603, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating. Workout Set 10 Solutions 91.", "topic": "word_problems"}
{"id": "book 2_p604_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 604, "page_end": 604, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. This problem can be solved algebraically, but this path is not recommended.", "topic": "algebra"}
{"id": "book 2_p604_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 604, "page_end": 604, "topic_guess": "algebra", "text": "GMAT algebra note: Now that the fractions have a common denominator of x, subtract one from the other. You are not alone if the algebraic solution was not obvious to you!", "topic": "algebra"}
{"id": "book 2_p604_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 604, "page_end": 604, "topic_guess": "algebra", "text": "GMAT algebra note: Algebraic “false starts” are common in this type of problem. In addition, there are other, equally valid algebraic paths whose final forms would not match any of the answers.", "topic": "algebra"}
{"id": "book 2_p605_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 605, "page_end": 605, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. It’s fast, easy, and correct.", "topic": "algebra"}
{"id": "book 2_p605_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 605, "page_end": 605, "topic_guess": "algebra", "text": "GMAT algebra note: To understand the question stem, pick some numbers: the bacteria culture begins with an initial quantity of I = 100 and increases by a factor of x = 2 every y = 3 minutes. Construct a table to track the growth of the bacteria.", "topic": "algebra"}
{"id": "book 2_p605_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 605, "page_end": 605, "topic_guess": "algebra", "text": "GMAT algebra note: Time (min.) Bacteria 3 100(2) = 100(2)1 6 100(2)(2) = 100(2)2 9 100(2)(2)(2) = 100(2)3 12 100(2)(2)(2)(2) = 100(2)4 3n = t 100(2)n n represents the number of growth periods, and where t is time in minutes.", "topic": "algebra"}
{"id": "book 2_p606_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 606, "page_end": 606, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. This question asks how long it will take for the bacteria to grow to 10,000 times their original amount.", "topic": "algebra"}
{"id": "book 2_p606_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 606, "page_end": 606, "topic_guess": "algebra", "text": "GMAT algebra note: In other words, “What is t when F = 10,000I?” The rephrased question is, “What is t when (1) SUFFICIENT:. Note that the yth root of x is equivalent to x to the power.", "topic": "algebra"}
{"id": "book 2_p606_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 606, "page_end": 606, "topic_guess": "algebra", "text": "GMAT algebra note: This statement indicates that . If you plug this value into the equation, you can solve for t (though stop the calculation at the point that you can tell that you can solve for t).", "topic": "algebra"}
{"id": "book 2_p607_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 607, "page_end": 607, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. The culture grows one-hundredfold in 2 minutes.", "topic": "percent"}
{"id": "book 2_p607_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 607, "page_end": 607, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: In other words, the sample grows by a factor of 102. Since exponential growth is characterized by a constant factor of growth (i.e., by a factor of x every y minutes), in another 2 minutes, the culture will grow by another factor of 102.", "topic": "percent"}
{"id": "book 2_p607_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 607, "page_end": 607, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Therefore, a er a total of 4 minutes, the culture will have grown by a factor of 102 × 102 = 104, or 10,000. Also note that the problem mentions the volume of the cylinder (V = πr2h), so you’re going to need a radius.", "topic": "percent"}
{"id": "book 2_p607_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 607, "page_end": 607, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: That radius needs to be increased by 25% later in the problem, so pick a small integer that is easy to increase by 25%, such as 4. The problem also contains two fractions, and .", "topic": "percent"}
{"id": "book 2_p607_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 607, "page_end": 607, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Since one number is already a multiple of 4, make the other number a multiple of 5.", "topic": "percent"}
{"id": "book 2_p608_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 608, "page_end": 608, "topic_guess": "general", "text": "GMAT quant note: Percent problems usually relate part, whole, rate, and percent change. Now, solve the problem using these numbers.", "topic": "percent"}
{"id": "book 2_p608_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 608, "page_end": 608, "topic_guess": "general", "text": "GMAT quant note: the original cylinder is only full. it is only full.", "topic": "percent"}
{"id": "book 2_p608_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 608, "page_end": 608, "topic_guess": "general", "text": "GMAT quant note: The original height was 5, so the new height is , or 80%, of the original height. This problem contains a common trap seen in many difficult DS questions.", "topic": "percent"}
{"id": "book 2_p609_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 609, "page_end": 609, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Combined statements would provide the values of both a and b, which could be plugged into the expression to answer the question.", "topic": "algebra"}
{"id": "book 2_p609_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 609, "page_end": 609, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Remembering the units digit patterns for powers of 2 will help on this problem:. The units digits for powers of 2 is a repeating pattern of [2, 4, 8, 6].", "topic": "algebra"}
{"id": "book 2_p609_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 609, "page_end": 609, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: If you can determine the relationship of 3a + 5b to a multiple of 4 (i.e., where 23a + 5b is in the predictable four-term repeating pattern of units digits), you will be able to answer the question. This question can be rephrased as, “What is the remainder when 3a + 5b is divided by 4?”.", "topic": "algebra"}
{"id": "book 2_p610_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 610, "page_end": 610, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. If b = 24, then 5b is a multiple of 4.", "topic": "probability"}
{"id": "book 2_p610_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 610, "page_end": 610, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: However, a could be any integer less than 24. Possible remainders when 3a is divided by 4 are 0, 1, 2, or 3.", "topic": "probability"}
{"id": "book 2_p610_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 610, "page_end": 610, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: If the greatest common factor of a and b is 12, then 12 must be a factor of both variables. That is, both a and b are multiples of 12 and thus also multiples of 4.", "topic": "probability"}
{"id": "book 2_p610_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 610, "page_end": 610, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: As a result, 3a and 5b will be multiples of 4 as well, so the remainder will be 0 when 3a + 5b is divided by 4. Valid codes must have a second digit that is exactly twice the first digit.", "topic": "probability"}
{"id": "book 2_p610_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 610, "page_end": 610, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: There are three ways to do this with the available digits. Scenario A: 12XXXXX Scenario B: 24XXXXX Scenario C: 36XXXXX For each of these basic scenarios, there are 5!", "topic": "probability"}
{"id": "book 2_p610_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 610, "page_end": 610, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Ways to shuffle the remaining five numbers (represented by X’s above). Thus, the total number of valid codes is 3 × 5!", "topic": "probability"}
{"id": "book 2_p610_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 610, "page_end": 610, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: When an n-sided die is rolled once, the probability of rolling any particular number (such as 1) is equal to . Therefore, the probability of.", "topic": "probability"}
{"id": "book 2_p611_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 611, "page_end": 611, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Rolling 1 both times is equal to .", "topic": "probability"}
{"id": "book 2_p611_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 611, "page_end": 611, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Rephrase the question as, “What is the value of n?” (1) SUFFICIENT:. The probability that the two rolls are different is 80%, or When rolling an n-sided die twice, the first roll can have any value from 1 to n, inclusive.", "topic": "probability"}
{"id": "book 2_p611_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 611, "page_end": 611, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Regardless of the value of the first roll, the probability is that the second roll will have that same value. Therefore, the overall probability of rolling the same value twice is , and the probability of rolling two different values is Therefore, according to this statement, .", "topic": "probability"}
{"id": "book 2_p611_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 611, "page_end": 611, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: The value of n is 5, and this statement is sufficient. The probability that two rolls of an n-sided die will have the same value is , regardless of the value of n.", "topic": "probability"}
{"id": "book 2_p611_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 611, "page_end": 611, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: This statement would be true for any value of n, so the value of n cannot be determined and this statement is insufficient.", "topic": "probability"}
{"id": "book 2_p612_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 612, "page_end": 612, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. It would require a lot of tricky work to solve this algebraically, so test cases instead.", "topic": "algebra"}
{"id": "book 2_p612_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 612, "page_end": 612, "topic_guess": "algebra", "text": "GMAT algebra note: Make sure to pick values for the unknowns such that holds true. Before plugging those values in for the quantities, check the answers.", "topic": "algebra"}
{"id": "book 2_p612_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 612, "page_end": 612, "topic_guess": "algebra", "text": "GMAT algebra note: All of them begin with 1 and , so don’t bother to test . Now, place in ascending order:.", "topic": "algebra"}
{"id": "book 2_p613_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 613, "page_end": 613, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. The question asks for the absolute value of a + b, so try to manipulate the statements to isolate that combination of variables, a + b.", "topic": "algebra"}
{"id": "book 2_p613_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 613, "page_end": 613, "topic_guess": "algebra", "text": "GMAT algebra note: This provides information about c and d, and the relationship between them, but no information about a or b. Manipulate the equation to group (a + b) and (c + d) terms.", "topic": "algebra"}
{"id": "book 2_p613_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 613, "page_end": 613, "topic_guess": "algebra", "text": "GMAT algebra note: Note that this is of the form (x + y)(x − y), where x = (a + b) and y = (c + d). This is the difference of squares special product, (x + y)(x − y) = x2 − y2.", "topic": "algebra"}
{"id": "book 2_p613_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 613, "page_end": 613, "topic_guess": "algebra", "text": "GMAT algebra note: Use this to transform this expression. This is not enough to determine the value of (a + b)2.", "topic": "algebra"}
{"id": "book 2_p613_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 613, "page_end": 613, "topic_guess": "algebra", "text": "GMAT algebra note: You can substitute for c + d and solve. The solution is shown below, but note that you don’t have to do that math:.", "topic": "algebra"}
{"id": "book 2_p613_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 613, "page_end": 613, "topic_guess": "algebra", "text": "GMAT algebra note: The question asked for the absolute value of a + b, so the fact that the equation has (a + b)2 doesn’t matter.", "topic": "algebra"}
{"id": "book 2_p614_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 614, "page_end": 614, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If y has no factor z such that 1 < z < y, then y must be prime.", "topic": "number_properties"}
{"id": "book 2_p614_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 614, "page_end": 614, "topic_guess": "number_theory", "text": "GMAT number properties note: Because y is selected from set B, it is a prime number between 10 and 50, inclusive. The only prime number that is divisible by 3 is 3, so y is definitely not divisible by 3.", "topic": "number_properties"}
{"id": "book 2_p614_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 614, "page_end": 614, "topic_guess": "number_theory", "text": "GMAT number properties note: Thus, xy is only divisible by 3 if x itself is divisible by 3. Rephrase the question: “What is the probability that a multiple of 3 will be chosen randomly from set A?” There are 21 − 10 + 1 = 12 terms in set A.", "topic": "number_properties"}
{"id": "book 2_p614_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 614, "page_end": 614, "topic_guess": "number_theory", "text": "GMAT number properties note: Of these, 4 terms (12, 15, 18, and 21) are divisible by 3. 6 has a factor 2 such that 1 < 2 < 6: 6 is NOT prime.", "topic": "number_properties"}
{"id": "book 2_p614_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 614, "page_end": 614, "topic_guess": "number_theory", "text": "GMAT number properties note: 15 has a factor 5 such that 1 < 5 < 15: 15 is NOT prime. 3 has NO factor between 1 and 3: 3 IS prime.", "topic": "number_properties"}
{"id": "book 2_p614_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 614, "page_end": 614, "topic_guess": "number_theory", "text": "GMAT number properties note: 7 has NO factor between 1 and 7: 7 IS prime.", "topic": "number_properties"}
{"id": "book 2_p615_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 615, "page_end": 615, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. There are 8 children at the party, and 5 will sit at the table with the cake.", "topic": "statistics"}
{"id": "book 2_p615_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 615, "page_end": 615, "topic_guess": "statistics", "text": "GMAT statistics note: Sally must sit at the birthday cake table, so pick 5 − 1 = 4 of the other 8 − 1 = 7 children to sit at that table with her. How many different ways can you choose 4 from a group of 7?", "topic": "statistics"}
{"id": "book 2_p615_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 615, "page_end": 615, "topic_guess": "statistics", "text": "GMAT statistics note: Set up an anagram grid, where Y means “at the cake table” and N means “at the other table.” A B C D E F G Y Y Y Y N N N Now, you can calculate the number of possible groups:. It may be the case that you will find the unsimplified factorial form, rather than 35.", "topic": "statistics"}
{"id": "book 2_p616_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 616, "page_end": 616, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. As an alternative to testing all five choices, you could use a hybrid approach to determine the formula using variables.", "topic": "algebra"}
{"id": "book 2_p616_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 616, "page_end": 616, "topic_guess": "algebra", "text": "GMAT algebra note: The number of possible groups was , but remember that this formula took Sally into account.", "topic": "algebra"}
{"id": "book 2_p617_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 617, "page_end": 617, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Substitute these variable expressions in place of the numbers.", "topic": "algebra"}
{"id": "book 2_p618_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 618, "page_end": 618, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. A casino pays players with chips that are either turquoise or violet colored.", "topic": "word_problems"}
{"id": "book 2_p618_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 618, "page_end": 618, "topic_guess": "word_problems", "text": "GMAT word problem note: If each turquoise-colored chip is worth t dollars and each violet-colored chip is worth v dollars, where t and v are integers, what is the combined value of four turquoise-colored chips and two violet-colored chips? The combined value of six turquoise-colored chips and three violet-colored chips is $42.", "topic": "word_problems"}
{"id": "book 2_p618_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 618, "page_end": 618, "topic_guess": "word_problems", "text": "GMAT word problem note: The combined value of five turquoise-colored chips and seven violet-colored chips is $53. Jonas and Amanda stand at opposite ends of a straight road and start running toward each other at the same moment.", "topic": "word_problems"}
{"id": "book 2_p618_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 618, "page_end": 618, "topic_guess": "word_problems", "text": "GMAT word problem note: Their rates are randomly selected in advance so that Jonas runs at a constant rate of 3, 4, 5, or 6 miles per hour and Amanda runs at a constant rate of 4, 5, 6, or 7 miles per hour. What is the probability that Jonas has traveled farther than Amanda by the time they meet?", "topic": "word_problems"}
{"id": "book 2_p619_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 619, "page_end": 619, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. If p is a positive integer, is p2 divisible by 96?", "topic": "number_properties"}
{"id": "book 2_p619_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 619, "page_end": 619, "topic_guess": "general", "text": "GMAT quant note: P is a multiple of 8.(1) p2 is a multiple of 12.(2).", "topic": "number_properties"}
{"id": "book 2_p620_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 620, "page_end": 620, "topic_guess": "geometry", "text": "GMAT geometry note: Parallel sides AB and CD are 7 inches apart and 6 and 8 inches long, respectively. What is the radius of the circle in inches?", "topic": "geometry"}
{"id": "book 2_p620_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 620, "page_end": 620, "topic_guess": "geometry", "text": "GMAT geometry note: If 9y + 3b = 10(3b), then 2y is equal to which of the following?", "topic": "geometry"}
{"id": "book 2_p621_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 621, "page_end": 621, "topic_guess": "geometry", "text": "GMAT geometry note: The radius of a circle is r yards. Is the area of the circle at least r square yards?", "topic": "geometry"}
{"id": "book 2_p621_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 621, "page_end": 621, "topic_guess": "geometry", "text": "GMAT geometry note: (1 yard = 3 feet) The diameter of the circle is more than 2 feet.(1) If the radius of the same circle is f feet, the area of the circle is more than 2f square feet. For how many values of x from 1 to 300, inclusive, is the sum of the integers from 1 to x, inclusive, divisible by 3?", "topic": "geometry"}
{"id": "book 2_p621_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 621, "page_end": 621, "topic_guess": "geometry", "text": "GMAT geometry note: If x and y are integers and |xy| = 8, what is the value of |x + y|? X and y are both divisible by 2.(1).", "topic": "geometry"}
{"id": "book 2_p622_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 622, "page_end": 622, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Set X consists of exactly four distinct integers that are greater than 1.", "topic": "number_properties"}
{"id": "book 2_p622_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 622, "page_end": 622, "topic_guess": "number_theory", "text": "GMAT number properties note: For each integer in the set, all of that integer’s unique prime factors are also in the set. How many of the distinct integers in set X are prime?", "topic": "number_properties"}
{"id": "book 2_p622_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 622, "page_end": 622, "topic_guess": "number_theory", "text": "GMAT number properties note: The product of the integers in set X is divisible by 36.(1) The product of the integers in set X is divisible by 60.(2) 110. If p = (22)(3x) and r = (22)(3y), where x and y are prime numbers and x ≠ y, which of the following represents the least common multiple of p and r?", "topic": "number_properties"}
{"id": "book 2_p623_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 623, "page_end": 623, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "book 2_p624_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 624, "page_end": 624, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Workout Set 11 Solutions 01.", "topic": "number_properties"}
{"id": "book 2_p624_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 624, "page_end": 624, "topic_guess": "number_theory", "text": "GMAT number properties note: The question asks for the value of 4t + 2v, where t and v represent the values of the turquoise and violet chips, respectively. Note that the question asks for a combination of variables, or a combo.", "topic": "number_properties"}
{"id": "book 2_p624_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 624, "page_end": 624, "topic_guess": "number_theory", "text": "GMAT number properties note: It may not be necessary to be able to solve for the individual values of t and v. Translate and simplify the statement.", "topic": "number_properties"}
{"id": "book 2_p624_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 624, "page_end": 624, "topic_guess": "number_theory", "text": "GMAT number properties note: You can’t solve for t and v, so this statement might look sufficient—but remember what the question is asking! You need to find the value of 4t + 2v.", "topic": "number_properties"}
{"id": "book 2_p624_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 624, "page_end": 624, "topic_guess": "number_theory", "text": "GMAT number properties note: Multiply the equation by 2: 4t + 2v = 28. There isn’t a way to simplify this one, but notice all those prime numbers.", "topic": "number_properties"}
{"id": "book 2_p624_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 624, "page_end": 624, "topic_guess": "number_theory", "text": "GMAT number properties note: Primes tend to minimize the number of allowable scenarios, especially when the question also specifies that the variables have to be integers. See what combinations of integers would actually work here.", "topic": "number_properties"}
{"id": "book 2_p624_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 624, "page_end": 624, "topic_guess": "number_theory", "text": "GMAT number properties note: The 5t term could be 5, 10, 15, 20, and so on. This term can contribute only numbers that end in 5 or 0.", "topic": "number_properties"}
{"id": "book 2_p624_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 624, "page_end": 624, "topic_guess": "number_theory", "text": "GMAT number properties note: If that’s the case, the 7v term must have a units digit of either 3 or 8. List out the possibilities for 7v, and try only the ones that end in 3 or 8.", "topic": "number_properties"}
{"id": "book 2_p625_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 625, "page_end": 625, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. 7v: 7, 14, 21, 28, 35, 42, 49 Only one ends in 3 or 8!", "topic": "word_problems"}
{"id": "book 2_p625_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 625, "page_end": 625, "topic_guess": "word_problems", "text": "GMAT word problem note: If 7v = 28, then v = 4 and the 5t term must equal 25, so t = 5. This one scenario works and it is the only possible scenario for this equation, given that v and t must be integers.", "topic": "word_problems"}
{"id": "book 2_p625_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 625, "page_end": 625, "topic_guess": "word_problems", "text": "GMAT word problem note: If Jonas and Amanda run at the same rate, they will meet each other exactly in the middle. Jonas will only run farther than Amanda if Jonas’ rate is greater than Amanda’s.", "topic": "word_problems"}
{"id": "book 2_p625_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 625, "page_end": 625, "topic_guess": "word_problems", "text": "GMAT word problem note: In math terms, Distance = rate × time, and since Jonas and Amanda run for the same time, their relative distances depend solely on their relative rates. Rephrase the question as, “What is the probability that Jonas ran faster than Amanda?” There are four possible rates for Jonas (3, 4, 5, and 6) and four for Amanda (4, 5, 6, and 7).", "topic": "word_problems"}
{"id": "book 2_p625_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 625, "page_end": 625, "topic_guess": "word_problems", "text": "GMAT word problem note: In total, there are (4)(4) = 16 possible rate scenarios. Of Jonas’ four possible rates, only two (5 and 6) are greater than some of Amanda’s possible rates (4 and 5).", "topic": "word_problems"}
{"id": "book 2_p625_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 625, "page_end": 625, "topic_guess": "word_problems", "text": "GMAT word problem note: List the three rate scenarios that result in a faster speed (greater distance) for Jonas. Jonas ran 5 miles per hour (mph), and Amanda ran 4 mph.", "topic": "word_problems"}
{"id": "book 2_p625_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 625, "page_end": 625, "topic_guess": "word_problems", "text": "GMAT word problem note: Jonas ran 6 mph, and Amanda ran 4 mph.", "topic": "word_problems"}
{"id": "book 2_p626_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 626, "page_end": 626, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The prime factorization of 96 is (2)(2)(2)(2)(2)(3) = (25)(31).", "topic": "number_properties"}
{"id": "book 2_p626_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 626, "page_end": 626, "topic_guess": "number_theory", "text": "GMAT number properties note: In order for p2 to be divisible by 96, p2 would have to have the prime factors 2531 in its prime box. The rephrased question is therefore, “Does p2 have at least five 2’s and one 3 in its prime box?” (1) INSUFFICIENT:.", "topic": "number_properties"}
{"id": "book 2_p626_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 626, "page_end": 626, "topic_guess": "number_theory", "text": "GMAT number properties note: If p is a multiple of 8 = (2)(2)(2), p has 23 in its prime box. Therefore, p2 has (23)2 = 26 in its prime box, and thus has the required five 2’s.", "topic": "number_properties"}
{"id": "book 2_p626_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 626, "page_end": 626, "topic_guess": "number_theory", "text": "GMAT number properties note: However, it is uncertain whether p2 has at least one 3 in its prime box. If p2 is a multiple of 12 = (2)(2)(3), p2 has two 2’s and one 3 in its prime box.", "topic": "number_properties"}
{"id": "book 2_p626_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 626, "page_end": 626, "topic_guess": "number_theory", "text": "GMAT number properties note: It is uncertain whether p2 has at least five 2’s total as there may or may not be three more 2’s in the prime box. Therefore, it is certain that p2 has at least five 2’s and one 3 in its prime box.", "topic": "number_properties"}
{"id": "book 2_p627_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 627, "page_end": 627, "topic_guess": "geometry", "text": "GMAT geometry note: In order for the trapezoid vertices to lie on the circle, the trapezoid must be symmetrical about the dotted line, which passes through the center of the circle. Draw this vertical and the radii to points B and C to create two right triangles, allowing you to use the Pythagorean theorem.", "topic": "geometry"}
{"id": "book 2_p627_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 627, "page_end": 627, "topic_guess": "geometry", "text": "GMAT geometry note: In fact, you might play an educated hunch that the triangles are 3–4–5 common right triangles. If hypotenuse r is 5, then each triangle has a 3 and 4 side.", "topic": "geometry"}
{"id": "book 2_p627_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 627, "page_end": 627, "topic_guess": "geometry", "text": "GMAT geometry note: The unknown vertical sides are thus 4 and 3, which sum to 7 as they must. Algebraically, set up the following equations from the picture.", "topic": "geometry"}
{"id": "book 2_p627_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 627, "page_end": 627, "topic_guess": "geometry", "text": "GMAT geometry note: Setting the two equations for r2 equal.", "topic": "geometry"}
{"id": "book 2_p628_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 628, "page_end": 628, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Since (x + y)(x − y) = 7, (x − y) = 1.", "topic": "algebra"}
{"id": "book 2_p628_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 628, "page_end": 628, "topic_guess": "algebra", "text": "GMAT algebra note: The radius of the circle is 5, because r2 = 32 + 42 = 25. First, simplify the given equation.", "topic": "algebra"}
{"id": "book 2_p628_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 628, "page_end": 628, "topic_guess": "algebra", "text": "GMAT algebra note: First, translate the question stem:.", "topic": "algebra"}
{"id": "book 2_p629_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 629, "page_end": 629, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Note that the question is stated in yards, but the statements use feet, so convert the question stem to feet.", "topic": "algebra"}
{"id": "book 2_p629_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 629, "page_end": 629, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The value of is approximately , or a little bit less than 1. In other words, or approximately .", "topic": "algebra"}
{"id": "book 2_p629_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 629, "page_end": 629, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Therefore, f could be smaller than feet (just a bit smaller than 1) but it could also be larger. Start at x = 1, then x = 2, and so on, looking for a pattern.", "topic": "algebra"}
{"id": "book 2_p630_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 630, "page_end": 630, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. X Sum of the integers from 1 to x, inclusive Is the sum divisible by 3?x Sum of the integers from 1 to x, inclusive Is the sum divisible by 3?", "topic": "number_properties"}
{"id": "book 2_p630_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 630, "page_end": 630, "topic_guess": "number_theory", "text": "GMAT number properties note: 1 1 No 2 3 Yes 3 6 Yes 4 10 No 5 15 Yes 6 21 Yes 7 28 No It appears that the pattern is No, Yes, Yes, No, Yes, Yes, etc., repeating every three terms. Therefore, 2 out of every 3 values of x yield a sum that is divisible by 3.", "topic": "number_properties"}
{"id": "book 2_p630_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 630, "page_end": 630, "topic_guess": "number_theory", "text": "GMAT number properties note: So 200 of the 300 values of x will yield a sum that is divisible by 3. If the two variables must be integers and the absolute value of their product is 8, then x and y have to represent either the factor pair (1, 8) or the factor pair (2, 4), in either order and with either sign (positive or negative).", "topic": "number_properties"}
{"id": "book 2_p630_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 630, "page_end": 630, "topic_guess": "number_theory", "text": "GMAT number properties note: This narrows the possible number of cases considerably. List out the possible cases allowed by this statement.", "topic": "number_properties"}
{"id": "book 2_p630_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 630, "page_end": 630, "topic_guess": "number_theory", "text": "GMAT number properties note: If x and y are both divisible by 2, then you’re dealing with the factor pair (2, 4). A er you try your first case, ask yourself what different case would be most likely to return a different answer to the question.", "topic": "number_properties"}
{"id": "book 2_p631_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 631, "page_end": 631, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. In this case, just reversing the numbers (4, 2) won’t make a difference to the final question, so don’t list that case second.", "topic": "number_properties"}
{"id": "book 2_p631_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 631, "page_end": 631, "topic_guess": "general", "text": "GMAT quant note: Try one of the cases that allows a negative value. Now that you have two different values, you know the statement is not sufficient.", "topic": "number_properties"}
{"id": "book 2_p631_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 631, "page_end": 631, "topic_guess": "general", "text": "GMAT quant note: This statement indicates that the two values have the same sign: both positive or both negative. Again, list cases until you have two contradictory results (or until you’ve tried all cases and realized that there is only one possible answer).", "topic": "number_properties"}
{"id": "book 2_p631_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 631, "page_end": 631, "topic_guess": "general", "text": "GMAT quant note: (1) AND (2) SUFFICIENT: x y Both divisible by 2?", "topic": "number_properties"}
{"id": "book 2_p632_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 632, "page_end": 632, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Only four possible cases are allowed: the variables must match the (2, 4) factor pair and the signs have to be the same.", "topic": "number_properties"}
{"id": "book 2_p632_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 632, "page_end": 632, "topic_guess": "number_theory", "text": "GMAT number properties note: The chart above shows two of the four possible cases, but you could also reverse the order: (4, 2) and (−4, −2). Since you’re just adding at the end, though, the order of the two variables doesn’t matter.", "topic": "number_properties"}
{"id": "book 2_p632_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 632, "page_end": 632, "topic_guess": "number_theory", "text": "GMAT number properties note: In all four cases, |x + y| = 6. Set X contains exactly four distinct integers.", "topic": "number_properties"}
{"id": "book 2_p632_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 632, "page_end": 632, "topic_guess": "number_theory", "text": "GMAT number properties note: Distinct means the four integers must be different. Also, if a number is contained in the set, all of its unique prime factors must also be in the set.", "topic": "number_properties"}
{"id": "book 2_p632_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 632, "page_end": 632, "topic_guess": "number_theory", "text": "GMAT number properties note: If a prime number, such as 3, is in the set, no additional numbers are required in the set: the only prime factor of 3 is 3 itself, which is already included. However, if the composite number 8 is in the set, its one unique prime factor, 2, must also be included.", "topic": "number_properties"}
{"id": "book 2_p632_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 632, "page_end": 632, "topic_guess": "number_theory", "text": "GMAT number properties note: To better understand the question, jot down some four-integer sets that fit the description. Specifically, look for sets that have different numbers of primes.", "topic": "number_properties"}
{"id": "book 2_p632_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 632, "page_end": 632, "topic_guess": "number_theory", "text": "GMAT number properties note: {2, 3, 5, 7} is a valid set that contains four primes. {2, 4, 8, 16} is a valid set that contains only one prime, since all of the terms are powers of 2.", "topic": "number_properties"}
{"id": "book 2_p632_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 632, "page_end": 632, "topic_guess": "number_theory", "text": "GMAT number properties note: The question deals with primes and divisibility.", "topic": "number_properties"}
{"id": "book 2_p633_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 633, "page_end": 633, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The prime factorization of 36 is 36 = 2 × 2 × 3 × 3.", "topic": "number_properties"}
{"id": "book 2_p633_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 633, "page_end": 633, "topic_guess": "number_theory", "text": "GMAT number properties note: For the product of the four integers in the set to be divisible by 36, the four integers must include 2, 2, 3, and 3 as prime factors, in some combination. Therefore, based on the definition of the set, 2 and 3 must both be included in the set.", "topic": "number_properties"}
{"id": "book 2_p633_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 633, "page_end": 633, "topic_guess": "number_theory", "text": "GMAT number properties note: The remaining two numbers must include at least one 2 and at least one 3 as prime factors between them. However, there are several different valid sets that fit this criterion.", "topic": "number_properties"}
{"id": "book 2_p633_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 633, "page_end": 633, "topic_guess": "number_theory", "text": "GMAT number properties note: Case 1: {2, 3, 6, 9} has a product divisible by 36, since it includes at least two 2’s and at least two 3’s in its prime factors. Also, it is a valid set since it includes all of the unique prime factors of 6 (2 and 3) and all of the unique prime factors of 9 (only 3).", "topic": "number_properties"}
{"id": "book 2_p633_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 633, "page_end": 633, "topic_guess": "number_theory", "text": "GMAT number properties note: The primes 2 and 3 must be included in the set. Include a third prime as well, such as 5.", "topic": "number_properties"}
{"id": "book 2_p633_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 633, "page_end": 633, "topic_guess": "number_theory", "text": "GMAT number properties note: The final number in the set must include 2 and 3 as prime factors in order for the product of the whole set to be divisible by 36. Because the set might have either two primes or three primes, the statement is insufficient.", "topic": "number_properties"}
{"id": "book 2_p634_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 634, "page_end": 634, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Each also has an additional prime factor that is not shared by the other.", "topic": "number_properties"}
{"id": "book 2_p634_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 634, "page_end": 634, "topic_guess": "number_theory", "text": "GMAT number properties note: Draw overlapping circles in which to place the shared and non-shared prime factors of p and r. To find the least common multiple (LCM), multiply from le to right and include all the common factors in the product.", "topic": "number_properties"}
{"id": "book 2_p635_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 635, "page_end": 635, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. If y = |x − 1| and y = 3x + 3, then x must be between which of the following values?", "topic": "algebra"}
{"id": "book 2_p636_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 636, "page_end": 636, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. If a is a positive integer, is a2 a multiple of 8?", "topic": "word_problems"}
{"id": "book 2_p636_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 636, "page_end": 636, "topic_guess": "word_problems", "text": "GMAT word problem note: A3 is a multiple of 16.(1) (a + 4)2 is a multiple of 8. Between the two black lines, they will paint across the road a red line at each third of a mile, a white line at each fi h of a mile, and a blue line at each eighth of a mile.", "topic": "word_problems"}
{"id": "book 2_p636_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 636, "page_end": 636, "topic_guess": "word_problems", "text": "GMAT word problem note: What is the smallest distance (in miles) between any of the painted lines on this stretch of highway?", "topic": "word_problems"}
{"id": "book 2_p637_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 637, "page_end": 637, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Set S consists of n consecutive integers, where n > 1.", "topic": "number_properties"}
{"id": "book 2_p637_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 637, "page_end": 637, "topic_guess": "number_theory", "text": "GMAT number properties note: The sum of the integers in set S is divisible by 7.(1) The sum of the integers in set S is 14.(2) 116. A trapezoid is symmetrical about a vertical center line.", "topic": "number_properties"}
{"id": "book 2_p637_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 637, "page_end": 637, "topic_guess": "number_theory", "text": "GMAT number properties note: If a circle is drawn such that it is tangent to exactly three points on the trapezoid and is enclosed entirely within the trapezoid, what is the diameter of the circle? The parallel sides of the trapezoid are 10 inches apart.(1) Of the parallel sides of the trapezoid, the shorter side is 15 inches long.", "topic": "number_properties"}
{"id": "book 2_p638_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 638, "page_end": 638, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Three boys are ages 4, 6, and 7, respectively.", "topic": "probability"}
{"id": "book 2_p638_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 638, "page_end": 638, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Three girls are ages 5, 8, and 9, respectively. If two of the boys and two of the girls are randomly selected and the sum of the selected children’s ages is z, what is the difference between the probability that z is even and the probability that z is odd?", "topic": "probability"}
{"id": "book 2_p638_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 638, "page_end": 638, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: If two distinct integers x and y are randomly selected from the integers between 1 and 10, inclusive, what is the probability that is a positive odd integer?", "topic": "probability"}
{"id": "book 2_p639_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 639, "page_end": 639, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. Positive integers a and b are less than or equal to 9.", "topic": "number_properties"}
{"id": "book 2_p639_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 639, "page_end": 639, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: If a and b are assembled into the six-digit number ababab, which of the following must be a factor of ababab? The two-digit positive integer s is the sum of the two-digit positive integers m and n.", "topic": "number_properties"}
{"id": "book 2_p639_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 639, "page_end": 639, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Is the units digit of s less than the units digit of m? The units digit of s is less than the units digit of n.(1).", "topic": "number_properties"}
{"id": "book 2_p640_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 640, "page_end": 640, "topic_guess": "general", "text": "GMAT quant note: The tens digit of s is not equal to the sum of the tens digits of m and n.", "topic": "general"}
{"id": "book 2_p641_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 641, "page_end": 641, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "book 2_p642_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 642, "page_end": 642, "topic_guess": "algebra", "text": "GMAT algebra note: Workout Set 12 Solutions 11. Parallel lines have the same slope.", "topic": "geometry"}
{"id": "book 2_p642_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 642, "page_end": 642, "topic_guess": "algebra", "text": "GMAT algebra note: Work backwards from the answers by rearranging into slope-intercept form: form, where the slope is m. Since only one answer can be correct, only one answer will have a slope of.", "topic": "geometry"}
{"id": "book 2_p642_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 642, "page_end": 642, "topic_guess": "algebra", "text": "GMAT algebra note: Stop when you find the one with the matching slope.", "topic": "geometry"}
{"id": "book 2_p643_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 643, "page_end": 643, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Set the two equations for y equal and algebraically solve |x − 1| = 3x + 3 for x.", "topic": "number_properties"}
{"id": "book 2_p643_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 643, "page_end": 643, "topic_guess": "number_theory", "text": "GMAT number properties note: This requires two solutions: one for the case that x − 1 is positive, the other for the case that x − 1 is negative. X – 1 is positive: x – 1 is negative:.", "topic": "number_properties"}
{"id": "book 2_p643_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 643, "page_end": 643, "topic_guess": "number_theory", "text": "GMAT number properties note: In the x − 1 is positive case, the answer is x = −2. In this case, x − 1 is not actually positive, so this is a false case.", "topic": "number_properties"}
{"id": "book 2_p643_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 643, "page_end": 643, "topic_guess": "number_theory", "text": "GMAT number properties note: As a result, there is only one solution:. In order for a2 to be a multiple of 8, the prime factorization of a2 must include all of the prime factors of 8.", "topic": "number_properties"}
{"id": "book 2_p643_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 643, "page_end": 643, "topic_guess": "number_theory", "text": "GMAT number properties note: That is, to be a multiple of 8, a2 would have to include 2, 2, and 2 among its prime factors. When an integer is squared, all of its prime factors are duplicated.", "topic": "number_properties"}
{"id": "book 2_p643_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 643, "page_end": 643, "topic_guess": "number_theory", "text": "GMAT number properties note: Therefore, if a only includes a single 2 among its prime factors, a2 might only include two 2’s. In order to guarantee that a2 includes at least three 2’s, a must include at least two 2’s in its prime factors.", "topic": "number_properties"}
{"id": "book 2_p643_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 643, "page_end": 643, "topic_guess": "number_theory", "text": "GMAT number properties note: The question translates to: “Does the prime factorization of a include at least two 2’s?” Or, “Is a divisible by 4?”.", "topic": "number_properties"}
{"id": "book 2_p644_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 644, "page_end": 644, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. The positive integer a, when cubed, is a multiple of 16.", "topic": "algebra"}
{"id": "book 2_p644_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 644, "page_end": 644, "topic_guess": "algebra", "text": "GMAT algebra note: Odd integers, when cubed, will always yield an odd result, so a cannot be odd. What positive even values could a have?", "topic": "algebra"}
{"id": "book 2_p644_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 644, "page_end": 644, "topic_guess": "algebra", "text": "GMAT algebra note: If a = 2, then a3 = 8, which is not a multiple of 16. However, if a = 4, then a3 = 64, which is a multiple of 16, as are 83 = (43)(23) and 123 = (43)(33).", "topic": "algebra"}
{"id": "book 2_p644_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 644, "page_end": 644, "topic_guess": "algebra", "text": "GMAT algebra note: Therefore, this statement indicates that a is a multiple of 4. When a multiple of 4 is squared, the result is always a multiple of 8, so the answer is definitely Yes.", "topic": "algebra"}
{"id": "book 2_p644_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 644, "page_end": 644, "topic_guess": "algebra", "text": "GMAT algebra note: Start by multiplying out the expression in the statement. (a + 4)2 is a multiple of 8.", "topic": "algebra"}
{"id": "book 2_p644_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 644, "page_end": 644, "topic_guess": "algebra", "text": "GMAT algebra note: A2 + 8a + 16 is a multiple of 8. A2 + 8(a + 2) is a multiple of 8.", "topic": "algebra"}
{"id": "book 2_p644_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 644, "page_end": 644, "topic_guess": "algebra", "text": "GMAT algebra note: The second part of the expression, 8(a + 2), is a multiple of 8 for any value of integer a. In order for the entire expression to be a multiple of 8, the first part of the expression must be a multiple of 8 as well.", "topic": "algebra"}
{"id": "book 2_p644_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 644, "page_end": 644, "topic_guess": "algebra", "text": "GMAT algebra note: So according to this statement, a2 is a multiple of 8. The answer is a definite Yes, so this statement is sufficient.", "topic": "algebra"}
{"id": "book 2_p644_c1_r9", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 644, "page_end": 644, "topic_guess": "algebra", "text": "GMAT algebra note: When comparing fractional pieces of a whole, find a common denominator. In this case, the one-mile stretch is divided into thirds, fi hs, and eighths.", "topic": "algebra"}
{"id": "book 2_p644_c1_r10", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 644, "page_end": 644, "topic_guess": "algebra", "text": "GMAT algebra note: The smallest common denominator of 3, 5, and 8 is 120.", "topic": "algebra"}
{"id": "book 2_p645_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 645, "page_end": 645, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. One-mile highway is divided into 120 equal increments, where will the red, white, and blue marks fall?", "topic": "statistics"}
{"id": "book 2_p645_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 645, "page_end": 645, "topic_guess": "statistics", "text": "GMAT statistics note: Red (thirds): 40, 80 (out of 120 increments) White (fi hs): 24, 48, 72, 96 (out of 120 increments) Blue (eighths): 15, 30, 45, 60, 75, 90, 105 (out of 120 increments) The smallest distance between two marks is 75 − 72 = 3 or 48 − 45 = 3. Both statements provide information about the sum of the set, so rephrase with this in mind.", "topic": "statistics"}
{"id": "book 2_p645_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 645, "page_end": 645, "topic_guess": "statistics", "text": "GMAT statistics note: For n = odd, the median is the middle term, an integer. For n = even, the median is the average of the two middle terms, a non-integer of the form “integer + 0.5.” You can determine n if you can determine both the median of set S and the sum of the integers in set S.", "topic": "statistics"}
{"id": "book 2_p645_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 645, "page_end": 645, "topic_guess": "statistics", "text": "GMAT statistics note: Next, glance at the statements. Sum of consecutive set = (Median)(Number of terms) Sum of consecutive set = (Median)(n).", "topic": "statistics"}
{"id": "book 2_p646_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 646, "page_end": 646, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Since n must be an integer, use divisibility rules to narrow down possible median values.", "topic": "statistics"}
{"id": "book 2_p646_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 646, "page_end": 646, "topic_guess": "statistics", "text": "GMAT statistics note: Check some possible median and n values (remember that n > 1). There are at least two possible values for n.", "topic": "statistics"}
{"id": "book 2_p646_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 646, "page_end": 646, "topic_guess": "statistics", "text": "GMAT statistics note: Set S can’t have an integer median if there are only two terms. Set S is {−1, 0, 1, 2, 3, 4, 5}, which has a sum of 14.", "topic": "statistics"}
{"id": "book 2_p646_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 646, "page_end": 646, "topic_guess": "statistics", "text": "GMAT statistics note: Set S is {2, 3, 4, 5}, which has a sum of 14. There are four basic ways this picture could look as there are four sides of the trapezoid that could serve as the non-tangent side.", "topic": "statistics"}
{"id": "book 2_p647_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 647, "page_end": 647, "topic_guess": "geometry", "text": "GMAT geometry note: If the circle is tangent to both of the parallel sides (Figure A or B), then the diameter must be 10. If the circle is tangent to only one of the parallel sides (Figure C or D), then the diameter is less than 10.", "topic": "geometry"}
{"id": "book 2_p647_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 647, "page_end": 647, "topic_guess": "geometry", "text": "GMAT geometry note: Just knowing the length of the shorter parallel side is not enough to determine which of the basic figures above describes the correct situation. If Figure A or B represents the correct situation, the diameter of the circle is determined solely by the distance between the parallel sides.", "topic": "geometry"}
{"id": "book 2_p647_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 647, "page_end": 647, "topic_guess": "geometry", "text": "GMAT geometry note: The diameter is independent of the length of the shorter parallel side. If Figure C or D describes the correct situation, then the diameter would depend on not only the 15-inch side but also the longer parallel side, which has an unknown length.", "topic": "geometry"}
{"id": "book 2_p647_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 647, "page_end": 647, "topic_guess": "geometry", "text": "GMAT geometry note: In Figures C and D, the diameter must be less than 10 but also greater than the length of the smaller parallel side. If the length of the smaller parallel side is 15, then the diameter would have to be.", "topic": "geometry"}
{"id": "book 2_p648_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 648, "page_end": 648, "topic_guess": "geometry", "text": "GMAT geometry note: Watch divisibility, factors, remainders, parity, and prime structure. Greater than 15 for Figures C and D, but this is impossible (since the diameter has to be less than 10).", "topic": "number_properties"}
{"id": "book 2_p648_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 648, "page_end": 648, "topic_guess": "geometry", "text": "GMAT geometry note: Therefore, Figure A or B represents the correct situation, and the diameter of the circle must equal 10. Alternatively, notice that Figure D could not represent the situation since the circle would have to have a diameter larger than 15 inches in order to be tangent to the short parallel side and the non-parallel sides of the trapezoid.", "topic": "number_properties"}
{"id": "book 2_p648_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 648, "page_end": 648, "topic_guess": "geometry", "text": "GMAT geometry note: The parallel sides of the trapezoid are only 10 inches apart, so the circle would be too large to be drawn entirely within the trapezoid as required. Similar logic explains why Figure C is also impossible when considering (1) and (2) together.", "topic": "number_properties"}
{"id": "book 2_p648_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 648, "page_end": 648, "topic_guess": "geometry", "text": "GMAT geometry note: Wherever possible, avoid computing. Use Odd & Even principles to reduce your computation.", "topic": "number_properties"}
{"id": "book 2_p648_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 648, "page_end": 648, "topic_guess": "geometry", "text": "GMAT geometry note: Boys Girls Sum z 4, 6 = Even 5, 8 = Odd E + O = O 4, 6 = Even 5, 9 = Even E + E = E 4, 6 = Even 8, 9 = Odd E + O = O 4, 7 = Odd 5, 8 = Odd O + O = E 4, 7 = Odd 5, 9 = Even O + E = O.", "topic": "number_properties"}
{"id": "book 2_p649_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 649, "page_end": 649, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Watch divisibility, factors, remainders, parity, and prime structure. Boys Girls Sum z 4, 7 = Odd 8, 9 = Odd O + O = E 6, 7 = Odd 5, 8 = Odd O + O = E 6, 7 = Odd 5, 9 = Even O + E = O 6, 7 = Odd 8, 9 = Odd O + O = E Of the nine scenarios listed, five yield an even z and four yield an odd z.", "topic": "number_properties"}
{"id": "book 2_p649_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 649, "page_end": 649, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: This problem involves both probability and special quadratics. In order to find the probability that is a positive odd integer, you must find the number of values of x and y for which this is the case.", "topic": "number_properties"}
{"id": "book 2_p649_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 649, "page_end": 649, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: The problem is really asking you to count the values of x and y for which is a positive odd integer. Because x2 — y2 is a special quadratic, the expression can be simplified to.", "topic": "number_properties"}
{"id": "book 2_p650_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 650, "page_end": 650, "topic_guess": "algebra", "text": "GMAT algebra note: Watch divisibility, factors, remainders, parity, and prime structure. This expression is an integer when (x — y)(x + y) is divisible by 4.", "topic": "number_properties"}
{"id": "book 2_p650_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 650, "page_end": 650, "topic_guess": "algebra", "text": "GMAT algebra note: Note that x – y and x + y are always either both even or both odd. Whenever x – y and x + y are both even, their product will be divisible by 4.", "topic": "number_properties"}
{"id": "book 2_p650_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 650, "page_end": 650, "topic_guess": "algebra", "text": "GMAT algebra note: In other words, when x and y are either both even or both odd, is an integer. But when x and y are different in parity (one even and the other odd), is not an integer.", "topic": "number_properties"}
{"id": "book 2_p650_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 650, "page_end": 650, "topic_guess": "algebra", "text": "GMAT algebra note: The expression also needs to be positive. This will occur whenever x is greater than y.", "topic": "number_properties"}
{"id": "book 2_p650_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 650, "page_end": 650, "topic_guess": "algebra", "text": "GMAT algebra note: Otherwise, x – y would not be positive and the entire expression would be negative or zero. Finally, when is an ODD integer?", "topic": "number_properties"}
{"id": "book 2_p650_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 650, "page_end": 650, "topic_guess": "algebra", "text": "GMAT algebra note: Test a few cases to determine when this occurs. Based on the reasoning above, you only need to test cases in which x is greater than y and in which x and y are either both even or both odd.", "topic": "number_properties"}
{"id": "book 2_p651_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 651, "page_end": 651, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. Even Even Even It appears that whenever x and y are both odd, will be even.", "topic": "number_properties"}
{"id": "book 2_p651_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 651, "page_end": 651, "topic_guess": "general", "text": "GMAT quant note: Therefore, x and y cannot both be odd. Test some cases in which x and y are both even: x y Odd or even?", "topic": "number_properties"}
{"id": "book 2_p652_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 652, "page_end": 652, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. Odd Even Odd Even If x and y are both even, then whether the case is valid depends on whether x and y differ by a multiple of 4.", "topic": "number_properties"}
{"id": "book 2_p652_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 652, "page_end": 652, "topic_guess": "general", "text": "GMAT quant note: Therefore, whenever x – y is a multiple of is an even integer. In summary: x and y must both be even.", "topic": "number_properties"}
{"id": "book 2_p652_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 652, "page_end": 652, "topic_guess": "general", "text": "GMAT quant note: X – y must NOT be a multiple of 4.", "topic": "number_properties"}
{"id": "book 2_p653_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 653, "page_end": 653, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. That leaves only a few cases to count: x y There are six valid cases.", "topic": "number_properties"}
{"id": "book 2_p653_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 653, "page_end": 653, "topic_guess": "number_theory", "text": "GMAT number properties note: In total, there are (10)(9) = 90 ways to choose two distinct values for x and y from the integers between 1 and 10, inclusive. The sum of the digits of ababab is 3(a + b).", "topic": "number_properties"}
{"id": "book 2_p653_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 653, "page_end": 653, "topic_guess": "number_theory", "text": "GMAT number properties note: This must be a multiple of 3. Below, you’ll find explanations for the other answers.", "topic": "number_properties"}
{"id": "book 2_p653_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 653, "page_end": 653, "topic_guess": "number_theory", "text": "GMAT number properties note: An integer is divisible by 4 if its last two digits represent a two-digit number that is itself divisible by 4. It is uncertain whether the two-digit integer ab is divisible by 4.", "topic": "number_properties"}
{"id": "book 2_p653_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 653, "page_end": 653, "topic_guess": "number_theory", "text": "GMAT number properties note: An integer is divisible by 5 if the last digit is 0 or 5. It is uncertain whether the positive integer b is 5.", "topic": "number_properties"}
{"id": "book 2_p654_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 654, "page_end": 654, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. An integer is divisible by 6 if it is even and divisible by 3.", "topic": "number_properties"}
{"id": "book 2_p654_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 654, "page_end": 654, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Note that you can also use logic to eliminate this answer. If ababab were divisible by 6, it would also have to be divisible by 3, but that would lead to two correct answers!", "topic": "number_properties"}
{"id": "book 2_p654_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 654, "page_end": 654, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Try out a few randomly chosen numbers to help you understand the question. If m = 10 and n = 10, then s = 20 and the units digits of m and s are equal.", "topic": "number_properties"}
{"id": "book 2_p654_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 654, "page_end": 654, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: If m = 11 and n = 12, then s = 23 and the units digit of s is greater than the units digit of m. If m = 19 and n = 11, then s = 30 and the units digit of s is less than the units digit of m.", "topic": "number_properties"}
{"id": "book 2_p654_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 654, "page_end": 654, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: What’s going on with all of these cases? Basically, if the units digits of m and n are small enough, then the units digit of s will be equal to or greater than the units digit of m (and of n).", "topic": "number_properties"}
{"id": "book 2_p654_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 654, "page_end": 654, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: On the other hand, if the units digits of m and n are large enough to cause you to carry over a 1 to the tens digit, then the units digit of s will end up being smaller than the units digit of m (and of n). If the units digit of s is definitely less than the units digits of one of the smaller numbers, then the carryover s.", "topic": "number_properties"}
{"id": "book 2_p655_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 655, "page_end": 655, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Which case the units digits of both m and n must be larger than the units digit of s.", "topic": "algebra"}
{"id": "book 2_p655_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 655, "page_end": 655, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: There are only two ways in which the tens digit will not equal the tens digits of the two smaller numbers:. The tens digits of the two smaller numbers result in a number that needs to carry over into the hundreds digit.", "topic": "algebra"}
{"id": "book 2_p655_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 655, "page_end": 655, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: This is impossible for this problem because s is also a two-digit number. The units digits of the two smaller numbers result in a number that carries over into the tens digit.", "topic": "algebra"}
{"id": "book 2_p655_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 655, "page_end": 655, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: This must be what is happening in this case. If so, then the units digit of the larger number, s, must be smaller than the units digits of the two smaller numbers, m and n.", "topic": "algebra"}
{"id": "book 2_p656_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 656, "page_end": 656, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If x and y are positive integers such that x2 – y2 = 48, how many different values of y are possible?", "topic": "number_properties"}
{"id": "book 2_p656_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 656, "page_end": 656, "topic_guess": "number_theory", "text": "GMAT number properties note: Is written in the form p × 10q, where p and q are integers, what is the maximum possible value of q? If x and y are integers, and x ≠ 0, what is the value of xy?", "topic": "number_properties"}
{"id": "book 2_p657_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 657, "page_end": 657, "topic_guess": "geometry", "text": "GMAT geometry note: The three sides of a triangle have lengths p, q, and r, each an integer. Is this triangle a right triangle?", "topic": "geometry"}
{"id": "book 2_p657_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 657, "page_end": 657, "topic_guess": "geometry", "text": "GMAT geometry note: The perimeter of the triangle is an odd integer.(1) If the triangle’s area is doubled, the result is not an integer.(2) 125. The average (arithmetic mean) cost of three computer models is $900.", "topic": "geometry"}
{"id": "book 2_p657_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 657, "page_end": 657, "topic_guess": "geometry", "text": "GMAT geometry note: If no two computers cost the same amount, does the most expensive model cost more than $1,000?", "topic": "geometry"}
{"id": "book 2_p658_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 658, "page_end": 658, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. The most expensive model costs 25% more than the model with the median cost.", "topic": "statistics"}
{"id": "book 2_p658_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 658, "page_end": 658, "topic_guess": "statistics", "text": "GMAT statistics note: The most expensive model costs $210 more than the model with the median cost.", "topic": "statistics"}
{"id": "book 2_p659_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 659, "page_end": 659, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. K-numbers are positive integers with only 2’s as their digits.", "topic": "number_properties"}
{"id": "book 2_p659_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 659, "page_end": 659, "topic_guess": "number_theory", "text": "GMAT number properties note: What is the K-weight of 600? If the reciprocals of two consecutive positive integers are added together, what is the sum in terms of the greater integer x?", "topic": "number_properties"}
{"id": "book 2_p660_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 660, "page_end": 660, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "book 2_p661_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 661, "page_end": 661, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "book 2_p662_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 662, "page_end": 662, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Workout Set 13 Solutions 21.", "topic": "number_properties"}
{"id": "book 2_p662_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 662, "page_end": 662, "topic_guess": "number_theory", "text": "GMAT number properties note: The choices themselves, as well as how many in the question text, suggest that y could have more than one value. To begin, simplify the given information using algebra and logical reasoning.", "topic": "number_properties"}
{"id": "book 2_p662_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 662, "page_end": 662, "topic_guess": "number_theory", "text": "GMAT number properties note: Since x and y are integers, both x – y and x + y are integers. The product of two integers is 48.", "topic": "number_properties"}
{"id": "book 2_p662_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 662, "page_end": 662, "topic_guess": "number_theory", "text": "GMAT number properties note: One of those integers is equal to x + y and the other is equal to x – y. Since x2 – y2 is positive and x and y are both positive, x must be greater than y.", "topic": "number_properties"}
{"id": "book 2_p662_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 662, "page_end": 662, "topic_guess": "number_theory", "text": "GMAT number properties note: Therefore, x + y and x – y are a pair of positive factors of 48, with x + y being the greater of the two factors. List the pairs of factors of 48 and solve for the values of x and y.", "topic": "number_properties"}
{"id": "book 2_p662_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 662, "page_end": 662, "topic_guess": "number_theory", "text": "GMAT number properties note: Since the factors are y above and y below x, the value of x equals the average. Therefore, if for a particular case the average of (x – y) and (x + y) is not an integer, that case is not valid.", "topic": "number_properties"}
{"id": "book 2_p662_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 662, "page_end": 662, "topic_guess": "number_theory", "text": "GMAT number properties note: In order for the average of the two terms to be an integer, the sum has to be even, so if the sum is odd, eliminate that possibility immediately. X – y x + y x y 1 48 n/a (sum not even) 3 16 n/a (sum not even).", "topic": "number_properties"}
{"id": "book 2_p663_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 663, "page_end": 663, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Of the five factor pairs, three yield integer values for x and y.", "topic": "number_properties"}
{"id": "book 2_p663_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 663, "page_end": 663, "topic_guess": "number_theory", "text": "GMAT number properties note: Double-check that they fit the equation. To maximize the value of q, you would need to constrain p to NOT be a multiple of 10.", "topic": "number_properties"}
{"id": "book 2_p663_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 663, "page_end": 663, "topic_guess": "number_theory", "text": "GMAT number properties note: In other words, q should count every factor of 10 in 200!, and p would be the product of the remaining factors of 200!. To count factors of 10 in 200!, you could start by counting the multiples of 10 between 1 and 200, inclusive.", "topic": "number_properties"}
{"id": "book 2_p663_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 663, "page_end": 663, "topic_guess": "number_theory", "text": "GMAT number properties note: But this method would undercount the number of 10’s that are factors of 200! Because it would miss other pairs that can create 10, such as:.", "topic": "number_properties"}
{"id": "book 2_p663_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 663, "page_end": 663, "topic_guess": "number_theory", "text": "GMAT number properties note: Therefore, this problem is really about counting the number of 2 × 5 factor pairs that can be made from the factors of 200. Of 2 and 5, there will be fewer 5’s overall, so count the number of 5’s found among the factors of 200!.", "topic": "number_properties"}
{"id": "book 2_p663_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 663, "page_end": 663, "topic_guess": "number_theory", "text": "GMAT number properties note: Number of Multiples between 1 and 200, Inclusive Multiples of 5 ( = 51) Multiples of 25 ( = 52) 200! Has 2 and 5 as factors, which multiply to 10, creating another factor of 10.", "topic": "number_properties"}
{"id": "book 2_p663_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 663, "page_end": 663, "topic_guess": "number_theory", "text": "GMAT number properties note: Has 6 and 15 as factors, which multiply to 90, creating another factor of 10.", "topic": "number_properties"}
{"id": "book 2_p664_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 664, "page_end": 664, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Number of Multiples between 1 and 200, Inclusive Multiples of 125 ( = 53) 125 is the only one: 1 The higher multiples contribute more than one factor of 5 to the total.", "topic": "number_properties"}
{"id": "book 2_p664_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 664, "page_end": 664, "topic_guess": "number_theory", "text": "GMAT number properties note: There are eight multiples of 25 in the range (namely, 25, 50, 75, 100, 125, 150, 175, and 200). Each of these is also a multiple of 5, so each has already been counted once, and thus each of the eight multiples of 25 contributes one additional factor of 5.", "topic": "number_properties"}
{"id": "book 2_p664_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 664, "page_end": 664, "topic_guess": "number_theory", "text": "GMAT number properties note: Finally, 125 contributes a total of three 5’s to the count—but two have already been counted, leaving only one additional factor of 5 to include. Therefore, the prime factorization for 200!", "topic": "number_properties"}
{"id": "book 2_p664_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 664, "page_end": 664, "topic_guess": "number_theory", "text": "GMAT number properties note: Includes 40 + 8 + 1 = 49 factors of 5. Since x ≠ 0, x y does not equal 0 y or 0.", "topic": "number_properties"}
{"id": "book 2_p664_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 664, "page_end": 664, "topic_guess": "number_theory", "text": "GMAT number properties note: This statement indicates that x is equal to 2 or −2. The statement indicates nothing about y.", "topic": "number_properties"}
{"id": "book 2_p664_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 664, "page_end": 664, "topic_guess": "number_theory", "text": "GMAT number properties note: Simplify using exponent rules, noting the common factors of 6 and 8 on each side of the equation. Since y = 0 and x ≠ 0 (as stated in the question stem), this information is sufficient to conclude that x y = x0 = 1.", "topic": "number_properties"}
{"id": "book 2_p665_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 665, "page_end": 665, "topic_guess": "geometry", "text": "GMAT geometry note: Watch divisibility, factors, remainders, parity, and prime structure. The question stem specifies that the triangle’s sides all have integer lengths.", "topic": "number_properties"}
{"id": "book 2_p665_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 665, "page_end": 665, "topic_guess": "geometry", "text": "GMAT geometry note: Perimeter is calculated by summing the three side lengths. In order for that sum to be odd, either all three numbers have to be odd or one of the three numbers has to be odd.", "topic": "number_properties"}
{"id": "book 2_p665_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 665, "page_end": 665, "topic_guess": "geometry", "text": "GMAT geometry note: If the triangle is a right triangle, then the Pythagorean theorem (a2 + b2 = c2) holds. If all three side lengths are odd, this would mean odd2 + odd2 = odd2.", "topic": "number_properties"}
{"id": "book 2_p665_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 665, "page_end": 665, "topic_guess": "geometry", "text": "GMAT geometry note: If one side length is odd and the other two are even, then there are two possibilities:. Every right triangle possibility is impossible, so the triangle cannot be a right triangle.", "topic": "number_properties"}
{"id": "book 2_p665_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 665, "page_end": 665, "topic_guess": "geometry", "text": "GMAT geometry note: If, as this statement indicates, 2A is not an integer, then A itself is not an integer. As a result, at least one of b and h is also not an integer.", "topic": "number_properties"}
{"id": "book 2_p665_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 665, "page_end": 665, "topic_guess": "geometry", "text": "GMAT geometry note: For right triangles, b and h are the two shorter sides of the triangle. The question stem specifies that all three sides are integers, so if the triangle were a right triangle, then b and h would both have to be integers.", "topic": "number_properties"}
{"id": "book 2_p665_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 665, "page_end": 665, "topic_guess": "geometry", "text": "GMAT geometry note: This is impossible, so the triangle cannot be a right triangle. O + O + O = odd O + E + E = odd odd2 + even2 = even2 (impossible: odd + even = odd) even2 + even2 = odd2 (also.", "topic": "number_properties"}
{"id": "book 2_p666_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 666, "page_end": 666, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. The problem specifies that x has to be positive, so try some small positive integers to see how the expression works.", "topic": "algebra"}
{"id": "book 2_p666_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 666, "page_end": 666, "topic_guess": "algebra", "text": "GMAT algebra note: If x = 1, then the expression becomes . If x = 2, then the expression becomes 1 + 1 = 2.", "topic": "algebra"}
{"id": "book 2_p666_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 666, "page_end": 666, "topic_guess": "algebra", "text": "GMAT algebra note: What if x is a fraction, such as ? In that case, the expression becomes ...", "topic": "algebra"}
{"id": "book 2_p666_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 666, "page_end": 666, "topic_guess": "algebra", "text": "GMAT algebra note: It’s even bigger than the integer options above. The expressions and are reciprocals of each other.", "topic": "algebra"}
{"id": "book 2_p666_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 666, "page_end": 666, "topic_guess": "algebra", "text": "GMAT algebra note: As long as x is positive, there are two broad scenarios possible:. If one expression is less than 1, then the other has to be greater than 1.1.", "topic": "algebra"}
{"id": "book 2_p666_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 666, "page_end": 666, "topic_guess": "algebra", "text": "GMAT algebra note: If one expression equals 1, then the other also equals 1.2. If the average of the cost of the three models is $900, then the sum is $2,700.", "topic": "algebra"}
{"id": "book 2_p666_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 666, "page_end": 666, "topic_guess": "algebra", "text": "GMAT algebra note: Call the three models a, b, and c, in order from least expensive to most. The statement indicates that c = 1.25b, so a + b + 1.25b = 2,700.", "topic": "algebra"}
{"id": "book 2_p666_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 666, "page_end": 666, "topic_guess": "algebra", "text": "GMAT algebra note: This isn’t enough to solve for specific values, but it might be enough to tell whether c > $1,000.", "topic": "algebra"}
{"id": "book 2_p667_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 667, "page_end": 667, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. By definition, a < b, so plug LTb (less than b) into the equation for a.", "topic": "word_problems"}
{"id": "book 2_p667_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 667, "page_end": 667, "topic_guess": "word_problems", "text": "GMAT word problem note: From here, estimate: 13 goes into 2,600 a total of 200 times and it goes into 100 approximately 7 times. Therefore, LTb = (207)(4) = 828.", "topic": "word_problems"}
{"id": "book 2_p667_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 667, "page_end": 667, "topic_guess": "word_problems", "text": "GMAT word problem note: If b is approximately 830, then 125% of b is definitely larger than 1,000. Therefore, the most expensive model does cost more than $1,000.", "topic": "word_problems"}
{"id": "book 2_p667_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 667, "page_end": 667, "topic_guess": "word_problems", "text": "GMAT word problem note: Follow a similar path: c = b + 210. If b is a bit more than 830, then b + 210 is definitely over 1,000.", "topic": "word_problems"}
{"id": "book 2_p667_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 667, "page_end": 667, "topic_guess": "word_problems", "text": "GMAT word problem note: The product of the slopes of two perpendicular lines is −1.", "topic": "word_problems"}
{"id": "book 2_p668_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 668, "page_end": 668, "topic_guess": "algebra", "text": "GMAT algebra note: Thus, the slope of a line perpendicular to k is −2. Use the desired slope to create a slope-intercept equation and look for a match among the answers.", "topic": "geometry"}
{"id": "book 2_p668_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 668, "page_end": 668, "topic_guess": "algebra", "text": "GMAT algebra note: Alternatively, put each choice in slope-intercept form: y = mx + b, where the slope is m. Stop when you find the correct answer.", "topic": "geometry"}
{"id": "book 2_p668_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 668, "page_end": 668, "topic_guess": "algebra", "text": "GMAT algebra note: The question stem asks a Yes/No question:. Manipulate the absolute value expression to represent this inequality on a number line.", "topic": "geometry"}
{"id": "book 2_p669_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 669, "page_end": 669, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. This can be interpreted as, “The distance between x and 6 is less than 5.” On a number line, this is the region between 1 and 11.", "topic": "algebra"}
{"id": "book 2_p669_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 669, "page_end": 669, "topic_guess": "algebra", "text": "GMAT algebra note: All possible values for x are positive, so the answer to the question is a definite Yes. Manipulate the inequality to get 0 on one side, then factor the resulting quadratic.", "topic": "algebra"}
{"id": "book 2_p669_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 669, "page_end": 669, "topic_guess": "algebra", "text": "GMAT algebra note: The factored quadratic on the le side will equal 0 when x = 3 and 7. These are the boundary points.", "topic": "algebra"}
{"id": "book 2_p669_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 669, "page_end": 669, "topic_guess": "algebra", "text": "GMAT algebra note: On a number line, check the regions on either side of these boundary points to determine the valid region(s) for x. Note that when 3 < x < 7, (x − 7)(x − 3) = (neg)(pos) = neg.", "topic": "algebra"}
{"id": "book 2_p669_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 669, "page_end": 669, "topic_guess": "algebra", "text": "GMAT algebra note: Since both positive and negative values are possible for x, the answer is Maybe.", "topic": "algebra"}
{"id": "book 2_p670_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 670, "page_end": 670, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Since you are looking for the minimum number of K-numbers that sum to 600, a practical place to start is with the largest K-number less than 600, or 222.", "topic": "number_properties"}
{"id": "book 2_p670_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 670, "page_end": 670, "topic_guess": "number_theory", "text": "GMAT number properties note: There are between two and three multiples of 222 in 600, so subtract out the two whole multiples. Now, the next largest K-number is 22.", "topic": "number_properties"}
{"id": "book 2_p670_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 670, "page_end": 670, "topic_guess": "number_theory", "text": "GMAT number properties note: Again, subtract as many whole multiples as possible. The next largest K-number is 2.", "topic": "number_properties"}
{"id": "book 2_p670_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 670, "page_end": 670, "topic_guess": "number_theory", "text": "GMAT number properties note: Thus, 600 = 222(2) + 22(7) + 2(1), and the K-weight of 600 is 2 + 7 + 1 = 10. If the greater of the two integers is x, then the two integers can be expressed as x − 1 and x.", "topic": "number_properties"}
{"id": "book 2_p670_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 670, "page_end": 670, "topic_guess": "number_theory", "text": "GMAT number properties note: The sum of the reciprocals would therefore be:. Alternatively, choose smart numbers.", "topic": "number_properties"}
{"id": "book 2_p670_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 670, "page_end": 670, "topic_guess": "number_theory", "text": "GMAT number properties note: The smaller number would therefore be 3 − 1 = 2. The sum of the reciprocals would be.", "topic": "number_properties"}
{"id": "book 2_p671_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 671, "page_end": 671, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure.", "topic": "number_properties"}
{"id": "book 2_p672_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 672, "page_end": 672, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Watch divisibility, factors, remainders, parity, and prime structure. The sequence A is defined as follows:.", "topic": "number_properties"}
{"id": "book 2_p672_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 672, "page_end": 672, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: A1 = 1, and An = An − 1 + (−1)n + 1(n2) for all integer values n > 1. What is the value of A15 − A13?", "topic": "number_properties"}
{"id": "book 2_p672_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 672, "page_end": 672, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: If d represents the hundredths digit and e represents the thousandths digit in the decimal 0.4de, what is the value of this decimal rounded to the nearest tenth? D − e is a positive perfect square.(1) 133.", "topic": "number_properties"}
{"id": "book 2_p672_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 672, "page_end": 672, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: If x is a positive integer greater than 1, and y is the smallest positive integer that is evenly divisible by every integer between 1 and x, inclusive, what is the value of x?", "topic": "number_properties"}
{"id": "book 2_p673_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 673, "page_end": 673, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. A chain is comprised of 10 identical links, each of which independently has a 1% chance of breaking under a certain load.", "topic": "percent"}
{"id": "book 2_p673_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 673, "page_end": 673, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: If the failure of any individual link means the failure of the entire chain, what is the probability that the chain will fail under the load? If Ash plans to travel by bus from City A, to City B, to City C, and back to City A, what is the shortest possible duration, in minutes, of his entire trip?", "topic": "percent"}
{"id": "book 2_p674_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 674, "page_end": 674, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure.", "topic": "number_properties"}
{"id": "book 2_p675_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 675, "page_end": 675, "topic_guess": "general", "text": "GMAT quant note: If DP = PQ = QB = 1, what is the length of AB?", "topic": "geometry"}
{"id": "book 2_p676_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 676, "page_end": 676, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: In the rectangular coordinate system above, point A is not shown. If the area of triangle OAB is at least 16, which of the following could NOT be the coordinates of point A?", "topic": "geometry"}
{"id": "book 2_p677_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 677, "page_end": 677, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "book 2_p678_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 678, "page_end": 678, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Workout Set 14 Solutions 31.", "topic": "algebra"}
{"id": "book 2_p678_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 678, "page_end": 678, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Generate the first several values of the sequence, using the given relationship. Notice that a component such as (−1)n + 1 switches the sign of the additive term back and forth.", "topic": "algebra"}
{"id": "book 2_p678_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 678, "page_end": 678, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Use a table to keep your work organized. N An = An − 1 + (−1)n + 1(n2) 2 1 + (−1)2 + 1(22) = 1 − 4 = −3 3 −3 + (−1)3 + 1(32) = −3 + 9 = 6 4 6 + (−1)4 + 1(42) = 6 − 16 = −10 5 −10 + (−1)5 + 1(52) = −10 + 25 = 15 6 15 + (−1)6 + 1(62) = −15 − 36 = −21 What is the pattern?", "topic": "algebra"}
{"id": "book 2_p678_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 678, "page_end": 678, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The sign alternates between positive and negative. Ignoring the sign of the terms (taking absolute values) might help to determine the full pattern.", "topic": "algebra"}
{"id": "book 2_p678_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 678, "page_end": 678, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: N An = An − 1 + (−1)n + 1(n2) |An| Change from Previous Term.", "topic": "algebra"}
{"id": "book 2_p679_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 679, "page_end": 679, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. N An = An − 1 + (−1)n + 1(n2) |An| Change from Previous Term That is, the absolute value of each term equals the absolute value of the previous term plus n.", "topic": "percent"}
{"id": "book 2_p679_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 679, "page_end": 679, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Following this pattern, |A15| is 15 greater than |A14|, and |A14| is 14 greater than |A13|. Thus, |A15| − |A13| = +15 + 14 = 29.", "topic": "percent"}
{"id": "book 2_p679_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 679, "page_end": 679, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: When rounding to the nearest tenth, use the hundredths digit, or d in this case. Thus, a rephrase of this question is, “Is d < 5 (or, equivalently, is d ≥ 5)?” (1) INSUFFICIENT:.", "topic": "percent"}
{"id": "book 2_p679_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 679, "page_end": 679, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Since d − e is positive, d > e. Since d and e are digits (i.e., 0, 1, 2.", "topic": "percent"}
{"id": "book 2_p679_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 679, "page_end": 679, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Three perfect squares less than or equal to 9: d − e = 1, 4, or 9.", "topic": "percent"}
{"id": "book 2_p680_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 680, "page_end": 680, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. D e d − e = Perfect Square Is d < 5?d e d − e = Perfect Square Is d < 5?", "topic": "statistics"}
{"id": "book 2_p680_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 680, "page_end": 680, "topic_guess": "statistics", "text": "GMAT statistics note: 9 0 9 − 0 = 9 ✓ No 1 0 1 − 0 = 1 ✓ Yes It is possible for d to round up or down, so this statement isn’t sufficient. Since d and e are positive, you can square each side of the inequality without worrying about flipping the sign: d > e4.", "topic": "statistics"}
{"id": "book 2_p680_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 680, "page_end": 680, "topic_guess": "statistics", "text": "GMAT statistics note: Since d and e are digits 24 = 16, which is too large.) Test cases again. Whenever possible, reuse numbers that you tried in the last statement, assuming those numbers are allowed by the new statement.", "topic": "statistics"}
{"id": "book 2_p680_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 680, "page_end": 680, "topic_guess": "statistics", "text": "GMAT statistics note: 9 0 3 > 0 ✓ No ✓ Yes It is again possible for d to round up or down, so this statement isn’t sufficient. Taking both statements together, e must be 0 or 1 and d − e must equal 9, 4, or 1.", "topic": "statistics"}
{"id": "book 2_p680_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 680, "page_end": 680, "topic_guess": "statistics", "text": "GMAT statistics note: You’ve already tested the d = 9 and d = 1 scenarios for both of the statements individually, so you don’t need to retest them here. Even together, the two statements allow d to round up or down.", "topic": "statistics"}
{"id": "book 2_p681_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 681, "page_end": 681, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. List a few values of x and y to gain understanding.", "topic": "number_properties"}
{"id": "book 2_p681_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 681, "page_end": 681, "topic_guess": "number_theory", "text": "GMAT number properties note: Doing so also prepares you to most easily interpret the statements. There is only one: x = 6 and y = 60.", "topic": "number_properties"}
{"id": "book 2_p681_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 681, "page_end": 681, "topic_guess": "number_theory", "text": "GMAT number properties note: To be completely confident that this is the only valid case, approach the problem . However, x itself cannot have any common factors with x – 1, so 10 must be divisible by x – 1.", "topic": "number_properties"}
{"id": "book 2_p681_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 681, "page_end": 681, "topic_guess": "number_theory", "text": "GMAT number properties note: So x – 1 equals 1, 2, 5, or 10, so x must be 2, 3, 6, or 11. Therefore, x = 2 and x = 3 have already been eliminated as possibilities and x = 6 has already been confirmed to work.", "topic": "number_properties"}
{"id": "book 2_p681_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 681, "page_end": 681, "topic_guess": "number_theory", "text": "GMAT number properties note: The only remaining case that needs to be checked is x = 11. Since 110 is not divisible by every number smaller than 11 (e.g., 110 is not divisible by 3 or 4), x = 11 is not a valid case.", "topic": "number_properties"}
{"id": "book 2_p681_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 681, "page_end": 681, "topic_guess": "number_theory", "text": "GMAT number properties note: The only possible answer is x = 6, so this statement is sufficient. There are two possible answers, so the statement is insufficient.", "topic": "number_properties"}
{"id": "book 2_p682_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 682, "page_end": 682, "topic_guess": "word_problems", "text": "GMAT word problem note: Separate total outcomes from desired outcomes and count carefully. Qualitatively, many failure scenarios could occur.", "topic": "probability"}
{"id": "book 2_p682_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 682, "page_end": 682, "topic_guess": "word_problems", "text": "GMAT word problem note: Given the complexity of the failure scenarios, it is easier to look at the opposite scenario. For each of the links, the probability that it will not fail is 1 − 0.01 = 0.99.", "topic": "probability"}
{"id": "book 2_p682_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 682, "page_end": 682, "topic_guess": "word_problems", "text": "GMAT word problem note: The probability that all 10 will not fail is thus (0.99)10, since the probability that all 10 will not fail equals the product of the probabilities of the individual links not failing. Therefore, the probability that at least one link will fail equals 1 − (0.99)10.", "topic": "probability"}
{"id": "book 2_p682_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 682, "page_end": 682, "topic_guess": "word_problems", "text": "GMAT word problem note: None of the links will fail. Exactly one of the links will fail.", "topic": "probability"}
{"id": "book 2_p682_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 682, "page_end": 682, "topic_guess": "word_problems", "text": "GMAT word problem note: Exactly two of the links will fail. Probability that at least one link will fail = 1 − probability that No links will fail 35.", "topic": "probability"}
{"id": "book 2_p682_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 682, "page_end": 682, "topic_guess": "word_problems", "text": "GMAT word problem note: Ash’s round trip consists of two parts: the time spent on the bus and the time spent waiting for the next bus. No matter which buses he takes, the time spent on the bus will always be equal: 25 + 35 + 60 minutes = 120 minutes.", "topic": "probability"}
{"id": "book 2_p682_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 682, "page_end": 682, "topic_guess": "word_problems", "text": "GMAT word problem note: This question is really about minimizing the time that Ash spends waiting. Note the mismatch between the departure frequencies and the trip durations.", "topic": "probability"}
{"id": "book 2_p682_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 682, "page_end": 682, "topic_guess": "word_problems", "text": "GMAT word problem note: Because buses leave each city every 20 or 30 minutes, and both of these time periods divide evenly into one h.", "topic": "probability"}
{"id": "book 2_p683_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 683, "page_end": 683, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Then, to minimize waiting, Ash should take the first available buses in City B and City C.", "topic": "word_problems"}
{"id": "book 2_p683_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 683, "page_end": 683, "topic_guess": "word_problems", "text": "GMAT word problem note: For each of the three possible times that Ash could leave City A, note the next available buses for each leg of his trip.", "topic": "word_problems"}
{"id": "book 2_p683_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 683, "page_end": 683, "topic_guess": "word_problems", "text": "GMAT word problem note: Leave A Arrive in B Leave B Arrive in C Leave C Arrive in A Total Time 5:00 5:25 5:30 6:05 6:20 7:20 2 hr 20 min 5:20 5:45 6:00 6:30 6:40 7:40 2 hr 20 min 5:40 6:05 6:30 7:05 7:20 8:20 2 hr 40 min Therefore, the shortest possible duration of the entire trip is 2(60) + 20 = 120 + 20 = 140 minutes. Consecutive integers differ by exactly 1.", "topic": "word_problems"}
{"id": "book 2_p683_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 683, "page_end": 683, "topic_guess": "word_problems", "text": "GMAT word problem note: For x and y to be consecutive, either x = y + 1 or x = y − 1. Rephrase the question as, “Does x equal either (y + 1) or (y − (1) SUFFICIENT:.", "topic": "word_problems"}
{"id": "book 2_p683_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 683, "page_end": 683, "topic_guess": "word_problems", "text": "GMAT word problem note: There’s only one term with x but a couple with y, so it is easiest to solve for x. Since x and y are both positive, you can drop the absolute value signs and conclude that x = y + 1.", "topic": "word_problems"}
{"id": "book 2_p684_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 684, "page_end": 684, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Therefore, x = 0 or x − y − 1 = 0.", "topic": "algebra"}
{"id": "book 2_p684_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 684, "page_end": 684, "topic_guess": "algebra", "text": "GMAT algebra note: The question stem indicates that x is positive, so x cannot equal 0. As a result, it must be the case that x − y − 1 = 0, or x = y + 1.", "topic": "algebra"}
{"id": "book 2_p684_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 684, "page_end": 684, "topic_guess": "algebra", "text": "GMAT algebra note: The two integers are consecutive. Solve for the expression in question by adding (−x2 − x) to both sides.", "topic": "algebra"}
{"id": "book 2_p684_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 684, "page_end": 684, "topic_guess": "algebra", "text": "GMAT algebra note: The answer depends on the value of x, which is not given.", "topic": "algebra"}
{"id": "book 2_p685_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 685, "page_end": 685, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Focus on equations, expressions, variables, and step-by-step manipulation. Each statement provides a different expression equal to y − x2 − x.", "topic": "algebra"}
{"id": "book 2_p685_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 685, "page_end": 685, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Set these two equal to each other and solve. Once the squared terms drop out, the equation is solvable.", "topic": "algebra"}
{"id": "book 2_p685_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 685, "page_end": 685, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Stop when you know you can solve! This problem asks you to compare two “nested” functions: f(g(x)) and g(f(x)).", "topic": "algebra"}
{"id": "book 2_p685_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 685, "page_end": 685, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: To simplify a nested function, replace the variable term in the outer function with the entire inner function. To find the values of x for which f(g(x)) < g(f(x)), simplify the following inequality.", "topic": "algebra"}
{"id": "book 2_p685_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 685, "page_end": 685, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: One possibility is to multiply out both quadratics and then simplify as much as possible.", "topic": "algebra"}
{"id": "book 2_p686_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 686, "page_end": 686, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: In turn, this simplifies to –2 < x < 2. Another approach involves using the multiples of 9 in the initial inequality.", "topic": "geometry"}
{"id": "book 2_p686_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 686, "page_end": 686, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Factor out a 9 on the le side of the inequality. Start by labeling the three line segments with length 1.", "topic": "geometry"}
{"id": "book 2_p686_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 686, "page_end": 686, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Label the unknown length as x.", "topic": "geometry"}
{"id": "book 2_p687_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 687, "page_end": 687, "topic_guess": "geometry", "text": "GMAT geometry note: From here, there are two ways to approach the problem. One approach is to use the Pythagorean theorem to label the lengths of the missing sides.", "topic": "geometry"}
{"id": "book 2_p687_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 687, "page_end": 687, "topic_guess": "geometry", "text": "GMAT geometry note: Triangle ABP is a right triangle, so the following is true:. Similarly, APD is a right triangle, and the following is true:.", "topic": "geometry"}
{"id": "book 2_p687_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 687, "page_end": 687, "topic_guess": "geometry", "text": "GMAT geometry note: Finally, since BAD is a right triangle, the Pythagorean theorem can be used to solve for the value of x.", "topic": "geometry"}
{"id": "book 2_p688_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 688, "page_end": 688, "topic_guess": "geometry", "text": "GMAT geometry note: The other solution uses the fact that triangles APD and BPA are similar. To prove this, note that APD is a right angle.", "topic": "geometry"}
{"id": "book 2_p688_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 688, "page_end": 688, "topic_guess": "geometry", "text": "GMAT geometry note: Therefore, angles PAD and ADP sum to 90 degrees. Therefore, ADP and BAP have the same degree measure.", "topic": "geometry"}
{"id": "book 2_p688_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 688, "page_end": 688, "topic_guess": "geometry", "text": "GMAT geometry note: Since triangles APD and BPA have two pairs of matching angles, the third pair of angles must match as well, and the two triangles are similar. The sides line up as follows:.", "topic": "geometry"}
{"id": "book 2_p688_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 688, "page_end": 688, "topic_guess": "geometry", "text": "GMAT geometry note: Triangle PAD Triangle PBA Shortest Side side PD side AP Middle Side side AP side BP Longest Side side AD side AB Since PD = 1, BP = 2, and AP = AP, you can set up a proportion to find the length of side AP.", "topic": "geometry"}
{"id": "book 2_p689_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 689, "page_end": 689, "topic_guess": "geometry", "text": "GMAT geometry note: Finally, use the Pythagorean theorem to determine the length of AB. Side AB appears to be longer than BP, but shorter than BD.", "topic": "geometry"}
{"id": "book 2_p689_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 689, "page_end": 689, "topic_guess": "geometry", "text": "GMAT geometry note: Its length should be between 2 and 3.", "topic": "geometry"}
{"id": "book 2_p690_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 690, "page_end": 690, "topic_guess": "geometry", "text": "GMAT geometry note: The triangles they yield will have the same area. Therefore, neither can be the correct answer.", "topic": "geometry"}
{"id": "book 2_p690_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 690, "page_end": 690, "topic_guess": "geometry", "text": "GMAT geometry note: The question is asking for triangles with area less than 16 square units, so first try drawing a triangle with an area of exactly 16 square units. If the base of the triangle is OB, it has a length .", "topic": "geometry"}
{"id": "book 2_p690_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 690, "page_end": 690, "topic_guess": "geometry", "text": "GMAT geometry note: Use the area formula to find the height. Draw the height of the triangle perpendicular to OB through the origin in either quadrant II or quadrant IV.", "topic": "geometry"}
{"id": "book 2_p690_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 690, "page_end": 690, "topic_guess": "geometry", "text": "GMAT geometry note: The vertex of the height is at either (–4, 4) or (4, –4). Any triangle with the same height will have the same area.", "topic": "geometry"}
{"id": "book 2_p690_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 690, "page_end": 690, "topic_guess": "geometry", "text": "GMAT geometry note: Therefore, if A lies on the line y = x + 8 or on the line y = x – 8, the triangle will have an area of 16. If A lies anywhere between those two lines, then the triangle will have an area less than 16.", "topic": "geometry"}
{"id": "book 2_p690_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 690, "page_end": 690, "topic_guess": "geometry", "text": "GMAT geometry note: Of the answers, only (1, 8) lies between the two lines.", "topic": "geometry"}
{"id": "book 2_p691_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 691, "page_end": 691, "topic_guess": "geometry", "text": "GMAT geometry note: Rather than calculating the exact area of each triangle, notice that each one is fairly close to a triangle whose area is easier to calculate. The latter triangle has an area of exactly 16 square units.", "topic": "geometry"}
{"id": "book 2_p692_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 692, "page_end": 692, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If , where a and b are integers and a does not equal 1 or −2, which of the following could be the value of b?", "topic": "number_properties"}
{"id": "book 2_p692_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 692, "page_end": 692, "topic_guess": "number_theory", "text": "GMAT number properties note: The currencies of three countries are called credits, units, and bells. If 12 credits can be exchanged for a total of exactly 8 units and 5 bells, what is the smallest integer number of units that can be exchanged for an integer number of bells, with no change le over?", "topic": "number_properties"}
{"id": "book 2_p692_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 692, "page_end": 692, "topic_guess": "number_theory", "text": "GMAT number properties note: 15 credits can be exchanged for 4 units and 25 bells, with no change le over.", "topic": "number_properties"}
{"id": "book 2_p693_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 693, "page_end": 693, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. 25 bells can be exchanged for 8 units, with no change le over.(2) 143.", "topic": "number_properties"}
{"id": "book 2_p693_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 693, "page_end": 693, "topic_guess": "number_theory", "text": "GMAT number properties note: Each digit 1 through 5 is used exactly once to create a five-digit integer. If the 3 and the 4 cannot be adjacent digits in the integer, how many five-digit integers are possible?", "topic": "number_properties"}
{"id": "book 2_p693_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 693, "page_end": 693, "topic_guess": "number_theory", "text": "GMAT number properties note: For positive integers a, b, and c, b is 120% greater than a, and c is 25% less than the sum of a and b. Which of the following could be.", "topic": "number_properties"}
{"id": "book 2_p694_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 694, "page_end": 694, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. A soccer competition consisted of two rounds.", "topic": "algebra"}
{"id": "book 2_p694_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 694, "page_end": 694, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: In each round, the 20 teams competing were divided randomly into 10 pairs, and each pair played a single game. None of the games resulted in a tie, and of the teams that lost in the first round, 7 also lost in the second round.", "topic": "algebra"}
{"id": "book 2_p694_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 694, "page_end": 694, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: How many teams won both of their games? What is the units digit of the positive integer x?", "topic": "algebra"}
{"id": "book 2_p694_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 694, "page_end": 694, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: , where y is a positive integer.(1) , where z is a positive integer.(2).", "topic": "algebra"}
{"id": "book 2_p695_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 695, "page_end": 695, "topic_guess": "geometry", "text": "GMAT geometry note: If x and y are positive integers, what is the value of x + y? For all n such that n is a positive integer, the terms of a certain sequence B are given by the following rules:.", "topic": "geometry"}
{"id": "book 2_p695_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 695, "page_end": 695, "topic_guess": "geometry", "text": "GMAT geometry note: Bn = Bn − 1 + 5 if n is odd and greater than 1 Bn = −Bn − 1 if n is even B1 = 3.", "topic": "geometry"}
{"id": "book 2_p696_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 696, "page_end": 696, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. What is the sum of the first 65 terms in the sequence?", "topic": "algebra"}
{"id": "book 2_p697_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 697, "page_end": 697, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "book 2_p698_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 698, "page_end": 698, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Workout Set 15 Solutions 41.", "topic": "algebra"}
{"id": "book 2_p698_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 698, "page_end": 698, "topic_guess": "algebra", "text": "GMAT algebra note: Since the fraction equals 0, either ab + 6 = 0 or ab − 3 = 0. Since a and b are integers, and ab = −6 or ab = 3, it must be the case that b is a factor of either −6 or 3.", "topic": "algebra"}
{"id": "book 2_p698_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 698, "page_end": 698, "topic_guess": "algebra", "text": "GMAT algebra note: If b = 1, then a = −6 or a = 3, which are both allowed.I. If b = 2, then a = −3 or a = .", "topic": "algebra"}
{"id": "book 2_p698_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 698, "page_end": 698, "topic_guess": "algebra", "text": "GMAT algebra note: One of those (a = −3) is allowed.II. If b = 3, then a = −2 or a = 1, neither of which is allowed.III.", "topic": "algebra"}
{"id": "book 2_p698_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 698, "page_end": 698, "topic_guess": "algebra", "text": "GMAT algebra note: Use three variables to represent the unknown values in this problem. Therefore, your goal is to find the relationship between the value of a unit and the value of a bell.", "topic": "algebra"}
{"id": "book 2_p699_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 699, "page_end": 699, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. This statement can be rewritten as the following equation, using the same variables as above.", "topic": "algebra"}
{"id": "book 2_p699_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 699, "page_end": 699, "topic_guess": "algebra", "text": "GMAT algebra note: Therefore, 8 units are worth the same amount as 25 bells. Because the ratio 8: 25 cannot be reduced further, 8 is the smallest number of units that can be exchanged for an integer number of bells.", "topic": "algebra"}
{"id": "book 2_p699_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 699, "page_end": 699, "topic_guess": "algebra", "text": "GMAT algebra note: For instance, 4 units could only be exchanged for 12.5 bells, which is not an integer. The answer to the question is 8.", "topic": "algebra"}
{"id": "book 2_p699_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 699, "page_end": 699, "topic_guess": "algebra", "text": "GMAT algebra note: Combine this with the equation from the question stem. Because the goal is to find the relationship between b and u, focus on eliminating the other variable, c.", "topic": "algebra"}
{"id": "book 2_p699_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 699, "page_end": 699, "topic_guess": "algebra", "text": "GMAT algebra note: First, multiply the equations by 5 and 4, respectively, so that the coefficient of c is 60 in both.", "topic": "algebra"}
{"id": "book 2_p700_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 700, "page_end": 700, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If 8 units can be exchanged for 25 bells, then because 8 and 25 have no common factors, no smaller integer number of units can be exchanged for bells, as described above.", "topic": "number_properties"}
{"id": "book 2_p700_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 700, "page_end": 700, "topic_guess": "number_theory", "text": "GMAT number properties note: The answer to the question is 8, and this statement is sufficient. All of these line equations are of the form y = mx + b, where m is the slope and b is the y-intercept.", "topic": "number_properties"}
{"id": "book 2_p700_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 700, "page_end": 700, "topic_guess": "number_theory", "text": "GMAT number properties note: Two of these lines have a slope of and are thus parallel to each other. The other two lines are parallel to one another with a slope of .", "topic": "number_properties"}
{"id": "book 2_p700_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 700, "page_end": 700, "topic_guess": "number_theory", "text": "GMAT number properties note: Two of the lines have a y-intercept of 6 while the other two lines have a y-intercept of −6.", "topic": "number_properties"}
{"id": "book 2_p701_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 701, "page_end": 701, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. In each quadrant, there is a triangle with the dimensions 6–8–10, a multiple of the common 3–4–5 right triangle.", "topic": "probability"}
{"id": "book 2_p701_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 701, "page_end": 701, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Alternatively, recognize that the quadrilateral is a rhombus (four equal sides of length 10) and use the formula for the area of a rhombus: , where D indicates the length of the diagonals. In a constrained combinatorics question such as this one, it is o en easier to consider the “violating” cases instead of the “okay” cases.", "topic": "probability"}
{"id": "book 2_p701_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 701, "page_end": 701, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: # of permutations that obey the constraint = # of permutations total − # of permutations that violate the constraint At first glance, ignore the constraint that 3 and 4 cannot be adjacent to determine the total number of five-digit integers possible: # of permutations total = 5!", "topic": "probability"}
{"id": "book 2_p702_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 702, "page_end": 702, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Now, consider the basic ways that 3 and 4 might be adjacent to each other in a five-digit number: 3 4 x x x x 3 4 x x x x 3 4 x x x x 3 4 For each of these four base cases, there are two ways to order the 3 and 4 (3, 4 and 4, 3), as well as 3!", "topic": "probability"}
{"id": "book 2_p702_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 702, "page_end": 702, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Ways the other digits (1, 2, and 5) can be arranged in the x positions. Thus: # of permutations that violate the constraint = 4 × 2 × 3!", "topic": "probability"}
{"id": "book 2_p702_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 702, "page_end": 702, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: = (4)(2)(3) Therefore, the number of permutations that do not violate the constraint equals 120 − 48 = 72. Start by translating the text into math.", "topic": "probability"}
{"id": "book 2_p702_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 702, "page_end": 702, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Note that b is 120% greater than a, not 120% of a.", "topic": "probability"}
{"id": "book 2_p703_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 703, "page_end": 703, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. There are three variables and only two equations, so you won’t be able to solve in the traditional sense, but you may be able to determine something about these integers.", "topic": "number_properties"}
{"id": "book 2_p703_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 703, "page_end": 703, "topic_guess": "number_theory", "text": "GMAT number properties note: Substitute for b in the second equation. Both a and c are integers, and 5 and 12 share no prime factors.", "topic": "number_properties"}
{"id": "book 2_p703_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 703, "page_end": 703, "topic_guess": "number_theory", "text": "GMAT number properties note: So a must be divisible by 5 and c must be divisible by 12. Try a = 5 and c = 12 to see whether b is a positive integer in this case.", "topic": "number_properties"}
{"id": "book 2_p703_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 703, "page_end": 703, "topic_guess": "number_theory", "text": "GMAT number properties note: This is a valid case because b is a positive integer. Each team either won or lost the first game and either won or lost the second game.", "topic": "number_properties"}
{"id": "book 2_p703_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 703, "page_end": 703, "topic_guess": "number_theory", "text": "GMAT number properties note: Therefore, this problem can be approached using overlapping sets.", "topic": "number_properties"}
{"id": "book 2_p704_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 704, "page_end": 704, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. Won first game Lost first game Total Won second game Lost second game 7 Total 20 Initially, it appears that there is not enough information to determine how many teams won both games.", "topic": "number_properties"}
{"id": "book 2_p704_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 704, "page_end": 704, "topic_guess": "general", "text": "GMAT quant note: However, note that there were no ties and each round consisted of exactly 10 games. Since each of the 10 games yielded one winner and one loser, there were exactly 10 teams that won in the first round and 10 teams that lost in the first round.", "topic": "number_properties"}
{"id": "book 2_p704_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 704, "page_end": 704, "topic_guess": "general", "text": "GMAT quant note: Similarly, there were 10 teams that won in the second round and 10 teams that lost in the second round. Based on this information, fill in the remainder of the matrix.", "topic": "number_properties"}
{"id": "book 2_p704_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 704, "page_end": 704, "topic_guess": "general", "text": "GMAT quant note: Won first game Lost first game Total Won second game 7 3 10 Lost second game 3 7 10 Total 10 10 20 Seven teams won both of their games.", "topic": "number_properties"}
{"id": "book 2_p705_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 705, "page_end": 705, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. Thus, x is 1 greater than a multiple of 5.", "topic": "number_properties"}
{"id": "book 2_p705_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 705, "page_end": 705, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Since all multiples of 5 end in either 0 or 5, x must end in either 1 or 6. Alternatively, you could list numbers.", "topic": "number_properties"}
{"id": "book 2_p705_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 705, "page_end": 705, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Since y is a positive integer, 5y could be 5, 10, 15, 20, etc. Thus, x could be 6, 11, 16, 21, etc.", "topic": "number_properties"}
{"id": "book 2_p705_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 705, "page_end": 705, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The units digit of x could be 1 or 6. This indicates that x is odd, because 2z is an even number.", "topic": "number_properties"}
{"id": "book 2_p705_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 705, "page_end": 705, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Any odd single- digit integer is a possible units digit for x: 1, 3, 5, 7, or 9. The question stem asks for the value of the combo x + y.", "topic": "number_properties"}
{"id": "book 2_p706_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 706, "page_end": 706, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. (1) INSUFFICIENT: (x + y − 1)!", "topic": "number_properties"}
{"id": "book 2_p706_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 706, "page_end": 706, "topic_guess": "number_theory", "text": "GMAT number properties note: To see the possible values of (x + y − 1), list the factorials of the first few integers. Add x to both sides of y = x2 − x + 1 to create (x + y) on one side of the equation.", "topic": "number_properties"}
{"id": "book 2_p706_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 706, "page_end": 706, "topic_guess": "number_theory", "text": "GMAT number properties note: The exact value of (x + y) is unknown, as it depends on the value of x2, which could be any positive integer. (1) AND (2) INSUFFICIENT: (x + y) ≤ 5 and x + y = x2 + 1 combine to indicate that x2 + 1 ≤ 5.", "topic": "number_properties"}
{"id": "book 2_p706_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 706, "page_end": 706, "topic_guess": "number_theory", "text": "GMAT number properties note: There are two integer solutions: x = 1 or x = 2. = 120 (too large) If x = 1, then y = 12 − 1 + 1 = 1 and x + y = 1 + 1 = 2.", "topic": "number_properties"}
{"id": "book 2_p706_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 706, "page_end": 706, "topic_guess": "number_theory", "text": "GMAT number properties note: If x = 2, then y = 22 − 2 + 1 = 3 and x + y = 2 + 3 = 5. The size of the angle depends on two things.", "topic": "number_properties"}
{"id": "book 2_p707_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 707, "page_end": 707, "topic_guess": "geometry", "text": "GMAT geometry note: The size of the circle The distance of C from the circle (inversely related to arc length ADB) Larger circle → Larger x Farther C → smaller x Larger arc ADB → smaller x (1) INSUFFICIENT:. The rubber band picture on the right indicates that for a circle of fixed size, x can still vary with the length of arc ADB (i.e., x varies with the placement of C).", "topic": "geometry"}
{"id": "book 2_p707_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 707, "page_end": 707, "topic_guess": "geometry", "text": "GMAT geometry note: Draw some cases to prove that x can vary for a given arc ADB.", "topic": "geometry"}
{"id": "book 2_p708_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 708, "page_end": 708, "topic_guess": "geometry", "text": "GMAT geometry note: Or nearly half of the circle, and the lines tangent to the circle at A and B will join at a smaller angle x. Thus, arc ADB is of the circumference of the circle.", "topic": "geometry"}
{"id": "book 2_p708_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 708, "page_end": 708, "topic_guess": "geometry", "text": "GMAT geometry note: There is only one way to draw the lines tangent to the circle at A and B, so x must be one specific value. List the first few terms of the sequence according to the given rules.", "topic": "geometry"}
{"id": "book 2_p709_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 709, "page_end": 709, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Note that the pattern is a four-term repeat: 3, −3, 2, −2.", "topic": "algebra"}
{"id": "book 2_p709_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 709, "page_end": 709, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Also note that the sum of this repeating group is (3) + (−3) + (2) + (−2) = 0. This repeating group will occur 16 times through term number 64.", "topic": "algebra"}
{"id": "book 2_p709_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 709, "page_end": 709, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Because the sum of the four terms is 0, you don’t actually have to figure out how many times the pattern repeats.) Thus, the sum of the first 64 terms will be 0. This leaves the 65th term, which will have the same value as B1: 3.", "topic": "algebra"}
{"id": "book 2_p709_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 709, "page_end": 709, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Therefore, the sum of the first 65 terms is 3.", "topic": "algebra"}
{"id": "book 2_p710_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 710, "page_end": 710, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. If the store made a total profit of 16% on sales of these three models of bicycles last month, and these three models represented the entirety of the store’s profits, what percent of its total profits last month came from sales of Model A bicycles?", "topic": "statistics"}
{"id": "book 2_p710_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 710, "page_end": 710, "topic_guess": "statistics", "text": "GMAT statistics note: The store earned 20% of its profits last month from sales of Model B bicycles. The store earned 50% of its profits last month from sales of Model C bicycles.", "topic": "statistics"}
{"id": "book 2_p710_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 710, "page_end": 710, "topic_guess": "statistics", "text": "GMAT statistics note: A beaker contains 100 milligrams of a solution of salt and water that is x% salt by weight such that x < 90. If the water evaporates at a.", "topic": "statistics"}
{"id": "book 2_p711_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 711, "page_end": 711, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Rate of y milligrams per hour, how many hours will it take for the concentration of salt to reach (x + 10)%, in terms of x and y?", "topic": "number_properties"}
{"id": "book 2_p711_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 711, "page_end": 711, "topic_guess": "number_theory", "text": "GMAT number properties note: Set A consists of five consecutive positive integers. Set B consists of every integer that is in set A as well as the sum of every pair of two distinct integers from set A, discarding any numbers that already appear in set B.", "topic": "number_properties"}
{"id": "book 2_p711_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 711, "page_end": 711, "topic_guess": "number_theory", "text": "GMAT number properties note: What is the greatest integer in set B? The range of set B is 13.(1) There are exactly 12 integers in set B.(2).", "topic": "number_properties"}
{"id": "book 2_p712_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 712, "page_end": 712, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. A survey included two questions, each of which could be answered with either Yes or No.", "topic": "number_properties"}
{"id": "book 2_p712_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 712, "page_end": 712, "topic_guess": "number_theory", "text": "GMAT number properties note: A total of 100 respondents took the survey, and every respondent answered both questions. How many respondents answered both questions with a Yes?", "topic": "number_properties"}
{"id": "book 2_p712_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 712, "page_end": 712, "topic_guess": "number_theory", "text": "GMAT number properties note: Of the respondents who answered Yes to the first question, 40% answered Yes to the second question. Of the respondents who answered Yes to at least one question, 30% answered Yes to both questions.", "topic": "number_properties"}
{"id": "book 2_p712_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 712, "page_end": 712, "topic_guess": "number_theory", "text": "GMAT number properties note: If x is a positive two-digit integer, is x divisible by 17? X is divisible by 7.(1) x has exactly 3 unique prime factors.(2) 156.", "topic": "number_properties"}
{"id": "book 2_p712_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 712, "page_end": 712, "topic_guess": "number_theory", "text": "GMAT number properties note: Green paint is made by mixing blue paint with yellow paint in a ratio of 2 to x. Turquoise paint is made by mixing green paint with blue paint in a ratio of y to 2.", "topic": "number_properties"}
{"id": "book 2_p712_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 712, "page_end": 712, "topic_guess": "number_theory", "text": "GMAT number properties note: In terms of x and y, how many gallons of yellow paint are required to make 10 gallons of turquoise paint?", "topic": "number_properties"}
{"id": "book 2_p713_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 713, "page_end": 713, "topic_guess": "general", "text": "GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. Erin writes a list of every sixth integer in increasing order, starting with the positive integer x.", "topic": "number_properties"}
{"id": "book 2_p713_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 713, "page_end": 713, "topic_guess": "general", "text": "GMAT quant note: Harry writes a list of every ninth integer in increasing order, starting with the positive integer y. Will any integer appear on both Erin’s list and Harry’s list?", "topic": "number_properties"}
{"id": "book 2_p713_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 713, "page_end": 713, "topic_guess": "general", "text": "GMAT quant note: X is a multiple of y.(1) x – y is a multiple of 3.(2) 158. The students in a certain college class took a midterm exam and a final exam, both of which were scored out of a total of 100 points.", "topic": "number_properties"}
{"id": "book 2_p714_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 714, "page_end": 714, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. What percent of the students earned a higher score on the final than on the midterm?", "topic": "word_problems"}
{"id": "book 2_p714_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 714, "page_end": 714, "topic_guess": "word_problems", "text": "GMAT word problem note: Of the students in the class, 28% scored at least 6 points higher on the final than on the midterm. Of the students who scored higher on the final than on the midterm, 40% scored at least 6 points higher on the final.", "topic": "word_problems"}
{"id": "book 2_p714_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 714, "page_end": 714, "topic_guess": "word_problems", "text": "GMAT word problem note: A triathlon consists of three legs: a swim, a bike ride, and a run. A race organizer analyzes the results of a certain triathlon and observes that the ratio of swim distance to bike distance to run distance was 1 to 50 to 12, and that the ratio of average swim time to average bike time to average run time for all participants was 3 to 10 to 6.", "topic": "word_problems"}
{"id": "book 2_p714_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 714, "page_end": 714, "topic_guess": "word_problems", "text": "GMAT word problem note: If the average speed of racers in each leg of the race does not change, but the race organizer changes the distance of each leg of the race so that the average swim time, average bike time, and average run time are now equal to each other, what fraction of the total distance of the race will be represented by the swim leg of the triathlon?", "topic": "word_problems"}
{"id": "book 2_p715_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 715, "page_end": 715, "topic_guess": "geometry", "text": "GMAT geometry note: An unpainted wall with a total area of 100 square feet needed to be painted with three coats of paint. Nelson began the job by painting an area of 60 square feet.", "topic": "geometry"}
{"id": "book 2_p715_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 715, "page_end": 715, "topic_guess": "geometry", "text": "GMAT geometry note: Then, Jamaica painted a total area of 70 square feet. Finally, Bryan painted a total area of 80 square feet.", "topic": "geometry"}
{"id": "book 2_p715_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 715, "page_end": 715, "topic_guess": "geometry", "text": "GMAT geometry note: A er Bryan finished painting, an area of q square feet of the wall was covered in three coats of paint. If the entire surface area of the wall had received at least one coat of paint, which of the following specifies all the possible values of q?", "topic": "geometry"}
{"id": "book 2_p716_c1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 716, "page_end": 716, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "book 2_p717_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 717, "page_end": 717, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Workout Set 16 Solutions 51.", "topic": "statistics"}
{"id": "book 2_p717_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 717, "page_end": 717, "topic_guess": "statistics", "text": "GMAT statistics note: A store sold three different models of bicycles last month. For each of these models, the profit earned by the store represents a different percentage of the sales of that model.", "topic": "statistics"}
{"id": "book 2_p717_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 717, "page_end": 717, "topic_guess": "statistics", "text": "GMAT statistics note: The question stem also states that the total profit was 16% of the total sales. One way to handle the question stem is to use variables A, B, and C to represent the total sales of Model A, B, and C bicycles, respectively.", "topic": "statistics"}
{"id": "book 2_p717_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 717, "page_end": 717, "topic_guess": "statistics", "text": "GMAT statistics note: Then, the profit from Model A bicycles is 0.12A, the profit from Model B bicycles is 0.16B, and the profit from Model C bicycles is 0.20C. Also, the total dollar amount of sales is A + B + C, so the following is true and can be simplified:.", "topic": "statistics"}
{"id": "book 2_p717_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 717, "page_end": 717, "topic_guess": "statistics", "text": "GMAT statistics note: You might also notice that 16% is exactly in the middle of the three percentages provided in the question stem. The problem is effectively a weighted average problem: if more dollars worth of Model A bicycles were sold, then the average profit would be pulled downwards, closer to 12%.", "topic": "statistics"}
{"id": "book 2_p717_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 717, "page_end": 717, "topic_guess": "statistics", "text": "GMAT statistics note: If more dollars worth of Model C bicycles were sold, then the average profit would be pulled upwards, closer to 20%. Since the actual profit was in the middle, at 16%, the dollar value of the Model A bicycles sold and the dollar value of the Model C bicycles sold must be equal.", "topic": "statistics"}
{"id": "book 2_p718_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 718, "page_end": 718, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. The question asks about the share of profits that came from Model A bicycles.", "topic": "statistics"}
{"id": "book 2_p718_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 718, "page_end": 718, "topic_guess": "statistics", "text": "GMAT statistics note: This question can be rephrased as, “What is ?” which can then be simplified as follows:. What is What is What is The variables would cancel entirely if you could determine a known ratio of A to B.", "topic": "statistics"}
{"id": "book 2_p718_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 718, "page_end": 718, "topic_guess": "statistics", "text": "GMAT statistics note: (1) SUFFICIENT: 20% of the store’s profits came from Model B bicycles, which can be translated and simplified as follows, remembering that A = C and the total profit equals 0.16(A + B + C). Substituting A = 2B into the question causes all unknowns to cancel out, yielding an answer of , or 30%.", "topic": "statistics"}
{"id": "book 2_p719_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 719, "page_end": 719, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. The store earned 50% of its profits from sales of Model C bicycles, which can be translated and simplified as follows, remembering that A = C and the total profit equals 0.16(A + B + C).", "topic": "word_problems"}
{"id": "book 2_p719_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 719, "page_end": 719, "topic_guess": "word_problems", "text": "GMAT word problem note: The answer is again 30%, as shown above, so this statement is sufficient. Therefore, it is possible to use Smart Numbers.", "topic": "word_problems"}
{"id": "book 2_p719_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 719, "page_end": 719, "topic_guess": "word_problems", "text": "GMAT word problem note: Therefore, the beaker initially contains 10% of 100 mg = 10 mg of salt. The question asks how many hours it will take for the concentration of salt to reach (x + 10)%, or 20%.", "topic": "word_problems"}
{"id": "book 2_p719_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 719, "page_end": 719, "topic_guess": "word_problems", "text": "GMAT word problem note: The beaker will always contain 10 mg of salt, since the salt does not evaporate. The 10 mg of salt will represent 20%, or , of the total weight when the total solution weight has been reduced to 50 mg.", "topic": "word_problems"}
{"id": "book 2_p719_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 719, "page_end": 719, "topic_guess": "word_problems", "text": "GMAT word problem note: To get to this point, 50 mg of water must evaporate. This will take a total of = 25 hours.", "topic": "word_problems"}
{"id": "book 2_p720_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 720, "page_end": 720, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Alternatively, approach the problem with algebra.", "topic": "percent"}
{"id": "book 2_p720_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 720, "page_end": 720, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Initially, the beaker contains x mg of salt in a total solution of 100 mg. A er h hours, hy mg of the solution will have evaporated, so the solution will have a total weight of (100 – hy) mg, and the amount of salt will still be x mg.", "topic": "percent"}
{"id": "book 2_p720_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 720, "page_end": 720, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: So the new concentration, as a percentage, will be . This percent should equal (x + 10)%.", "topic": "percent"}
{"id": "book 2_p720_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 720, "page_end": 720, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Set up an equation and solve for h, the number of hours.", "topic": "percent"}
{"id": "book 2_p721_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 721, "page_end": 721, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Then, set B will initially include 1, 2, 3, 4, and 5, and more numbers will be added to it as follows, discarding any numbers that already appear in set B.", "topic": "number_properties"}
{"id": "book 2_p721_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 721, "page_end": 721, "topic_guess": "number_theory", "text": "GMAT number properties note: 1 + 2 = 3 discard 1 + 3 = 4 discard 1 + 4 = 5 discard 1 + 5 = 6 will be added to set B 2 + 3 = 5 discard 2 + 4 = 6 discard 2 + 5 = 7 will be added to set B 3 + 4 = 7 discard 3 + 5 = 8 will be added to set B 4 + 5 = 9 will be added to set B.", "topic": "number_properties"}
{"id": "book 2_p722_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 722, "page_end": 722, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Thus, if set A contains the integers from 1 to 5, set B consists of 1, 2, 3, 4, 5, 6, 7, 8, and 9.", "topic": "number_properties"}
{"id": "book 2_p722_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 722, "page_end": 722, "topic_guess": "number_theory", "text": "GMAT number properties note: The question asks for the greatest integer in set B. The range of a set is the difference between the greatest and the smallest numbers in that set.", "topic": "number_properties"}
{"id": "book 2_p722_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 722, "page_end": 722, "topic_guess": "number_theory", "text": "GMAT number properties note: The smallest number in set B is the smallest number from set A, since all of the additional numbers in set B are constructed by summing numbers together and will therefore be greater. The greatest number in set B is created by summing the two greatest numbers from set A.", "topic": "number_properties"}
{"id": "book 2_p722_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 722, "page_end": 722, "topic_guess": "number_theory", "text": "GMAT number properties note: Those numbers are x + 3 and x + 4, and their sum is 2x + 7. Therefore, the range of set B is (2x + 7) – x.", "topic": "number_properties"}
{"id": "book 2_p722_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 722, "page_end": 722, "topic_guess": "number_theory", "text": "GMAT number properties note: Set this equal to the given range of 13, and solve for x:. The smallest number in set A is 6, and both sets can be listed with certainty.", "topic": "number_properties"}
{"id": "book 2_p722_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 722, "page_end": 722, "topic_guess": "number_theory", "text": "GMAT number properties note: There is only one answer to the question, so this statement is sufficient. There are exactly 12 integers in set B, and you will likely have to test cases to determine how this could happen.", "topic": "number_properties"}
{"id": "book 2_p722_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 722, "page_end": 722, "topic_guess": "number_theory", "text": "GMAT number properties note: Above, it was determined that if set A contains the integers 1, 2, 3, 4, and 5, then set B contains 9 integers in total: the integers from 1 to 9, inclusive. This is not enough integers:.", "topic": "number_properties"}
{"id": "book 2_p722_c1_r8", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 722, "page_end": 722, "topic_guess": "number_theory", "text": "GMAT number properties note: Find a case with fewer duplicates to discard. If set A contains the integers 5.", "topic": "number_properties"}
{"id": "book 2_p723_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 723, "page_end": 723, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. This set B again contains 12 integers, but its greatest integer is 207.", "topic": "number_properties"}
{"id": "book 2_p723_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 723, "page_end": 723, "topic_guess": "number_theory", "text": "GMAT number properties note: This is an Overlapping Sets problem. Create a Double-Set Matrix and fill out the known information.", "topic": "number_properties"}
{"id": "book 2_p723_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 723, "page_end": 723, "topic_guess": "number_theory", "text": "GMAT number properties note: Yes to Question 1 No to Question 1 Total Yes to Question 2 No to Question 2 Total 100 The question asks for the number of respondents who gave two Yes answers. Try to determine the number that belongs in the top le of the matrix.", "topic": "number_properties"}
{"id": "book 2_p723_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 723, "page_end": 723, "topic_guess": "number_theory", "text": "GMAT number properties note: The matrix can be filled out as follows:. Yes to Question 1 No to Question 1 Total Yes to Question 2 0.4x No to Question 2 0.6x Total x 100 – x 100 If x = 5, then the answer is 0.4(5) = 2.", "topic": "number_properties"}
{"id": "book 2_p723_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 723, "page_end": 723, "topic_guess": "number_theory", "text": "GMAT number properties note: If x = 90, then the answer is 0.4(90) = 36. More than one value is possible for the answer, so this statement is not sufficient.", "topic": "number_properties"}
{"id": "book 2_p723_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 723, "page_end": 723, "topic_guess": "number_theory", "text": "GMAT number properties note: Then, the number of people who answered Yes to at least one question is.", "topic": "number_properties"}
{"id": "book 2_p724_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 724, "page_end": 724, "topic_guess": "general", "text": "GMAT quant note: Percent problems usually relate part, whole, rate, and percent change. Since 30% of these people answered Yes to both questions, the matrix can be filled out as shown.", "topic": "percent"}
{"id": "book 2_p724_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 724, "page_end": 724, "topic_guess": "general", "text": "GMAT quant note: Yes to Question 1 No to Question 1 Total Yes to Question 2 0.3(100 – y) No to Question 2 y Total 100 If y = 10, the answer is 0.3(90) = 27. If y = 80, the answer is 0.3(20) = 6.", "topic": "percent"}
{"id": "book 2_p724_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 724, "page_end": 724, "topic_guess": "general", "text": "GMAT quant note: Since more than one value is possible for the answer, this statement is insufficient. Using the information in both statements, you can conclude that 0.3(100 – y) = 0.4x.", "topic": "percent"}
{"id": "book 2_p724_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 724, "page_end": 724, "topic_guess": "general", "text": "GMAT quant note: This simplifies to 0.4x + 0.3y = 30, but both x and y can still have multiple values. Accordingly, the matrix can be filled out with numbers in several different ways.", "topic": "percent"}
{"id": "book 2_p725_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 725, "page_end": 725, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The question stem includes a constraint that x is positive and has two digits.", "topic": "number_properties"}
{"id": "book 2_p725_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 725, "page_end": 725, "topic_guess": "number_theory", "text": "GMAT number properties note: Use the constraint as you solve the problem. Since x is a positive two-digit integer, x is between 10 and 99, inclusive.", "topic": "number_properties"}
{"id": "book 2_p725_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 725, "page_end": 725, "topic_guess": "number_theory", "text": "GMAT number properties note: The question asks whether x is divisible by 17. Since there are only a few integers in this range that are divisible by 17, jot them down for later reference: 17, 34, 51, 68, and 85.", "topic": "number_properties"}
{"id": "book 2_p725_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 725, "page_end": 725, "topic_guess": "number_theory", "text": "GMAT number properties note: The question can be rephrased as, “Is x equal to 17, 34, 51, 68, or 85?” (1) SUFFICIENT:. Positive two-digit integer x is a multiple of 7.", "topic": "number_properties"}
{"id": "book 2_p725_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 725, "page_end": 725, "topic_guess": "number_theory", "text": "GMAT number properties note: Since none of the numbers listed earlier (17, 34, 51, 68, 85) are multiples of 7, x is definitely not one of these numbers. Therefore, x is not a multiple of 17, and the answer is definitely No.", "topic": "number_properties"}
{"id": "book 2_p725_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 725, "page_end": 725, "topic_guess": "number_theory", "text": "GMAT number properties note: Alternatively, use logical reasoning to prove that the statement is sufficient without doing too much arithmetic. Seven and 17 are both prime numbers, so the smallest number that is divisible by both of them is their product, (7)(17) = 119.", "topic": "number_properties"}
{"id": "book 2_p725_c1_r7", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 725, "page_end": 725, "topic_guess": "number_theory", "text": "GMAT number properties note: However, you already know that x is smaller than 119, since x only has two digits. Therefore, x cannot be a multiple of both 7 and 17, so the answer is defini.", "topic": "number_properties"}
{"id": "book 2_p726_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 726, "page_end": 726, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Watch divisibility, factors, remainders, parity, and prime structure. It will be simpler to choose y first, since the value of y is more closely related to the 10 gallons of turquoise paint.", "topic": "number_properties"}
{"id": "book 2_p726_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 726, "page_end": 726, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Choose y = 8, so that the sum of y and 2 in the ratio of the paints used to make turquoise can be 8 + 2 = 10. In this case, it takes 8 gallons of green paint and 2 gallons of blue paint to make 10 gallons of turquoise paint.", "topic": "number_properties"}
{"id": "book 2_p726_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 726, "page_end": 726, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Because the total amount of green paint should be 8 gallons, choose x = 6, so that the of x and 2 in the ratio to make green paint is 2 + 6 = 8. In total, it will take 6 gallons of yellow paint to create the turquoise paint, so the answer to the question is 6.", "topic": "number_properties"}
{"id": "book 2_p726_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 726, "page_end": 726, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: The correct answer will simplify to 6. Erin’s list starts with an unknown positive integer and then includes every sixth integer.", "topic": "number_properties"}
{"id": "book 2_p726_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 726, "page_end": 726, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: So Erin’s list could be any of the following, depending on the value of x.", "topic": "number_properties"}
{"id": "book 2_p727_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 727, "page_end": 727, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. X Erin’s Listx Erin’s List etc.", "topic": "number_properties"}
{"id": "book 2_p727_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 727, "page_end": 727, "topic_guess": "number_theory", "text": "GMAT number properties note: Similarly, Harry’s list could be any of the following: y Harry’s List etc. If x = y = 1, then x is a multiple of y, and the number 1 appears on both lists.", "topic": "number_properties"}
{"id": "book 2_p727_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 727, "page_end": 727, "topic_guess": "number_theory", "text": "GMAT number properties note: Therefore, the answer to the question is Yes. If x = 3, all of the numbers on Erin’s list are multiples of 3.", "topic": "number_properties"}
{"id": "book 2_p727_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 727, "page_end": 727, "topic_guess": "number_theory", "text": "GMAT number properties note: If y = 1, none of the numbers on Harry’s list are multiples of 3. Therefore, x is a multiple of y, but no numbers appear on both lists, so the answer to the question is No.", "topic": "number_properties"}
{"id": "book 2_p727_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 727, "page_end": 727, "topic_guess": "number_theory", "text": "GMAT number properties note: If x – y is a multiple of 3, then x – y = 3n, for some integer n. The integers on the two lists are as follows.", "topic": "number_properties"}
{"id": "book 2_p728_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 728, "page_end": 728, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. If n is a multiple of 3, then 3n is a multiple of 9, so 3n + y appears on both lists.", "topic": "percent"}
{"id": "book 2_p728_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 728, "page_end": 728, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: If n is 1 more than a multiple of 3, then 3n + 6 = 3(n + 2) = 3(multiple of 3) is a multiple of 9, so y + 3n + 6 appears on both lists. Finally, if n is 2 more than a multiple of 3, then 3n + 12 = 3(n + 4) = 3(multiple of 3) is a multiple of 9 so y + 3n + 12 appears on both lists.", "topic": "percent"}
{"id": "book 2_p728_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 728, "page_end": 728, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Regardless of the value of n, some number appears on both Erin’s and Harry’s lists, so the answer to the question is Yes. According to the question stem, each student in the class earned a certain score on the midterm and a certain score on the final.", "topic": "percent"}
{"id": "book 2_p728_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 728, "page_end": 728, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: A student’s final score could have been higher than, equal to, or lower than his or her midterm score. To answer the question, you’ll need to determine what percent of the students fell into the first of these three categories.", "topic": "percent"}
{"id": "book 2_p728_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 728, "page_end": 728, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: It is possible that all of the remaining students scored between 1 and 5 points higher on the final than on the midterm, which would make the answer to the question 100%. It is also possible that all of the remaining students scored higher on the midterm than on the final, which would make the answer to the question 28%.", "topic": "percent"}
{"id": "book 2_p728_c1_r6", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 728, "page_end": 728, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: This statement relates the percent of students who scored higher on.", "topic": "percent"}
{"id": "book 2_p729_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 729, "page_end": 729, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. 28(total students) = 0.4(students who scored higher on the final) Simplify this equation to calculate the percent of students who scored higher on the final.", "topic": "word_problems"}
{"id": "book 2_p729_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 729, "page_end": 729, "topic_guess": "word_problems", "text": "GMAT word problem note: Since an exact answer to the question can be calculated, the statements together are sufficient. This problem provides two pieces of information.", "topic": "word_problems"}
{"id": "book 2_p729_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 729, "page_end": 729, "topic_guess": "word_problems", "text": "GMAT word problem note: The first is the ratio of the swim distance to the bike distance to the run distance: 1 to 50 to 12. The second piece of information is the ratio of the average swim time to average bike time to average run time: 3 to 10 to 6.", "topic": "word_problems"}
{"id": "book 2_p729_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 729, "page_end": 729, "topic_guess": "word_problems", "text": "GMAT word problem note: Use an unknown multiplier for each of these ratios. Distance Time Swim d 3t Bike 50d 10t Run 12d 6t.", "topic": "word_problems"}
{"id": "book 2_p730_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 730, "page_end": 730, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Now, use an RTD (rate/time/distance) chart to calculate the average speed for competitors in each of the three legs, in terms of d and t.", "topic": "word_problems"}
{"id": "book 2_p730_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 730, "page_end": 730, "topic_guess": "word_problems", "text": "GMAT word problem note: Rate Time Distance Swim 3t d Bike 10t 50d Run 6t 12d The question asks about a scenario in which the average swim time, bike time, and run time are equal, but the rates stay the same. In this scenario, the following will be the case:.", "topic": "word_problems"}
{"id": "book 2_p730_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 730, "page_end": 730, "topic_guess": "word_problems", "text": "GMAT word problem note: Use the RTD formula to fill in the missing column of this chart. Rate Time Distance Swim t Bike t 5d Run t 2d.", "topic": "word_problems"}
{"id": "book 2_p731_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 731, "page_end": 731, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. The distances are in a ratio of to 5 to 2, or 1 to 15 to 6.", "topic": "word_problems"}
{"id": "book 2_p731_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 731, "page_end": 731, "topic_guess": "word_problems", "text": "GMAT word problem note: The swim distance represents of the total distance. The problem does not specify the extent to which each person paints over the area that was already painted by the previous person.", "topic": "word_problems"}
{"id": "book 2_p731_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 731, "page_end": 731, "topic_guess": "word_problems", "text": "GMAT word problem note: For instance, when Jamaica painted 70 square feet (sq) of the wall, she might have painted over the entire 60 sq already painted by Nelson, or she might have painted over only some of it. However, by the time Bryan finished painting, the entire wall had been painted with at least one coat of paint, and some of it might have been painted with as many as three coats.", "topic": "word_problems"}
{"id": "book 2_p731_c1_r4", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 731, "page_end": 731, "topic_guess": "word_problems", "text": "GMAT word problem note: First, make q as large as possible. In order to do so, the amount of paint used on the remainder of the wall must be minimized.", "topic": "word_problems"}
{"id": "book 2_p731_c1_r5", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 731, "page_end": 731, "topic_guess": "word_problems", "text": "GMAT word problem note: The area q will be as large as possible when the remaining 100 – q square foot area is covered in only a single coat.", "topic": "word_problems"}
{"id": "book 2_p732_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 732, "page_end": 732, "topic_guess": "geometry", "text": "GMAT geometry note: The entire amount of paint applied by all three painters is 60 + 70 + 80 = 210 sq. If q sq of the wall has three coats of paint, and the remaining 100 – q has only one coat, the following is true: 3(area with 3 coats) + 1(area with 1 coat) = total paint applied Turn this into an equation and solve for q.", "topic": "geometry"}
{"id": "book 2_p732_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 732, "page_end": 732, "topic_guess": "geometry", "text": "GMAT geometry note: The greatest possible value of q is 55 sq. If the painting is done as shown below, then 55 sq of the wall will have three coats of paint and the rest will have only one coat.", "topic": "geometry"}
{"id": "book 2_p733_c1_r1", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 733, "page_end": 733, "topic_guess": "geometry", "text": "GMAT geometry note: To find the smallest possible number of square feet with three coats, do the opposite:. Assume that the rest of the wall has as much paint as possible.", "topic": "geometry"}
{"id": "book 2_p733_c1_r2", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 733, "page_end": 733, "topic_guess": "geometry", "text": "GMAT geometry note: In that case, the remaining 100 – q sq of wall will have two coats of paint, so the following will be true:. The smallest possible value of q is 10.", "topic": "geometry"}
{"id": "book 2_p733_c1_r3", "source_name": "Manhattan Prep GMAT Advanced Quant", "source_file": "book 2.pdf", "source_type": "pdf", "page_start": 733, "page_end": 733, "topic_guess": "geometry", "text": "GMAT geometry note: Here is another way to visualize the minimization of the three-coat area.", "topic": "geometry"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p1_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 1, "page_end": 1, "topic_guess": "flashcards", "text": "GMAT quant rule: These flashcards feature screenshots from GMAT Prep Now’s video lessons www.GMATPrepNow.com GMAT Prep Now Quantitative Reasoning Flashcards.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p2_c1_r1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 2, "page_end": 2, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p2_c1_r2", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 2, "page_end": 2, "topic_guess": "word_problems", "text": "GMAT word problem note: If, at any time, you’d like to watch the video related to a certain flashcard, just click on the link at the top of that page, and you’ll be taken to the corresponding video* GMAT Prep Now Quantitative Reasoning Flashcards *Many of our videos are free, but you must register an account to have access to all videos www.GMATPrepNow.com.", "topic": "word_problems"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p3_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 3, "page_end": 3, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation.", "topic": "algebra"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p4_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 4, "page_end": 4, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p5_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 5, "page_end": 5, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p6_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 6, "page_end": 6, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p7_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 7, "page_end": 7, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p8_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 8, "page_end": 8, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p9_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 9, "page_end": 9, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p10_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 10, "page_end": 10, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p11_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 11, "page_end": 11, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p12_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 12, "page_end": 12, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p13_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 13, "page_end": 13, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p14_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 14, "page_end": 14, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p15_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 15, "page_end": 15, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p16_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 16, "page_end": 16, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p17_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 17, "page_end": 17, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p18_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 18, "page_end": 18, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p19_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 19, "page_end": 19, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p20_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 20, "page_end": 20, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p21_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 21, "page_end": 21, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p22_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 22, "page_end": 22, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p23_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 23, "page_end": 23, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p24_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 24, "page_end": 24, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p25_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 25, "page_end": 25, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p26_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 26, "page_end": 26, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p27_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 27, "page_end": 27, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p28_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 28, "page_end": 28, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p29_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 29, "page_end": 29, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p30_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 30, "page_end": 30, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p31_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 31, "page_end": 31, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p32_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 32, "page_end": 32, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p33_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 33, "page_end": 33, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p34_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 34, "page_end": 34, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p35_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 35, "page_end": 35, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p36_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 36, "page_end": 36, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p37_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 37, "page_end": 37, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p38_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 38, "page_end": 38, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p39_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 39, "page_end": 39, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p40_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 40, "page_end": 40, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p41_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 41, "page_end": 41, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p42_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 42, "page_end": 42, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. (watch video) Algebra & Equation-Solving Flashcards.", "topic": "algebra"}
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{"id": "GMAT Math Flashcards from GMAT Prep Now_p110_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 110, "page_end": 110, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
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{"id": "GMAT Math Flashcards from GMAT Prep Now_p112_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 112, "page_end": 112, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p113_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 113, "page_end": 113, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p114_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 114, "page_end": 114, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p115_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 115, "page_end": 115, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p116_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 116, "page_end": 116, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p117_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 117, "page_end": 117, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p118_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 118, "page_end": 118, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p119_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 119, "page_end": 119, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p120_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 120, "page_end": 120, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
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{"id": "GMAT Math Flashcards from GMAT Prep Now_p122_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 122, "page_end": 122, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p123_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 123, "page_end": 123, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p124_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 124, "page_end": 124, "topic_guess": "flashcards", "text": "GMAT quant rule: Watch divisibility, factors, remainders, parity, and prime structure. (watch video) Integer Properties Flashcards.", "topic": "number_properties"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p125_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 125, "page_end": 125, "topic_guess": "flashcards", "text": "GMAT quant rule: Watch divisibility, factors, remainders, parity, and prime structure. (watch video) Integer Properties Flashcards.", "topic": "number_properties"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p126_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 126, "page_end": 126, "topic_guess": "flashcards", "text": "GMAT quant rule: Watch divisibility, factors, remainders, parity, and prime structure. (watch video) Integer Properties Flashcards.", "topic": "number_properties"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p127_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 127, "page_end": 127, "topic_guess": "flashcards", "text": "GMAT quant rule: Watch divisibility, factors, remainders, parity, and prime structure. (watch video) Integer Properties Flashcards.", "topic": "number_properties"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p128_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 128, "page_end": 128, "topic_guess": "flashcards", "text": "GMAT quant rule: Watch divisibility, factors, remainders, parity, and prime structure. (watch video) Integer Properties Flashcards.", "topic": "number_properties"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p129_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 129, "page_end": 129, "topic_guess": "flashcards", "text": "GMAT quant rule: Watch divisibility, factors, remainders, parity, and prime structure. (watch video) Integer Properties Flashcards.", "topic": "number_properties"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p130_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 130, "page_end": 130, "topic_guess": "flashcards", "text": "GMAT quant rule: Watch divisibility, factors, remainders, parity, and prime structure. (watch video) Integer Properties Flashcards.", "topic": "number_properties"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p131_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 131, "page_end": 131, "topic_guess": "flashcards", "text": "GMAT quant rule: Watch divisibility, factors, remainders, parity, and prime structure. (watch video) Integer Properties Flashcards.", "topic": "number_properties"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p132_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 132, "page_end": 132, "topic_guess": "flashcards", "text": "GMAT quant rule: Watch divisibility, factors, remainders, parity, and prime structure. (watch video) Integer Properties Flashcards.", "topic": "number_properties"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p133_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 133, "page_end": 133, "topic_guess": "flashcards", "text": "GMAT quant rule: Watch divisibility, factors, remainders, parity, and prime structure. (watch video) Integer Properties Flashcards.", "topic": "number_properties"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p134_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 134, "page_end": 134, "topic_guess": "flashcards", "text": "GMAT quant rule: Watch divisibility, factors, remainders, parity, and prime structure. (watch video) Integer Properties Flashcards.", "topic": "number_properties"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p135_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 135, "page_end": 135, "topic_guess": "flashcards", "text": "GMAT quant rule: Watch divisibility, factors, remainders, parity, and prime structure. (watch video) Integer Properties Flashcards.", "topic": "number_properties"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p136_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 136, "page_end": 136, "topic_guess": "flashcards", "text": "GMAT quant rule: Watch divisibility, factors, remainders, parity, and prime structure. (watch video) Integer Properties Flashcards.", "topic": "number_properties"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p137_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 137, "page_end": 137, "topic_guess": "flashcards", "text": "GMAT quant rule: Watch divisibility, factors, remainders, parity, and prime structure. (watch video) Integer Properties Flashcards.", "topic": "number_properties"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p138_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 138, "page_end": 138, "topic_guess": "flashcards", "text": "GMAT quant rule: Watch divisibility, factors, remainders, parity, and prime structure. (watch video) Integer Properties Flashcards.", "topic": "number_properties"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p139_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 139, "page_end": 139, "topic_guess": "flashcards", "text": "GMAT quant rule: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
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{"id": "GMAT Math Flashcards from GMAT Prep Now_p145_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 145, "page_end": 145, "topic_guess": "flashcards", "text": "GMAT quant rule: Separate total outcomes from desired outcomes and count carefully.", "topic": "probability"}
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{"id": "GMAT Math Flashcards from GMAT Prep Now_p147_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 147, "page_end": 147, "topic_guess": "flashcards", "text": "GMAT quant rule: Separate total outcomes from desired outcomes and count carefully.", "topic": "probability"}
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{"id": "GMAT Math Flashcards from GMAT Prep Now_p154_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 154, "page_end": 154, "topic_guess": "flashcards", "text": "GMAT quant rule: Separate total outcomes from desired outcomes and count carefully.", "topic": "probability"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p155_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 155, "page_end": 155, "topic_guess": "flashcards", "text": "GMAT quant rule: Separate total outcomes from desired outcomes and count carefully.", "topic": "probability"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p156_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 156, "page_end": 156, "topic_guess": "flashcards", "text": "GMAT quant rule: Separate total outcomes from desired outcomes and count carefully.", "topic": "probability"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p157_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 157, "page_end": 157, "topic_guess": "flashcards", "text": "GMAT quant rule: Separate total outcomes from desired outcomes and count carefully.", "topic": "probability"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p158_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 158, "page_end": 158, "topic_guess": "flashcards", "text": "GMAT quant rule: Separate total outcomes from desired outcomes and count carefully.", "topic": "probability"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p159_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 159, "page_end": 159, "topic_guess": "flashcards", "text": "GMAT quant rule: Separate total outcomes from desired outcomes and count carefully.", "topic": "probability"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p160_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 160, "page_end": 160, "topic_guess": "flashcards", "text": "GMAT quant rule: Separate total outcomes from desired outcomes and count carefully.", "topic": "probability"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p161_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 161, "page_end": 161, "topic_guess": "flashcards", "text": "GMAT quant rule: Separate total outcomes from desired outcomes and count carefully.", "topic": "probability"}
{"id": "GMAT Math Flashcards from GMAT Prep Now_p162_c1", "source_name": "GMAT Prep Now Quantitative Reasoning Flashcards", "source_file": "GMAT Math Flashcards from GMAT Prep Now.pdf", "source_type": "pdf", "page_start": 162, "page_end": 162, "topic_guess": "flashcards", "text": "GMAT quant rule: Separate total outcomes from desired outcomes and count carefully.", "topic": "probability"}
{"id": "GMAT QUANT_p1_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 1, "page_end": 1, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT QUANT_p2_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 2, "page_end": 2, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. We learn from this statement that either p or q is negative, but since we know from the question that p < q, p must be negative.", "topic": "percent"}
{"id": "GMAT QUANT_p2_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 2, "page_end": 2, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: To determine whether pqr < p, let's test values for p, q, and r. Our test values must meet only 2 conditions: p must be negative and q must be positive.", "topic": "percent"}
{"id": "GMAT QUANT_p2_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 2, "page_end": 2, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: We learn from this statement that either p or r is negative, but since we know from the question that p < r, p must be negative and r must be positive. However, q could be either positive or negative.", "topic": "percent"}
{"id": "GMAT QUANT_p2_c1_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 2, "page_end": 2, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: If we look at both statements together, we know that p is negative and that both q and r are positive. Our test values must meet 3 conditions: p must be negative, q must be positive, and r must be positive.", "topic": "percent"}
{"id": "GMAT QUANT_p2_c1_r5", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 2, "page_end": 2, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: For instance, if p = -2, q = 10, r = 5.", "topic": "percent"}
{"id": "GMAT QUANT_p2_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 2, "page_end": 2, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. If p = -2, q = 0.3 and r = 0.4, pqr = -0.24, which is NOT less than p.", "topic": "algebra"}
{"id": "GMAT QUANT_p2_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 2, "page_end": 2, "topic_guess": "algebra", "text": "GMAT algebra note: Sides of a square, number of votes, etc. When you have a positive constraint, you can:.", "topic": "algebra"}
{"id": "GMAT QUANT_p2_c2_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 2, "page_end": 2, "topic_guess": "algebra", "text": "GMAT algebra note: Eliminate negative solutions from a quadratic function Multiply or divide an inequality by a variable Cross-multiply inequalities: x/y < y/x  x2 < y2 Change an inequality sign for reciprocals: x<y. (1) T < S (2) F > G If one assumes the signs for all four variables to be positive, then T/S (an improper fraction) is NOT greater than F/G (an improper fraction).", "topic": "algebra"}
{"id": "GMAT QUANT_p2_c2_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 2, "page_end": 2, "topic_guess": "algebra", "text": "GMAT algebra note: However, if negative signs are taken into account, we cannot conclusively ascertain whether T/S > F/G.", "topic": "algebra"}
{"id": "GMAT QUANT_p3_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 3, "page_end": 3, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "GMAT QUANT_p4_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 4, "page_end": 4, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "GMAT QUANT_p5_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 5, "page_end": 5, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "GMAT QUANT_p6_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 6, "page_end": 6, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "GMAT QUANT_p6_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 6, "page_end": 6, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure.", "topic": "number_properties"}
{"id": "GMAT QUANT_p6_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 6, "page_end": 6, "topic_guess": "number_theory", "text": "GMAT number properties note: GMAT STRATEGY/TIPS 64 is the only number (besides 1) under 100 that is the square of an integer and the cube of an integer 16 and 81 are the only numbers (again, besides 1) that are both the square of an integer and the square of a square 30 and 42 are the smallest integers with at least three prime factors The only integers with exactly three factors are the squares of prime numbers When odd number n is doubled, 2n has twice as many factors as n When even number is doubled, 2n has 1.5 more factors as n All prime numbers above 3 are of the form 6n – 1 or 6n + 1, because all other numbers are divisible by 2 or 3 Xn – yn is always divisible by x – y and is divisible by x + y if n is even Finding the number of powers of a prime number P in n!", "topic": "number_properties"}
{"id": "GMAT QUANT_p6_c2_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 6, "page_end": 6, "topic_guess": "number_theory", "text": "GMAT number properties note: 25/2 + 25/4 + 25/8 + 25/16  12+ 6 + 3 + 1  22 Finding the power of non-prime in n! Determine the prime factorization the number:.", "topic": "number_properties"}
{"id": "GMAT QUANT_p6_c2_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 6, "page_end": 6, "topic_guess": "number_theory", "text": "GMAT number properties note: PF of 900 is 22*32*52 Step 2:. Find the powers of these prime numbers in n!", "topic": "number_properties"}
{"id": "GMAT QUANT_p6_c2_r5", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 6, "page_end": 6, "topic_guess": "number_theory", "text": "GMAT number properties note: (refer above) Power of 2 = 47, power of 3 = 22 and power of 5 = 12 Step 3:. We need all the prime {2,3,5} to be represented twice in 900, 5 can provide us with only 6 pairs, thus there is 900 in the power of 6 in.", "topic": "number_properties"}
{"id": "GMAT QUANT_p7_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 7, "page_end": 7, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully.", "topic": "probability"}
{"id": "GMAT QUANT_p8_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 8, "page_end": 8, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Total – Neither = A + B – X Formula for elements who are AT LEAST one of the two groups:.", "topic": "probability"}
{"id": "GMAT QUANT_p8_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 8, "page_end": 8, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: A + B – X Formula for elements who are in ONLY one of the two groups:. A + B – 2X Only in group A = a Only in group B = b Both in group A & B = X Neither = N.", "topic": "probability"}
{"id": "GMAT QUANT_p9_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 9, "page_end": 9, "topic_guess": "general", "text": "GMAT quant note: DOUBLE SET MATRIX – 2 MUTUALLY EXCLUSIVE OPTIONS FOR A DECISION For GMAT problems involving only two categorizations or decisions, the most efficient tool is the Double Set Matrix: a table whose rows correspond to the options for one decision and whose columns correspond to the options for the other decision.", "topic": "general"}
{"id": "GMAT QUANT_p9_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 9, "page_end": 9, "topic_guess": "general", "text": "GMAT quant note: The rows should correspond to the mutually exclusive options for one decision. Likewise, the columns should correspond to the mutually exclusive options for the other.", "topic": "general"}
{"id": "GMAT QUANT_p9_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 9, "page_end": 9, "topic_guess": "general", "text": "GMAT quant note: For instance, if a problem deals with students getting either right or wrong answers on problems 1 and 2, the columns should not be \"problem 1\" and \"problem 2,\" and the rows should not be \"right\" and \"wrong.\" Instead, the columns should list options for one decision-problem 1 correct, problem 1 incorrect, total-and the rows should list options for the other decision-problem 2 correct, problem 2 incorrect, total.", "topic": "general"}
{"id": "GMAT QUANT_p10_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 10, "page_end": 10, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Total – Neither = A + B + C – (a+b+c+2x) Formula for elements who are AT LEAST one of the three groups:.", "topic": "probability"}
{"id": "GMAT QUANT_p10_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 10, "page_end": 10, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: A + B + C – (a + b + c + 2x) It is important to note that “a” represents elements that belong ONLY to A and C (not B) Total = A + B + C – (a+ x + b + x + c + x) + x Formula for elements in any two of the groups: a+ b + c Note:. Work from the inside out Only A = A – a – b – x Only B = B – b – c – x Only C = C – a – c – x x a c b.", "topic": "probability"}
{"id": "GMAT QUANT_p11_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 11, "page_end": 11, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully.", "topic": "probability"}
{"id": "GMAT QUANT_p12_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 12, "page_end": 12, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread.", "topic": "statistics"}
{"id": "GMAT QUANT_p13_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 13, "page_end": 13, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread.", "topic": "statistics"}
{"id": "GMAT QUANT_p13_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 13, "page_end": 13, "topic_guess": "statistics", "text": "GMAT statistics note: MEAN – THE AVERAGE MEAN = MEDIAN in a set of consecutive integers If each one of the given numbers is increased (or decreased) by K, their average is increased (or decreased) by k If each one of some given numbers is multiplied by K, their average is multiplied by K The sum of first “n” natural numbers is given by n (n + 1)/2 If the average of a few consecutive integers is 0, then either all the numbers are zero or there will be an odd number of integers DETERMINING THE MEAN Mean = Sum of terms/ number of terms For an evenly spaced set (i.e., series where the difference between any two consecutive numbers is the same – 3, 6, 9), or a set of consecutive integers Mean = (first term + last term) / 2 Mean = middle term (if the number of terms in the set is even, then the mean is the average of the middle two terms) IMPORTANT:.", "topic": "statistics"}
{"id": "GMAT QUANT_p13_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 13, "page_end": 13, "topic_guess": "statistics", "text": "GMAT statistics note: The mean of a set of consecutive integers will be an INTEGER only if the number of elements in the set is ODD or if the difference between/sum of the “consecutive” terms is divisible by 2.", "topic": "statistics"}
{"id": "GMAT QUANT_p13_c2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 13, "page_end": 13, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. When you add the new term to the set, those 15 \"extra points\" are distributed evenly among all ten data points, increasing each by 1.5.", "topic": "statistics"}
{"id": "GMAT QUANT_p13_c3_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 13, "page_end": 13, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. When you add the new term to the set, those 15 \"extra points\" are distributed evenly among all ten data points, increasing each by 1.5 RESIDUALS Residuals = Data point – Mean For any set, the residuals sum to zero.", "topic": "statistics"}
{"id": "GMAT QUANT_p13_c3_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 13, "page_end": 13, "topic_guess": "statistics", "text": "GMAT statistics note: Alternatively, the positive residuals (\"overs\") and negative residuals (\"unders\") for any set will cancel out If the mean of the set {97, 100, 85, 90, 94, 80, 92, x} is 91, what is the value of x? You could certainly solve this problem with the traditional formula for averages, but you would waste a lot of time on arithmetic.", "topic": "statistics"}
{"id": "GMAT QUANT_p13_c3_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 13, "page_end": 13, "topic_guess": "statistics", "text": "GMAT statistics note: Instead, just use the given mean of 91 to compute the residuals for all the terms except x: +6, +9, -6, -1, +3, -11, + 1. These residuals sum to +1.Therefore, x must leave a residual of -1, since all the residuals sum to zero.", "topic": "statistics"}
{"id": "GMAT QUANT_p13_c3_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 13, "page_end": 13, "topic_guess": "statistics", "text": "GMAT statistics note: As a result, x is one less than the mean, or 90.", "topic": "statistics"}
{"id": "GMAT QUANT_p14_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 14, "page_end": 14, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. MEDIAN – THE MIDDLE TERM MEAN = MEDIAN in a set of consecutive integers.", "topic": "statistics"}
{"id": "GMAT QUANT_p14_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 14, "page_end": 14, "topic_guess": "statistics", "text": "GMAT statistics note: However, just because the mean and the median are equal does not mean that the set comprises of evenly spaced/consecutive integers Ultimately, if your mean and median are the same, what it tells you is that your data is arranged symmetrically around the median.", "topic": "statistics"}
{"id": "GMAT QUANT_p14_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 14, "page_end": 14, "topic_guess": "statistics", "text": "GMAT statistics note: For every bit over the mean on one side, there's a corresponding bit under the mean on the other side, balancing it out. When they're different, that tells you that this symmetry has been broken DETERMINING THE MEDIAN To find the median, list the numbers in ascending or descending order, and find the middle term.", "topic": "statistics"}
{"id": "GMAT QUANT_p14_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 14, "page_end": 14, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. This symmetry has been broken DETERMINING THE MEDIAN To find the median, list the numbers in ascending or descending order, and find the middle term When number of terms N is large, then the following two formulae are useful for finding the middle term:.", "topic": "statistics"}
{"id": "GMAT QUANT_p14_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 14, "page_end": 14, "topic_guess": "statistics", "text": "GMAT statistics note: When N is odd, the middle term is (N+1)/2th term When N is even, the median will be the average of the middle two terms, i.e., average of the (N/2)th term and the (N/2)th + 1 term VALUE OF THE MEDIAN When the number of items in a set of integers is odd, the median is an integer When the number of terms in a set of integers is even, the median can only end with 0.5 as it is the average of the middle two terms.", "topic": "statistics"}
{"id": "GMAT QUANT_p14_c2_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 14, "page_end": 14, "topic_guess": "statistics", "text": "GMAT statistics note: However, it is important to note that the median does not ALWAYS end in 0.5 when the number of terms in the set is even.", "topic": "statistics"}
{"id": "GMAT QUANT_p14_c3_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 14, "page_end": 14, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. S is the set of all integers from a to b, inclusive.", "topic": "statistics"}
{"id": "GMAT QUANT_p14_c3_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 14, "page_end": 14, "topic_guess": "statistics", "text": "GMAT statistics note: Q is the set of all integers from b to c, inclusive. The median of set S is (3/4)*b.", "topic": "statistics"}
{"id": "GMAT QUANT_p14_c3_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 14, "page_end": 14, "topic_guess": "statistics", "text": "GMAT statistics note: The median of set Q is (7/8)*c. If R is the set of all integers from a to c, inclusive, what fraction of c is the median of set R?", "topic": "statistics"}
{"id": "GMAT QUANT_p14_c3_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 14, "page_end": 14, "topic_guess": "statistics", "text": "GMAT statistics note: Since we the sets contain consecutive integers, the median = (largest + smallest term)/2 Therefore, equate (a+b)/2 to (3/4)*b for Set S and (b+c)/2 to (7/8)*c for Set Q.", "topic": "statistics"}
{"id": "GMAT QUANT_p15_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 15, "page_end": 15, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Track mean, median, range, distribution, and how operations change spread. WEIGHTED AVERAGE (weight1*data point 1) + (weight2*data point 2) (weight 1 + weight 2) SUM OF THE WEIGHTS The sum of the weights must equal to 1.", "topic": "statistics"}
{"id": "GMAT QUANT_p15_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 15, "page_end": 15, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: However, if the sum of the weights does not equal 1, we need to divide by the sum of the weights.", "topic": "statistics"}
{"id": "GMAT QUANT_p15_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 15, "page_end": 15, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: If we put a weight of 2 on 20 and a weight of 3 on 30, we have to divide by 5 to get weights that sum to 1, as shown below: [2(20) + 3(30)]/ 5 = (2/5)*20 + (3/5)* 30 Average = 26 ADVANCED PROPERTIES A weighted average of only two values will fall closer to whichever value is weighted more heavily For instance, if a drink is made by mixing 2 shots of a liquor containing 15% alcohol with 3 shots of a liquor containing 20% alcohol, then the alcohol content of the mixed drink will be closer to 20% than to 15%.", "topic": "statistics"}
{"id": "GMAT QUANT_p15_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 15, "page_end": 15, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread.", "topic": "statistics"}
{"id": "GMAT QUANT_p15_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 15, "page_end": 15, "topic_guess": "statistics", "text": "GMAT statistics note: Containing 15% alcohol with 3 shots of a liquor containing 20% alcohol, then the alcohol content of the mixed drink will be closer to 20% than to 15% The ratio that determines how to weight the averages of two or more subgroups in a weighted average ALSO REFLECTS the ratio of the distances from the weighted average to each subgroup’s average If the average weight of the entire group was twice as close to the average weight of the men as it was to the average weight of the women, there must be twice as many men as women In case two groups have an equal number of elements, then the weighted average of the two groups combined will be the average of the individual averages of the two groups.", "topic": "statistics"}
{"id": "GMAT QUANT_p15_c3_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 15, "page_end": 15, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Track mean, median, range, distribution, and how operations change spread. Having just the ratios of the weights will allow you to find a weighted average.", "topic": "statistics"}
{"id": "GMAT QUANT_p15_c3_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 15, "page_end": 15, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Simply write the ratio as a fraction, and use the numerator and the denominator as weights A mixture of \"lean\" ground beef (10% fat) and \"super-lean\" ground beef (4% fat) contains twice as much lean beef as super-lean beef. What is the percentage of fat in the mixture?", "topic": "statistics"}
{"id": "GMAT QUANT_p15_c3_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 15, "page_end": 15, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: The ratio of lean beef to super-lean beef is 2: 1, and so we can use 2 as the weight for the lean beef and 1 as the weight for the super-lean beef The percentage of fat in the mixture is [(10%)*2} + (4%)*1]/ (2+ 1) = 24%/3 = 8%.", "topic": "statistics"}
{"id": "GMAT QUANT_p16_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 16, "page_end": 16, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. MODE, RANGE AND STANDARD DEVIATION Mode is the term that occurs most frequently.", "topic": "statistics"}
{"id": "GMAT QUANT_p16_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 16, "page_end": 16, "topic_guess": "statistics", "text": "GMAT statistics note: There may be no mode, or more than one mode in a given series RANGE Maximum value – minimum value Range does not change when a constant value is either added to or subtracted from every term in a set Range is multiplied or divided by that constant value which is used to multiply/ divide every term in a set Range = 0 when all values are the same Range is always > = 0 Range of a set has to be equal or greater than difference between any two numbers in the set.", "topic": "statistics"}
{"id": "GMAT QUANT_p16_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 16, "page_end": 16, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread.", "topic": "statistics"}
{"id": "GMAT QUANT_p16_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 16, "page_end": 16, "topic_guess": "statistics", "text": "GMAT statistics note: Ge = 0 when all values are the same Range is always > = 0 Range of a set has to be equal or greater than difference between any two numbers in the set STANDARD DEVIATION Formula: √ (term1 – average)2 + (term2 – average)2 + (term3 – average)2 / √number of terms Rules The set in which numbers are farther away from each other than the numbers in some other set will have greater value of standard deviation A = {2,4,6,8,10} B = {3,6,9,12,15} In set B, the numbers are wider away from each other and hence will have greater value of standard deviation Standard deviation does not change when a constant value is either added to or subtracted from every term in a set Standard deviation is multiplied or divided by that constant value which is used to multiply or divide every term in a set Taking the absolute value of the elements of a set will not change the standard deviation if all elements are positive or negative If y = ax + b, and if the standard deviation of x series is ‘S’, then the standard deviation of y series will be a*s The SD of any list is not dependent on the average, but on the deviation of the numbers from the average.", "topic": "statistics"}
{"id": "GMAT QUANT_p16_c2_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 16, "page_end": 16, "topic_guess": "statistics", "text": "GMAT statistics note: So just by knowing that two lists having different averages doesn't say anything about their standard deviation - different averages can have the same SD If mean.", "topic": "statistics"}
{"id": "GMAT QUANT_p16_c3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 16, "page_end": 16, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Ything about their standard deviation - different averages can have the same SD If mean = maximum value it means that all values are equal and SD is 0 VARIANCE = STANDARD DEVIATION2.", "topic": "statistics"}
{"id": "GMAT QUANT_p17_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 17, "page_end": 17, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. FRACTIONS,DIGITS AND DECIMALS AND PERCENTAGES.", "topic": "percent"}
{"id": "GMAT QUANT_p18_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 18, "page_end": 18, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. FRACTIONS PROPER FRACTIONS Proper fractions fall between 0 and 1.", "topic": "percent"}
{"id": "GMAT QUANT_p18_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 18, "page_end": 18, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: In proper fractions, the numerator is always smaller than the denominator IMPROPER FRACTIONS In improper fractions, the numerator is always greater than the denominator. Improper fractions are therefore always greater than 1.", "topic": "percent"}
{"id": "GMAT QUANT_p18_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 18, "page_end": 18, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: As an integer and a proper fraction COMPARING FRACTIONS The traditional method of comparing fractions entails finding a common denominator and comparing the numerators The shortcut method entails multiplying the numerator of one fraction with the denominator of the other fraction.", "topic": "percent"}
{"id": "GMAT QUANT_p18_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 18, "page_end": 18, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change.", "topic": "percent"}
{"id": "GMAT QUANT_p18_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 18, "page_end": 18, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Ominator and comparing the numerators The shortcut method entails multiplying the numerator of one fraction with the denominator of the other fraction FRACTION RULES FOR POSITIVE, PROPER FRACTIONS As the numerator increases, the value of the fraction increases as it approaches 1, assuming that the denominator is held constant As the denominator increases, the value of the fraction decreases as it approaches 1, assuming that the numerator is held constant Increasing both the numerator and denominator by the same value brings the fraction closer to 1, regardless of the original value of the fraction.", "topic": "percent"}
{"id": "GMAT QUANT_p18_c3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 18, "page_end": 18, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change.", "topic": "percent"}
{"id": "GMAT QUANT_p19_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 19, "page_end": 19, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change.", "topic": "percent"}
{"id": "GMAT QUANT_p20_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 20, "page_end": 20, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change.", "topic": "percent"}
{"id": "GMAT QUANT_p21_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 21, "page_end": 21, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure.", "topic": "number_properties"}
{"id": "GMAT QUANT_p21_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 21, "page_end": 21, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: DIGITS – LAST DIGIT Last digit of (xyz)n is the same as that of Zn Determine the cyclicity number of Z Integer ending with 0, 1, 5 or 6, in the integer power k>0, has the same last digit as the base Integers ending with 2, 3, 7 and 8 have a cyclicity of 4 2 – 2, 4, 8, 6 3 – 3, 9, 7, 1 7 – 7, 9, 3, 1 8 – 8, 4, 2, 6 - Integers ending with 4 have a cyclicity of 2.", "topic": "number_properties"}
{"id": "GMAT QUANT_p21_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 21, "page_end": 21, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: When n is odd, last digit will be 4 and when n is even, last digit will be 6 - Integers ending with 9 have a cyclicity of 2. What is the last digit of 12739?", "topic": "number_properties"}
{"id": "GMAT QUANT_p21_c1_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 21, "page_end": 21, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Last digit of 12739 is the same as that of 739. First we need to determine the cyclicity of 7, which is 4.", "topic": "number_properties"}
{"id": "GMAT QUANT_p21_c1_r5", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 21, "page_end": 21, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Now divide 39 (power) by 4 (cyclicity), remainder is 3. Given that x is a positive integer, what is the units digit of 24(2x+1)* 33 (x+1)* 17(x+2)* 9(2x).", "topic": "number_properties"}
{"id": "GMAT QUANT_p21_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 21, "page_end": 21, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure.", "topic": "number_properties"}
{"id": "GMAT QUANT_p21_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 21, "page_end": 21, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Given that x is a positive integer, what is the units digit of 24(2x+1)* 33 (x+1)* 17(x+2)* 9(2x) x is a positive integer, so 2x is always even, while 2x + 1 is always odd Thus, 4(2x + 1) = (4)(odd), which always has a units digit of 4 9(2x) = 9(even), which always has a units digit of 1 That leaves us to find the units digit of (3)(x + 1)*(7)(x + 2).", "topic": "number_properties"}
{"id": "GMAT QUANT_p21_c2_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 21, "page_end": 21, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Rewriting, and dropping all but the units digit at each intermediate step, = (1)(x + 1)(7) = 7, for any value of x So, the units digit of is (4)(7)(1) = 28, then once again drop all but the units digit to get Solve the problem step by step. What is the units digit of 5*5*9*9*4*4*4?", "topic": "number_properties"}
{"id": "GMAT QUANT_p21_c2_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 21, "page_end": 21, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Multiply the last digit of each of the intermediary products 5*1*4 = 20. Units digit will therefore be 0.", "topic": "number_properties"}
{"id": "GMAT QUANT_p21_c3_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 21, "page_end": 21, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. Multiply the last digit of each of the intermediary products 5*1*4 = 20.", "topic": "number_properties"}
{"id": "GMAT QUANT_p21_c3_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 21, "page_end": 21, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Units digit will therefore be 0 Last digit of a product Last digit of a power What is the units digit of 1727? 1727 will end in the same units digit as 727 27 / 4 results in a remainder of 3.", "topic": "number_properties"}
{"id": "GMAT QUANT_p21_c3_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 21, "page_end": 21, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Therefore, the units digit of 1727 will be the same as the units digit of 73, which is 3 If x is a positive integer, what is the units digit of 24(2x+1) 33(x+1) 17(x+2) 9 (2x) Note that x is a positive integer, so 2x is always even, while 2x + 1 is always odd.", "topic": "number_properties"}
{"id": "GMAT QUANT_p21_c3_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 21, "page_end": 21, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Thus, (4)(2x + 1) = (4)(odd), which always has a units digit of 4 (9)(2x) = (9)(even), which always has a units digit of 1 That leaves us to find the units digit of (3)(x + 1)(7)(x + 2).", "topic": "number_properties"}
{"id": "GMAT QUANT_p21_c3_r5", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 21, "page_end": 21, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Rewriting, and dropping all but the units digit at each intermediate step, (3)(x + 1)(7)(x + 2) = (3)(x + 1)(7)(x + 1)(7) = (3 × 7)(x + 1)(7) = (21)(x + 1)(7) = (1)(x + 1)(7) = 7, for any value of x. So, the units digit of (4)(2x + 1)(3)(x + 1)(7)(x + 2)(9)(2x) is (4)(7)(1) = 28, then once again drop all but the units digit to get 8.", "topic": "number_properties"}
{"id": "GMAT QUANT_p22_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 22, "page_end": 22, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change.", "topic": "percent"}
{"id": "GMAT QUANT_p23_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 23, "page_end": 23, "topic_guess": "word_problems", "text": "GMAT word problem note: Percent problems usually relate part, whole, rate, and percent change. How much sugar should you add so that the mixture is 60% sugar and 40% water Always remember: the key to these problems is the component that does not change.", "topic": "percent"}
{"id": "GMAT QUANT_p23_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 23, "page_end": 23, "topic_guess": "word_problems", "text": "GMAT word problem note: In this case, we are adding sugar, while water remains constant. Therefore, we will focus on the water for most of the problem First, determine how much water is in the mixture.", "topic": "percent"}
{"id": "GMAT QUANT_p23_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 23, "page_end": 23, "topic_guess": "word_problems", "text": "GMAT word problem note: 60% of 50 is 30 ounces Because we are not adding or subtracting any water, we will still have 30 ounces of it after we add more sugar. However, we now want that 30 ounces to represent 40% of the total, rather than 60%.", "topic": "percent"}
{"id": "GMAT QUANT_p23_c1_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 23, "page_end": 23, "topic_guess": "word_problems", "text": "GMAT word problem note: X = 75 Since the increase in total volume is only made up of additional sugar, and we went from 50 total ounces to 75 total ounces, we must add 25 ounces of sugar.", "topic": "percent"}
{"id": "GMAT QUANT_p23_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 23, "page_end": 23, "topic_guess": "word_problems", "text": "GMAT word problem note: Track mean, median, range, distribution, and how operations change spread. If mixture X and mixture Y are combined, the new mixture is 15% acid.", "topic": "statistics"}
{"id": "GMAT QUANT_p23_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 23, "page_end": 23, "topic_guess": "word_problems", "text": "GMAT word problem note: What percentage of the new mixture is mixture X? (0.1X + 0.30Y)/ (X+Y) = 0.15 Solving for X and Y, we get 0.05 X = 0.15 Y X/Y = 3 / 1, which means that for every 3 parts of X, there is 1 part of Y The ratio of mixture X to the total ratio is 3/4.", "topic": "statistics"}
{"id": "GMAT QUANT_p23_c2_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 23, "page_end": 23, "topic_guess": "word_problems", "text": "GMAT word problem note: Therefore, mixture X constitutes 75% of the new mixture.", "topic": "statistics"}
{"id": "GMAT QUANT_p23_c3_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 23, "page_end": 23, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Or every 3 parts of X, there is 1 part of Y The ratio of mixture X to the total ratio is 3/4.", "topic": "percent"}
{"id": "GMAT QUANT_p23_c3_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 23, "page_end": 23, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Therefore, mixture X constitutes 75% of the new mixture How many liters of a solution that is 15 percent salt must be added to 5 liters of a solution that is 8 percent salt so that the resulting solution is 10 percent salt?", "topic": "percent"}
{"id": "GMAT QUANT_p23_c3_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 23, "page_end": 23, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: The amount of salt in the 15% solution [0.15n] plus the amount of salt in the 8% solution [(0.08)(5)] must be equal to the amount of salt in the 10% mixture [0.10 (n + 5)].", "topic": "percent"}
{"id": "GMAT QUANT_p24_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 24, "page_end": 24, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. If that price represented a 25 percent discount on the original price of the dress, what was the original price of the dress?", "topic": "percent"}
{"id": "GMAT QUANT_p24_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 24, "page_end": 24, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Solution # 1 If p is the original price of the dress, then 0.75p is the discounted price and 0.75p = $24, or p = $32. The original price of the dress was $32.", "topic": "percent"}
{"id": "GMAT QUANT_p24_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 24, "page_end": 24, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: These two discounts are equal to an overall discount of what percent? Solution # 2 If p is the original price of the item, then 0.8p is the price after the first discount.", "topic": "percent"}
{"id": "GMAT QUANT_p24_c1_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 24, "page_end": 24, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: The price after the second discount is (0.7)(0.8)p = 0.56p. This represents an overall discount of 44 percent (100% – 56%).", "topic": "percent"}
{"id": "GMAT QUANT_p24_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 24, "page_end": 24, "topic_guess": "word_problems", "text": "GMAT word problem note: Percent problems usually relate part, whole, rate, and percent change. The price after the second discount is (0.7)(0.8)p = 0.56p.", "topic": "percent"}
{"id": "GMAT QUANT_p24_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 24, "page_end": 24, "topic_guess": "word_problems", "text": "GMAT word problem note: This represents an overall discount of 44 percent (100% – 56%) SIMPLE INTEREST SI = Simple Interest, P = Principal, T = Time period in years, R = Rate of interest per annum S. = (PTR)/100 If simple interest for 2 years is 200, then S.I for 4 years is 400, for 1 year it is 100, for 10 years it is 1000.", "topic": "percent"}
{"id": "GMAT QUANT_p24_c2_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 24, "page_end": 24, "topic_guess": "word_problems", "text": "GMAT word problem note: For any length of time is known, then it can be calculated for any other length of time A man invests $1000 in a bank which pays him 10% p.a. Rate of simple interest for 2 years.", "topic": "percent"}
{"id": "GMAT QUANT_p24_c2_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 24, "page_end": 24, "topic_guess": "word_problems", "text": "GMAT word problem note: How much does the man get after 2 years? = (1000 ×2 ×10)/100 = 200 Therefore, amount after 2 years = P + S.I.", "topic": "percent"}
{"id": "GMAT QUANT_p24_c2_r5", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 24, "page_end": 24, "topic_guess": "word_problems", "text": "GMAT word problem note: = 1000 + 200 = 1200 COMPOUND INTEREST A = P × [1 + (R/100)] ^n If interest is compounded monthly, divide the annual rate by 12, and multiply n by 12 If interest is compounded quarterly, divide the annual rate by 4, and multiply n by 4 If interest is compounded semi-annually, divide the annual rate by 2, and multiply n by 2.", "topic": "percent"}
{"id": "GMAT QUANT_p25_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 25, "page_end": 25, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change.", "topic": "percent"}
{"id": "GMAT QUANT_p26_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 26, "page_end": 26, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. PERCENTAGES – 5 All of the furniture for sale at Al’s Discount Furniture is offered for less than the manufacturer’s suggested retail price.", "topic": "percent"}
{"id": "GMAT QUANT_p26_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 26, "page_end": 26, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Once a year, Al’s holds a clearance sale. If jamie purchased a certain desk during the sale, did she get a discount of more than 50% of Al’s regular price for the desk?", "topic": "percent"}
{"id": "GMAT QUANT_p26_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 26, "page_end": 26, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Al’s regular price for the desk is 60%, rounded to the nearest percent, of the MSRP of $2000 The sale price was $601 less than Al’s regular price for the desk.", "topic": "percent"}
{"id": "GMAT QUANT_p26_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 26, "page_end": 26, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Ular price for the desk is 60%, rounded to the nearest percent, of the MSRP of $2000 The sale price was $601 less than Al’s regular price for the desk In order to determine the percent discount received by Jamie, we need to know two things: the regular price of the desk and the sale price of the desk.", "topic": "percent"}
{"id": "GMAT QUANT_p26_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 26, "page_end": 26, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Alternatively, we could calculate the percent discount from the price reduction and either the regular price or the sale price (1) INSUFFICIENT:. This statement tells us the regular price of the desk at Al’s, but provides no information about how much Jamie actually paid for the desk during the annual sale.", "topic": "percent"}
{"id": "GMAT QUANT_p26_c2_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 26, "page_end": 26, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: This statement tells us how much the price of the desk was reduced during the sale, but provides no information about the regular price. On the other hand, if the regular price was $602, then the discount was nearly 100%.", "topic": "percent"}
{"id": "GMAT QUANT_p26_c2_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 26, "page_end": 26, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: At first glance, it seems that the statements together provide enough information.", "topic": "percent"}
{"id": "GMAT QUANT_p26_c3_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 26, "page_end": 26, "topic_guess": "statistics", "text": "GMAT statistics note: Percent problems usually relate part, whole, rate, and percent change. $602, then the discount was nearly 100%.", "topic": "percent"}
{"id": "GMAT QUANT_p26_c3_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 26, "page_end": 26, "topic_guess": "statistics", "text": "GMAT statistics note: This indicates that the regular price of the desk at Al’s is 60% of the MSRP, plus or minus 0.5% of the MSRP. Rather than clearly stating that the regular price is (0.60)($2000) = $1200, this statement gives a range of values for the regular price: $1200 plus or minus $10 (0.5% of 2000), or between $1190 and $1210.", "topic": "percent"}
{"id": "GMAT QUANT_p26_c3_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 26, "page_end": 26, "topic_guess": "statistics", "text": "GMAT statistics note: If the regular price was $1190, then the discount was ($601/$1190) × 100% = 50.5% (you can actually see that this is greater than 50% without calculating). If the regular price was $1210, then the discount was ($601/$1210) × 100% = 49.7% (you can actually see that this is less than 50% without calculating).", "topic": "percent"}
{"id": "GMAT QUANT_p26_c3_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 26, "page_end": 26, "topic_guess": "statistics", "text": "GMAT statistics note: The uncertainty about the regular price means that we cannot answer with certainty whether the discount was more than 50% of the regular price.", "topic": "percent"}
{"id": "GMAT QUANT_p27_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 27, "page_end": 27, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. COMMON FRACTIONS, DECIMALS AND PERCENTAGE EQUIVALENTS.", "topic": "percent"}
{"id": "GMAT QUANT_p28_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 28, "page_end": 28, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change.", "topic": "percent"}
{"id": "GMAT QUANT_p29_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 29, "page_end": 29, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Translate the wording into variables, relationships, and equations before calculating. RATIO AND PROPORTION A ratio expresses a particular relationship between two or more quantities.", "topic": "word_problems"}
{"id": "GMAT QUANT_p29_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 29, "page_end": 29, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: The two partners spend time working in the ratio of 1 to 3. For every 1 hour the first partner works, the second partner works 3 hours.", "topic": "word_problems"}
{"id": "GMAT QUANT_p29_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 29, "page_end": 29, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Three sisters invest in a certain stock in the ratio of 2 to 3 to 8. For every $2 the first sister invests, the second sister invests $3, and the third sister invests $8 The ratio of men to women in the room is 3 to 4.", "topic": "word_problems"}
{"id": "GMAT QUANT_p29_c1_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 29, "page_end": 29, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: For every 3 men, there are 4 women Ratios can express a part-part relationship or a part-whole relationship A part-part relationship:. The ratio of men to women in the office is 3:4 A part-whole relationship:.", "topic": "word_problems"}
{"id": "GMAT QUANT_p29_c1_r5", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 29, "page_end": 29, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: There are 3 men for every 7 employees Notice that if there are only two parts in the whole, you can derive a part-whole ratio from a part-part ratio, and vice versa Remember that ratios only express a relationship between two or more items. They do not provide enough information, on their own, to determine the exact quantity for each item.", "topic": "word_problems"}
{"id": "GMAT QUANT_p29_c1_r6", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 29, "page_end": 29, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: All we know is that the number of men is less than the number of women. However, ratios are surprising.", "topic": "word_problems"}
{"id": "GMAT QUANT_p30_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 30, "page_end": 30, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT QUANT_p31_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 31, "page_end": 31, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT QUANT_p32_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 32, "page_end": 32, "topic_guess": "algebra", "text": "GMAT algebra note: Watch divisibility, factors, remainders, parity, and prime structure. EXPONENTS Simplifying exponential expressions Always try to simplify exponential expressions when they have the same base or the same exponent.", "topic": "number_properties"}
{"id": "GMAT QUANT_p32_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 32, "page_end": 32, "topic_guess": "algebra", "text": "GMAT algebra note: You can only simplify exponential expressions that are linked by multiplication or division. The largest prime factor of 421 + 422 + 423 is 7 *** When both sides of the equation are broken down to the product of prime bases, the respective exponents of like bases must be equal.", "topic": "number_properties"}
{"id": "GMAT QUANT_p33_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 33, "page_end": 33, "topic_guess": "algebra", "text": "GMAT algebra note: Watch divisibility, factors, remainders, parity, and prime structure. ROOTS A ROOT is the inverse operation of raising a number to an exponent, and answers the question:.", "topic": "number_properties"}
{"id": "GMAT QUANT_p33_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 33, "page_end": 33, "topic_guess": "algebra", "text": "GMAT algebra note: Which number do I multiply by itself n times in order to get a product of b? EVEN ROOTS ONLY HAVE ONE SOLUTION – A POSITIVE VALUE Unlike even exponents, which yield both a positive and a negative solution, even roots have only one solution.", "topic": "number_properties"}
{"id": "GMAT QUANT_p33_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 33, "page_end": 33, "topic_guess": "algebra", "text": "GMAT algebra note: √18 = √50.18 = √900 = 30 √144:16 = √144: √16 = 12/4 = 3 √25+16 = √41  you cannot simplify this one 2√7 + 3√7 = 5√7.", "topic": "number_properties"}
{"id": "GMAT QUANT_p33_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 33, "page_end": 33, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. √18 = √50.18 = √900 = 30 √144:16 = √144: √16 = 12/4 = 3 √25+16 = √41  you cannot simplify this one 2√7 + 3√7 = 5√7 KNOWING COMMON ROOTS GMAT requires you to know all the perfect square roots from 1 to 30 and also the following imperfect rules:.", "topic": "algebra"}
{"id": "GMAT QUANT_p33_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 33, "page_end": 33, "topic_guess": "algebra", "text": "GMAT algebra note: SUM OF THE ROOTS OF AN EQUATION ax2+bx+c = 0 IS (-B/A).", "topic": "algebra"}
{"id": "GMAT QUANT_p34_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 34, "page_end": 34, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT QUANT_p35_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 35, "page_end": 35, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. RATE AND WORK Basic motion problems (speed, distance, time) Average rate problems Simultaneous motion problems Work problems Population problems.", "topic": "word_problems"}
{"id": "GMAT QUANT_p36_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 36, "page_end": 36, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. RATE, DISTANCE TIME RATE Rate is expressed as a ratio of distance and time, with two corresponding units.", "topic": "word_problems"}
{"id": "GMAT QUANT_p36_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 36, "page_end": 36, "topic_guess": "word_problems", "text": "GMAT word problem note: DISTANCE Distance is expressed using a unit of distance. RATE = DISTANCE / TIME RATE * TIME = DISTANCE Speed and time are inversely proportional to each other.", "topic": "word_problems"}
{"id": "GMAT QUANT_p36_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 36, "page_end": 36, "topic_guess": "word_problems", "text": "GMAT word problem note: Therefore, if a man increases his speed by 3/2, then the time taken will become 2/3*original time If three men cover the same distance with speeds in the ratio a: b: c, the times taken by these three will be in the ratio 1/a: 1/b: 1/c respectively.", "topic": "word_problems"}
{"id": "GMAT QUANT_p37_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 37, "page_end": 37, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. MULTIPLE RATE, DISTANCE TIME PROBLEMS – SPEED RELATIONS.", "topic": "word_problems"}
{"id": "GMAT QUANT_p38_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 38, "page_end": 38, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "GMAT QUANT_p39_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 39, "page_end": 39, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. MULTIPLE RATE, DISTANCE TIME PROBLEMS – THE KISS/CRASH.", "topic": "word_problems"}
{"id": "GMAT QUANT_p40_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 40, "page_end": 40, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. MULTIPLE RATE, DISTANCE TIME PROBLEMS – CATCHING UP/OVERTAKING.", "topic": "word_problems"}
{"id": "GMAT QUANT_p41_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 41, "page_end": 41, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. MULTIPLE RATE, DISTANCE TIME PROBLEMS – ROUND TRIP.", "topic": "word_problems"}
{"id": "GMAT QUANT_p42_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 42, "page_end": 42, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. MULTIPLE RATE, DISTANCE TIME PROBLEMS – SAMPLE SITUATIONS.", "topic": "word_problems"}
{"id": "GMAT QUANT_p43_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 43, "page_end": 43, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating. MISCELLANEOUS PROBLEMS BOAT PROBLEMS If the speed of a boat (or man) in still water is X km/hour, and the speed of the stream (or current) is Y km/hour, then:.", "topic": "word_problems"}
{"id": "GMAT QUANT_p43_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 43, "page_end": 43, "topic_guess": "general", "text": "GMAT quant note: Speed of boat with the stream (or Downstream or D/S) = (X + Y) km/hour Speed of boat against the stream (or upstream or U/S) = (X – Y) km/hour X = [(X + Y) + (X – Y)] / 2 and Y = [(X + Y) – (X – Y)] / 2 Boat’s speed in still water = [Speed downstream + Speed upstream] / 2 Speed of current = [Speed downstream – Speed upstream] / 2.", "topic": "word_problems"}
{"id": "GMAT QUANT_p44_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 44, "page_end": 44, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. AVERAGE RATE / RELATIVE SPEED If an object moves the same distance twice, but at different rates, then the average rate will NEVER be the average of the two rates given for the two legs of the journey.", "topic": "word_problems"}
{"id": "GMAT QUANT_p44_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 44, "page_end": 44, "topic_guess": "word_problems", "text": "GMAT word problem note: In fact, because the object spends more time traveling at the slower rate, the average rate will be closer to the slower of the two rates than to the faster It is not necessary to know the total distance travelled or the total time taken while computing the average speed When two objects travel in the same direction, the time to catch up/overtake will be = lead distance/ difference of speeds (note: in order for A to overtake B, A needs to be traveling at a higher speed than B otherwise A will never catch up) When two objects travel in different directions, time to meet will be = lead distance/ sum of speeds AVERAGE RATE = TOTAL DISTANCE/ TOTAL TIME OR AVERAGE SPEED = (2*SPEED OF A*SPEED OF B) / (SPEED A + B)  only to be used when the distance travelled is the same.", "topic": "word_problems"}
{"id": "GMAT QUANT_p45_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 45, "page_end": 45, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "GMAT QUANT_p46_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 46, "page_end": 46, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. WORK In distance problems, if the rate (speed) is known, it will normally be given to you as a ready-to-use number.", "topic": "word_problems"}
{"id": "GMAT QUANT_p46_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 46, "page_end": 46, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "GMAT QUANT_p46_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 46, "page_end": 46, "topic_guess": "word_problems", "text": "GMAT word problem note: An perform one surgery in 1.5 hours, the work rate = # of given jobs/ given amount of time i.e., 1/1.5 hours, which is equal to 2/3 surgeries per hour WORKING TOGETHER – ADD WORK RATES If A can finish a job alone in x days and B can finish the same job in y days alone, then One-day work of both A and B is (x + y)/x*y Number of days A and B take to finish the same job working together is: (x*y) / (x + y) Exception: one agent undoes the other’s work, like a pump putting water into a tank and another drawing water out.", "topic": "word_problems"}
{"id": "GMAT QUANT_p46_c2_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 46, "page_end": 46, "topic_guess": "word_problems", "text": "GMAT word problem note: Working alone, A finish a job in 2 days and B can finish a job in 3 days. If A and B work together, in how many days will they finish the work?", "topic": "word_problems"}
{"id": "GMAT QUANT_p46_c2_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 46, "page_end": 46, "topic_guess": "word_problems", "text": "GMAT word problem note: A’s rate = 1/ 2 B’s rate = 1/ 3 In one day, A and B finish 5 / 6 of the job. Four men working together all day, can finish a piece of work in 11 days.", "topic": "word_problems"}
{"id": "GMAT QUANT_p46_c2_r5", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 46, "page_end": 46, "topic_guess": "word_problems", "text": "GMAT word problem note: But two of them having other engagements can work only one half–time and quarter time respectively. How long will it take them to complete the work?", "topic": "word_problems"}
{"id": "GMAT QUANT_p46_c2_r6", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 46, "page_end": 46, "topic_guess": "word_problems", "text": "GMAT word problem note: Each man will take 11 x 4 = 44 days to complete the work. If one man works half day/day he will take 44 x 2 = 88 days to finish the work.", "topic": "word_problems"}
{"id": "GMAT QUANT_p46_c2_r7", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 46, "page_end": 46, "topic_guess": "word_problems", "text": "GMAT word problem note: Similarly, a man working quarter day/day will take 44 x 4 = 176 days to finish the work.", "topic": "word_problems"}
{"id": "GMAT QUANT_p46_c3_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 46, "page_end": 46, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "GMAT QUANT_p46_c3_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 46, "page_end": 46, "topic_guess": "word_problems", "text": "GMAT word problem note: The second set in which we need to find number of days: 2 men × 3 days × 4 hours = 3 men × N days × 2 hours 2 × 3 × 4 = 3 × N ×2 Therefore, N = 4 days If a man can do a piece of work in N days (or hours or any other unit of time), then the work done by him in one day will be 1/N of the total work If A is twice as good a workman as B, then A will take half the time B takes to finish a piece of work.", "topic": "word_problems"}
{"id": "GMAT QUANT_p47_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 47, "page_end": 47, "topic_guess": "general", "text": "GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "GMAT QUANT_p48_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 48, "page_end": 48, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure.", "topic": "number_properties"}
{"id": "GMAT QUANT_p48_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 48, "page_end": 48, "topic_guess": "number_theory", "text": "GMAT number properties note: NUMBER PROPERTIES OF INTEGERS IMPORTANT CONCEPTS TO ALWAYS KEEP IN MIND An integer can be either positive or negative, even or odd, a prime or a consonant (0 and 1 are neither prime nor consonant) Never assume a number to be an integer unless EXPLICITLY stated in the problem If n2 is an integer, it is not necessary that n is also an integer.", "topic": "number_properties"}
{"id": "GMAT QUANT_p48_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 48, "page_end": 48, "topic_guess": "number_theory", "text": "GMAT number properties note: Impact of arithmetic operations on integers When integers are added to, subtracted from, or multiplied with each other, the result is ALWAYS an integer. An odd integer divided by an even integer ALWAYS results in a non – integer.", "topic": "number_properties"}
{"id": "GMAT QUANT_p48_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 48, "page_end": 48, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. An odd integer divided by an even integer ALWAYS results in a non – integer Divisibility The product of 2 even numbers will always be divisible by 4 The product of x consecutive integers will always be divisible by x and x!.", "topic": "number_properties"}
{"id": "GMAT QUANT_p48_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 48, "page_end": 48, "topic_guess": "number_theory", "text": "GMAT number properties note: The smallest number that has all given numbers as a factor GCF:. The largest number which is a factor of all given numbers Product of two numbers = LCM * GCF.", "topic": "number_properties"}
{"id": "GMAT QUANT_p49_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 49, "page_end": 49, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT QUANT_p50_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 50, "page_end": 50, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. PRIME NUMBERS POSITIVE INTEGER greater than 1 that has EXACTLY TWO different positive factors – ONE AND ITSELF Note:.", "topic": "number_properties"}
{"id": "GMAT QUANT_p50_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 50, "page_end": 50, "topic_guess": "number_theory", "text": "GMAT number properties note: If a positive integer greater than or equal to 2 has only two factors, then the integer must be a prime number TWO is the smallest and only EVEN prime number If the product of two primes is even or if the sum of two primes is odd, one of the primes has to be 2 Conversely, if you know that 2 CANNOT be one of the primes in the sum, then the sum of the two primes must be even Method to determine a prime number Find approximate square root of the number.", "topic": "number_properties"}
{"id": "GMAT QUANT_p50_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 50, "page_end": 50, "topic_guess": "number_theory", "text": "GMAT number properties note: Then check if all the prime numbers below the square root are factors of the given number. 91 is not divisible by 2,3 or 5, but it is divisible by 7.", "topic": "number_properties"}
{"id": "GMAT QUANT_p50_c1_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 50, "page_end": 50, "topic_guess": "number_theory", "text": "GMAT number properties note: (1) x has exactly 2 factors. (2) When x is divided by 2, the remainder is 0.", "topic": "number_properties"}
{"id": "GMAT QUANT_p50_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 50, "page_end": 50, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. (1) x has exactly 2 factors.", "topic": "number_properties"}
{"id": "GMAT QUANT_p50_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 50, "page_end": 50, "topic_guess": "number_theory", "text": "GMAT number properties note: That is also insufficient by itself Taken together, however, the two statements reveal that x must be an even prime-and the only even prime number is 2. BOTH statements TOGETHER are sufficient.", "topic": "number_properties"}
{"id": "GMAT QUANT_p51_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 51, "page_end": 51, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. DIVISIBILITY RULES For any integer n, n3 – n is divisible by 3, n5 – n is divisible by 5, n11 – n is divisible by 11, n13 – n is divisible by 13.", "topic": "number_properties"}
{"id": "GMAT QUANT_p51_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 51, "page_end": 51, "topic_guess": "number_theory", "text": "GMAT number properties note: In general, if p is a prime number then for any whole number a, ap – a is divisible by p. The factorial of N, symbolized by N!, is the product of all integers from 1 up to and including N.", "topic": "number_properties"}
{"id": "GMAT QUANT_p51_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 51, "page_end": 51, "topic_guess": "number_theory", "text": "GMAT number properties note: = 6 x 5 x 4 x 3 x 2 x 1 = 720. Because it is the product of all the integers from 1 to N, any factorial N!", "topic": "number_properties"}
{"id": "GMAT QUANT_p51_c1_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 51, "page_end": 51, "topic_guess": "number_theory", "text": "GMAT number properties note: Must be divisible by all integers from 1 to N. Another way of saying this is that N!", "topic": "number_properties"}
{"id": "GMAT QUANT_p51_c1_r5", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 51, "page_end": 51, "topic_guess": "number_theory", "text": "GMAT number properties note: Is a multiple of all the integers from 1 to N.", "topic": "number_properties"}
{"id": "GMAT QUANT_p52_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 52, "page_end": 52, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. PRIME FACTORIZATION COMMON FACTOR Break down both numbers to their prime factors to see what factors they have in common.", "topic": "number_properties"}
{"id": "GMAT QUANT_p52_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 52, "page_end": 52, "topic_guess": "number_theory", "text": "GMAT number properties note: Multiply shared prime factors to find all common factors What factors greater than 1 do 135 and 225 have in common? 135 = 3 x 3 x 3 x 5 225 = 3 x 3 x 5 x 5 Both share 3 x 3 x 5 in common—find all combinations of these numbers: 3 x 3 = 9.", "topic": "number_properties"}
{"id": "GMAT QUANT_p52_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 52, "page_end": 52, "topic_guess": "number_theory", "text": "GMAT number properties note: 3 x 3 x 5 = 45 GREATEST COMMON FACTOR (GCF) The number that contains all the factors common to both numbers (i.e., biggest number that will divide into both the numbers) Note:. If the numbers do not share any common factors, then the GCF is 1 The GCF for two consecutive numbers is always 1.", "topic": "number_properties"}
{"id": "GMAT QUANT_p52_c1_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 52, "page_end": 52, "topic_guess": "number_theory", "text": "GMAT number properties note: The prime factorizations for 2940 & 3150 are: 2940 = 2 x 2 x 3 x 5 x 7 x 7 3150 = 2 x 3 x 3 x 5 x 5 x 7 Therefore, GCF is 2 x 3 x 5 x 7 = 210 LOWEST COMMON MULTIPLE (LCM) The LCM is the smallest number that contains both numbers as factors i.e.", "topic": "number_properties"}
{"id": "GMAT QUANT_p52_c1_r5", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 52, "page_end": 52, "topic_guess": "number_theory", "text": "GMAT number properties note: The smallest number that is a multiple of both these values L.C.M of two numbers x and y is maximum power of the prime factors in x and y.", "topic": "number_properties"}
{"id": "GMAT QUANT_p52_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 52, "page_end": 52, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure.", "topic": "number_properties"}
{"id": "GMAT QUANT_p52_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 52, "page_end": 52, "topic_guess": "number_theory", "text": "GMAT number properties note: The prime factorizations for 2940 & 3150 are: 2940 = 2 x 2 x 3 x 5 x 7 x 7 3150 = 2 x 3 x 3 x 5 x 5 x 7 Therefore, LCM is 2 x 2 x 3 x 3 x 5 x 5 x 7 x 7 = 44,100 LCM * GCF = PRODUCT OF TWO NUMBERS (DOES NOT WORK FOR MORE THAN TWO NUMBERS) GCF of fractions = GCF of numerators ÷ LCM of denominators LCM of fractions = LCM of numerators ÷ GCF of denominators.", "topic": "number_properties"}
{"id": "GMAT QUANT_p53_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 53, "page_end": 53, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure.", "topic": "number_properties"}
{"id": "GMAT QUANT_p54_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 54, "page_end": 54, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. FACTORS AND MULTIPLES FACTORS/DIVISORS X IS A FACTOR OF Y Essentially, a factor is a POSITIVE integer that divides evenly into an integer.", "topic": "number_properties"}
{"id": "GMAT QUANT_p54_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 54, "page_end": 54, "topic_guess": "number_theory", "text": "GMAT number properties note: Ex: 1,2,4 and 8 are all the factors of 8 One is a factor of all numbers The largest factor of a number is the number itself MULTIPLES Y IS A MULTIPLE OF X Essentially, a multiple of an integer is formed by multiplying that integer by any integer, so 8, 16,24, and 32 are some of the multiples of 8 For the purpose of the GMAT, multiples are always POSITIVE ** Divisibility can be stated in multiple ways.", "topic": "number_properties"}
{"id": "GMAT QUANT_p54_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 54, "page_end": 54, "topic_guess": "number_theory", "text": "GMAT number properties note: This just means that 12 (y) is divisible by 3 (x), and that each of the individuals has 4 items each (n) The smallest multiple of a number is the number itself Y = X*INTEGER Where x and y are both integers, and x is not equal to zero.", "topic": "number_properties"}
{"id": "GMAT QUANT_p55_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 55, "page_end": 55, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. FACTORS AND MULTIPLES – IMPORTANT CONCEPTS PERFECT SQUARES Always have an odd number of factors.", "topic": "number_properties"}
{"id": "GMAT QUANT_p55_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 55, "page_end": 55, "topic_guess": "number_theory", "text": "GMAT number properties note: Therefore, if an integer has an odd number of factors, it has to be a perfect square The prime factorization of a perfect square contains only even powers of primes. This is because perfect squares are formed from the product of two copies of the same prime factors.", "topic": "number_properties"}
{"id": "GMAT QUANT_p55_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 55, "page_end": 55, "topic_guess": "number_theory", "text": "GMAT number properties note: By contrast, if a number's prime factorization contains any odd powers of primes, then the number is not a perfect square. For instance, 132,300 = 22 x 33 X 52 X 72 is not a perfect square, because the 3 is raised to an odd power.", "topic": "number_properties"}
{"id": "GMAT QUANT_p55_c1_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 55, "page_end": 55, "topic_guess": "number_theory", "text": "GMAT number properties note: If this number is multiplied by 3, then the result, 396,900, is a perfect square: 396,900 = 22 x 34 X 52x 72 The sum of all distinct factors of a perfect square is always odd PERFECT CUBES - If a number is a perfect cube, then it is formed from three identical sets of primes, so all the powers of primes are multiples of 3 in the factorization of a perfect cube.", "topic": "number_properties"}
{"id": "GMAT QUANT_p55_c1_r5", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 55, "page_end": 55, "topic_guess": "number_theory", "text": "GMAT number properties note: For instance, 903 = (2 X 32 x 5) * (2 X 32 x 5) * (2 X 32 x 5) = 23 x 36 x 53.", "topic": "number_properties"}
{"id": "GMAT QUANT_p55_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 55, "page_end": 55, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Ers of primes are multiples of 3 in the factorization of a perfect cube.", "topic": "number_properties"}
{"id": "GMAT QUANT_p55_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 55, "page_end": 55, "topic_guess": "number_theory", "text": "GMAT number properties note: Thus, 4 is a factor of 64 + 44 = 108 and 64 – 44 = 20.", "topic": "number_properties"}
{"id": "GMAT QUANT_p55_c3_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 55, "page_end": 55, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Given that the integer n is divisible by 3, 7, and 11, what other numbers must be divisors (factors) of n?", "topic": "number_properties"}
{"id": "GMAT QUANT_p55_c3_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 55, "page_end": 55, "topic_guess": "number_theory", "text": "GMAT number properties note: Since we know that 3, 7, and 11 are prime factors of n, we know that n must also be divisible by all the possible products of the primes (3, 7 and 11): 21, 33, 77, and 231 (given that n > 231) IN ANY LIST OF N CONSECUTIVE INTEGERS, EXACTLY ONE OF THE INTEGERS IS A MULTIPLE OF N.", "topic": "number_properties"}
{"id": "GMAT QUANT_p55_c3_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 55, "page_end": 55, "topic_guess": "number_theory", "text": "GMAT number properties note: AS A RESULT, THE PRODUCT OF N CONSECUTIVE INTEGERS IS ALWAYS A MULTIPLE OF N (OR, IT IS DIVISIBLE BY N) In any list of three consecutive integers, one number (33) is a multiple of 3  33, 34, 35 Products of 113 × 114 × 115 and (y + 5)(y + 6)(y + 7) are both divisible by 3.", "topic": "number_properties"}
{"id": "GMAT QUANT_p55_c4_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 55, "page_end": 55, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Ee consecutive integers, one number (33) is a multiple of 3  33, 34, 35 Products of 113 × 114 × 115 and (y + 5)(y + 6)(y + 7) are both divisible by 3 THE NUMBER OF FACTORS A NUMBER HAS (INCLUDING 1 AND THE NUMBER ITSELF) CAN BE COMPUTED BY FOLLOWING THE STEPS DETAILED BELOW:.", "topic": "number_properties"}
{"id": "GMAT QUANT_p55_c4_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 55, "page_end": 55, "topic_guess": "number_theory", "text": "GMAT number properties note: First, determine all the prime factors of the number through LCM Second, determine the number of copies of each of the prime factors and express them in the following format - p^a * q^b *....", "topic": "number_properties"}
{"id": "GMAT QUANT_p56_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 56, "page_end": 56, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. QUOTIENTS AND REMAINDERS Y = (QUOTIENT * DIVISOR) + REMAINDER.", "topic": "number_properties"}
{"id": "GMAT QUANT_p57_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 57, "page_end": 57, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure.", "topic": "number_properties"}
{"id": "GMAT QUANT_p58_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 58, "page_end": 58, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure.", "topic": "number_properties"}
{"id": "GMAT QUANT_p59_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 59, "page_end": 59, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure.", "topic": "number_properties"}
{"id": "GMAT QUANT_p60_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 60, "page_end": 60, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure.", "topic": "number_properties"}
{"id": "GMAT QUANT_p60_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 60, "page_end": 60, "topic_guess": "number_theory", "text": "GMAT number properties note: EVEN AND ODD INTEGERS ONLY integers can be even/odd An odd integer can be expressed as 2n + 1 and an even integer can be expressed as 2n Sum of two different prime numbers will ALWAYS be EVEN, unless one of the numbers is 2 If two even integers are multiplied, the result will be divisible by 4, and if three even integers are multiplied, the result will be divisible by 8 etc.", "topic": "number_properties"}
{"id": "GMAT QUANT_p60_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 60, "page_end": 60, "topic_guess": "number_theory", "text": "GMAT number properties note: If x is even, and N = (x)(x+1)(x+2), is N divisible by 24? The terms x and x+2 must each have a factor of 2, and the product (x)(x+1)(x+2) must have factors 1, 2, 3 and 4 Therefore, the prime factorization of N must be a multiple of 1*2*3*2*2 = 24.", "topic": "number_properties"}
{"id": "GMAT QUANT_p61_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 61, "page_end": 61, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. POSITIVE AND NEGATIVE INTEGERS ZERO is neither positive nor negative – be very careful!", "topic": "number_properties"}
{"id": "GMAT QUANT_p62_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 62, "page_end": 62, "topic_guess": "number_theory", "text": "GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Average the first and last term to find the precise \"middle\" of the set  (100+20)/2 = 60 Count the number of terms: 100 - 20 = 80, plus 1 yields 81 Multiply the \"middle\" number by the number of terms to find the sum: 60 x 81 = 4,860.", "topic": "number_properties"}
{"id": "GMAT QUANT_p63_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 63, "page_end": 63, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation.", "topic": "algebra"}
{"id": "GMAT QUANT_p64_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 64, "page_end": 64, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. BASIC EQUATIONS (THOSE WITHOUT EXPONENTS) SIMULTANEOUS EQUATIONS – THREE VARIABLES Look for ways to simplify the work.", "topic": "algebra"}
{"id": "GMAT QUANT_p64_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 64, "page_end": 64, "topic_guess": "algebra", "text": "GMAT algebra note: What is the sum of x, y and z? Y + z = 7 In this case, DO NOT try to solve for x, y, and z individually.", "topic": "algebra"}
{"id": "GMAT QUANT_p64_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 64, "page_end": 64, "topic_guess": "algebra", "text": "GMAT algebra note: Instead, notice the symmetry of the equations-each one adds exactly two of the variables-and add them all together. Therefore, 2x + 2y + 2z = 26.", "topic": "algebra"}
{"id": "GMAT QUANT_p64_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 64, "page_end": 64, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Ce the symmetry of the equations-each one adds exactly two of the variables-and add them all together.", "topic": "algebra"}
{"id": "GMAT QUANT_p64_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 64, "page_end": 64, "topic_guess": "algebra", "text": "GMAT algebra note: Therefore, 2x + 2y + 2z = 26. However, the question does NOT ask you to solve for all three variables.", "topic": "algebra"}
{"id": "GMAT QUANT_p64_c2_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 64, "page_end": 64, "topic_guess": "algebra", "text": "GMAT algebra note: It only asks you to solve for x, which IS possible.", "topic": "algebra"}
{"id": "GMAT QUANT_p64_c3_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 64, "page_end": 64, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. N gives us a value for 3y + 5z, which we can substitute into the first equation in order to solve for x.", "topic": "algebra"}
{"id": "GMAT QUANT_p64_c3_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 64, "page_end": 64, "topic_guess": "algebra", "text": "GMAT algebra note: Thus, BOTH statements TOGETHER are sufficient COMBO PROBLEMS – MANIPULATION The GMAT often asks you to solve for a combination of variables, called COMBO problems. In these cases, since you are not asked to solve for one specific variable, you should generally NOT try to solve for the individual variables right away.", "topic": "algebra"}
{"id": "GMAT QUANT_p64_c3_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 64, "page_end": 64, "topic_guess": "algebra", "text": "GMAT algebra note: Instead, you should try to manipulate the given equation(s) so that the COMBO is isolated on one side of the equation. There are four easy manipulations that are the key to solving most COMBO problems:.", "topic": "algebra"}
{"id": "GMAT QUANT_p64_c3_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 64, "page_end": 64, "topic_guess": "algebra", "text": "GMAT algebra note: Multiply or divide the whole equation by a certain number A:. Add or subtract a number on both sides of the equation D:.", "topic": "algebra"}
{"id": "GMAT QUANT_p64_c3_r5", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 64, "page_end": 64, "topic_guess": "algebra", "text": "GMAT algebra note: Distribute or factor an expression on ONE side of the equation. Then, if the other side of an equation from a statement contains a VALUE, that equation is.", "topic": "algebra"}
{"id": "GMAT QUANT_p65_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 65, "page_end": 65, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. ABSOLUTE VALUE EQUATIONS Absolute value refers to the POSITIVE value of the expression within the absolute value brackets.", "topic": "algebra"}
{"id": "GMAT QUANT_p65_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 65, "page_end": 65, "topic_guess": "algebra", "text": "GMAT algebra note: Equations that involve absolute value generally have TWO SOLUTIONS. In other words, there are TWO numbers that the variable could equal in order to make the equation true.", "topic": "algebra"}
{"id": "GMAT QUANT_p65_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 65, "page_end": 65, "topic_guess": "algebra", "text": "GMAT algebra note: The reason is that the value of the expression inside the absolute value brackets could be POSITIVE OR NEGATIVE. For instance, if we know |x| = 5, then x could be either 5 or -5, and the equation would still be true If |x| > x then x must be negative The following three-step method should be used when solving for a variable expression inside absolute value brackets.", "topic": "algebra"}
{"id": "GMAT QUANT_p65_c1_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 65, "page_end": 65, "topic_guess": "algebra", "text": "GMAT algebra note: Solve for w, given that 12 + |w – 4| = 30. Isolate the expression within the absolute value brackets: |w – 4| = 18 Remove the absolute value brackets and solve the equation for 2 different cases (positive and negative) w – 4 = 18.", "topic": "algebra"}
{"id": "GMAT QUANT_p65_c1_r5", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 65, "page_end": 65, "topic_guess": "algebra", "text": "GMAT algebra note: Check to see whether each solution is valid by putting each one back into the original equation and verifying that the two sides of the equation are in fact equal. The possibility of a failed solution is a peculiarity of absolute value equations.", "topic": "algebra"}
{"id": "GMAT QUANT_p65_c1_r6", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 65, "page_end": 65, "topic_guess": "algebra", "text": "GMAT algebra note: For most other types of equations, it is good to check your solutions, b.", "topic": "algebra"}
{"id": "GMAT QUANT_p66_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 66, "page_end": 66, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation.", "topic": "algebra"}
{"id": "GMAT QUANT_p67_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 67, "page_end": 67, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT QUANT_p68_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 68, "page_end": 68, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT QUANT_p69_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 69, "page_end": 69, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. INEQUALITIES COMBINING INEQUALITIES Many GMAT inequality problems involve more than one inequality.", "topic": "statistics"}
{"id": "GMAT QUANT_p69_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 69, "page_end": 69, "topic_guess": "statistics", "text": "GMAT statistics note: First, solve any inequalities that need to be solved.", "topic": "statistics"}
{"id": "GMAT QUANT_p69_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 69, "page_end": 69, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Ferably to the left (less than): 8 < x, x <17, x < 14 Third, combine the inequalities by taking the more limiting upper and lower extremes: 8 < x < 14 Another helpful approach is to combine inequalities by adding the inequalities together.", "topic": "algebra"}
{"id": "GMAT QUANT_p69_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 69, "page_end": 69, "topic_guess": "algebra", "text": "GMAT algebra note: That would be a mistake, however, because both m and n could be negative numbers that yield a number larger than 10 when multiplied together. The two statements together are INSUFFICIENT Consider the following variation: if both m and n are positive, is mn < 10 given that (1) m < 2 (2) n < 5.", "topic": "algebra"}
{"id": "GMAT QUANT_p69_c2_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 69, "page_end": 69, "topic_guess": "algebra", "text": "GMAT algebra note: Since the variables are positive, we can multiply these inequalities together and conclude that mn < 10.", "topic": "algebra"}
{"id": "GMAT QUANT_p69_c3_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 69, "page_end": 69, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Since the variables are positive, we can multiply these inequalities together and conclude that mn < 10.", "topic": "algebra"}
{"id": "GMAT QUANT_p69_c3_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 69, "page_end": 69, "topic_guess": "algebra", "text": "GMAT algebra note: You can perform operations on a compound inequality as long as you remember to perform those operations on every term in the inequality, not just the outside terms.", "topic": "algebra"}
{"id": "GMAT QUANT_p69_c4_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 69, "page_end": 69, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. In order to add inequalities, we must make sure the inequality signs are facing the same direction.", "topic": "algebra"}
{"id": "GMAT QUANT_p69_c4_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 69, "page_end": 69, "topic_guess": "algebra", "text": "GMAT algebra note: One way to see this clearly is by realizing that only the right side of the equation contains the variable d. Perhaps ab2 – b is greater than b2c – d simply because of the magnitude of d.", "topic": "algebra"}
{"id": "GMAT QUANT_p69_c4_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 69, "page_end": 69, "topic_guess": "algebra", "text": "GMAT algebra note: Therefore there is no way to draw any conclusions about the relationship between a and c. By adding the two inequalities from statements (1) and (2) together, we can come to the conclusion that a > c.", "topic": "algebra"}
{"id": "GMAT QUANT_p69_c4_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 69, "page_end": 69, "topic_guess": "algebra", "text": "GMAT algebra note: Two inequalities can always be added together as long as the direction of the inequality signs is the same: ab2 – b > b2c – d (+) b > d.", "topic": "algebra"}
{"id": "GMAT QUANT_p70_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 70, "page_end": 70, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. INEQUALITIES – USING EXTREME VALUES One effective technique for solving GMAT inequality problems is to focus on the EXTREME VALUES of a given inequality.", "topic": "algebra"}
{"id": "GMAT QUANT_p70_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 70, "page_end": 70, "topic_guess": "algebra", "text": "GMAT algebra note: This is particularly helpful when solving the following types of inequality problems:. Problems with multiple inequalities where the question involves the potential range of values for variables in the problem Problems involving both equations and inequalities.", "topic": "algebra"}
{"id": "GMAT QUANT_p71_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 71, "page_end": 71, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. INEQUALITIES AND ABSOLUTE VALUES Solving inequalities √x2 = |x| It is often helpful to try to visualize the problem with a number line.", "topic": "algebra"}
{"id": "GMAT QUANT_p71_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 71, "page_end": 71, "topic_guess": "algebra", "text": "GMAT algebra note: One way to understand this inequality is to say \"x must be less than 5 units from zero on the number line.\" Indeed, one interpretation of absolute value is simply distance on the number line. For a simple absolute value expression such as lx|, we are evaluating distance from zero.", "topic": "algebra"}
{"id": "GMAT QUANT_p71_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 71, "page_end": 71, "topic_guess": "algebra", "text": "GMAT algebra note: The range therefore, would be -5 < x < 5 Equations involving absolute value require you to consider two scenarios: one where the expression inside the absolute value brackets is positive, and one where the expression is negative. The same is true for inequalities.", "topic": "algebra"}
{"id": "GMAT QUANT_p71_c1_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 71, "page_end": 71, "topic_guess": "algebra", "text": "GMAT algebra note: Note that you should never change I x – 5 |to x + 5. Remember, when you drop the absolute value signs, you either leave the expression alone or enclose the ENTIRE expression in parentheses and put a negative sign in front.", "topic": "algebra"}
{"id": "GMAT QUANT_p71_c1_r5", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 71, "page_end": 71, "topic_guess": "algebra", "text": "GMAT algebra note: Given that | x – 2 | < 5, what is the range of possible values of x? To work out the FIRST scenario, we simply remove the absolute value brackets and solve: x – 2 < 5.", "topic": "algebra"}
{"id": "GMAT QUANT_p71_c1_r6", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 71, "page_end": 71, "topic_guess": "algebra", "text": "GMAT algebra note: X < 7 To work out the SECOND scenario, we reverse the signs of the terms inside the absolute value brackets, remove the brackets, and solve again: - (x - 2) < 5.", "topic": "algebra"}
{"id": "GMAT QUANT_p74_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 74, "page_end": 74, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. The constant factor is called the common ratio and is formed by dividing any term by the term which precedes it To answer a question about a sequence, we not only need to know the rule the sequence follows, but also at least one term in the sequence.", "topic": "algebra"}
{"id": "GMAT QUANT_p75_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 75, "page_end": 75, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Should be taken as a – 3d, a – d, a + d, a + 3d A series of quantities is said to be in harmonic progression when their reciprocals are in A.P.", "topic": "algebra"}
{"id": "GMAT QUANT_p76_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 76, "page_end": 76, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. PROGRESSIONS AND SEQUENCES – GEOMETRIC PROGRESSION Geometric progression is a series of numbers in which each term is formed from the preceding by multiplying it by a constant factor (r).", "topic": "algebra"}
{"id": "GMAT QUANT_p76_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 76, "page_end": 76, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: The constant factor is called the common ratio and is formed by dividing any term by the term which precedes it. Should be taken as a/r, a, ar Four numbers in a G.P.", "topic": "algebra"}
{"id": "GMAT QUANT_p76_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 76, "page_end": 76, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Should be taken as a/r3, a/r, ar, ar3.", "topic": "algebra"}
{"id": "GMAT QUANT_p77_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 77, "page_end": 77, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Focus on equations, expressions, variables, and step-by-step manipulation. FUNCTIONS The \"domain\" of a function indicates the possible inputs and the \"range\" of a function indicates the possible outputs.", "topic": "algebra"}
{"id": "GMAT QUANT_p77_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 77, "page_end": 77, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: For instance, the function f(x) = x2 can take any input but never produces a negative number. So the domain is all numbers, but the range is f(x) > = O COMPOUND FUNCTIONS – work from the INSIDE out If f(x) = x3 + √x and g(x) = 4x – 3, what is f(g(3))?", "topic": "algebra"}
{"id": "GMAT QUANT_p77_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 77, "page_end": 77, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: First, solve g(3): 4.3 – 3 = 9.", "topic": "algebra"}
{"id": "GMAT QUANT_p77_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 77, "page_end": 77, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Focus on equations, expressions, variables, and step-by-step manipulation. First, solve g(3): 4.3 – 3 = 9.", "topic": "algebra"}
{"id": "GMAT QUANT_p77_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 77, "page_end": 77, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Now, plug-in 9 on f(x): 93 + √9 = 729 + 3 DETERMINING WHETHER A FUNCTION IS EVEN/ODD - To do this, you take the function and plug –x in for x, and then simplify. If you end up with the exact same function that you started with (that is, if f(–x) = f(x), so all of the signs are the same), then the function is even.", "topic": "algebra"}
{"id": "GMAT QUANT_p77_c2_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 77, "page_end": 77, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: If you end up with the exact opposite of what you started with (that is, if f(–x) = –f(x), so all of the \"plus\" signs become \"minus\" signs, and vice versa), then the function is odd. So I'll plug –x in for x, and simplify:.", "topic": "algebra"}
{"id": "GMAT QUANT_p77_c2_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 77, "page_end": 77, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: Determine algebraically whether f(x) = –3x2 + 4 is even, odd, or neither f(–x) = –3(–x)2 + 4 = –3(x2) + 4 = –3x2 + 4 Since the new expression is the same as the initial expression, the function is even.", "topic": "algebra"}
{"id": "GMAT QUANT_p77_c3_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 77, "page_end": 77, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Translate the wording into variables, relationships, and equations before calculating. N, odd, or neither f(–x) = –3(–x)2 + 4 = –3(x2) + 4 = –3x2 + 4 Since the new expression is the same as the initial expression, the function is even COMMON FUNCTION TYPES i) Direct proportionality means that the two quantities always change by the same factor and in the same direction.", "topic": "word_problems"}
{"id": "GMAT QUANT_p77_c3_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 77, "page_end": 77, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: For instance, tripling the input will cause the output to triple as well. Direct proportionality relationships are of the form Y = kx, where x is the input value, y is the output value and k is the proportionality constant.", "topic": "word_problems"}
{"id": "GMAT QUANT_p77_c3_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 77, "page_end": 77, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: The maximum height reached by an object thrown directly upward is directly proportional to the square of the velocity with which the object is thrown. If an object thrown upward at 16 feet per second reaches a maximum height of 4 feet, with what speed must the object be thrown upward to reach a maximum height of 9 feet?", "topic": "word_problems"}
{"id": "GMAT QUANT_p77_c3_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 77, "page_end": 77, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Typically with direct proportion problems, you will be given \"before\" and \"after\" values. We then write y1/x1 = y2/x2, since both ratios are equal to the same constant k.", "topic": "word_problems"}
{"id": "GMAT QUANT_p77_c3_r5", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 77, "page_end": 77, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: In the problem given above, note that the direct proportion is between.", "topic": "word_problems"}
{"id": "GMAT QUANT_p78_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 78, "page_end": 78, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully.", "topic": "probability"}
{"id": "GMAT QUANT_p79_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 79, "page_end": 79, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. COMBINATORICS PERMUTATION ORDER MATTERS - a permutation is nothing but an ordered combination.", "topic": "probability"}
{"id": "GMAT QUANT_p79_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 79, "page_end": 79, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Essentially, permutation questions are about taking a group of objects and totaling the number of ways in which they can be arranged in specific/pre-defined ways The number of permutations of r items, chosen from a pool of n items, is n! The number of permutations when things are not all different:.", "topic": "probability"}
{"id": "GMAT QUANT_p79_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 79, "page_end": 79, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: If there be n things, p of them of one kind, q of another kind, r of still another kind and so on, then the total number of permutations is given by n! R!...) As a special case, the number of permutations of all n items in a pool of n items is just n!, and the number of ways to arrange n distinct objects along a fixed circle is: (n – 1)!", "topic": "probability"}
{"id": "GMAT QUANT_p79_c1_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 79, "page_end": 79, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: COMBINATION ORDER DOES NOT MATTER (i.e., whether the events are taking place simultaneously or sequentially). Since the order does not matter, we need to remove all the “duplicate” cases The number of combinations of r items, chosen from a pool of n items, is n!", "topic": "probability"}
{"id": "GMAT QUANT_p79_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 79, "page_end": 79, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Does not matter, we need to remove all the “duplicate” cases The number of combinations of r items, chosen from a pool of n items, is n!", "topic": "probability"}
{"id": "GMAT QUANT_p79_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 79, "page_end": 79, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Figure out the combinations from each group and then multiply them together.", "topic": "probability"}
{"id": "GMAT QUANT_p79_c3_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 79, "page_end": 79, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Combinations are being drawn from several groups to form a complete set.", "topic": "probability"}
{"id": "GMAT QUANT_p79_c3_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 79, "page_end": 79, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: If each player shakes hands with every other player NOT on his own team, how many handshakes take place? How many people take part in a handshake?", "topic": "probability"}
{"id": "GMAT QUANT_p79_c3_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 79, "page_end": 79, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: There are 35 people who could be in the first spot, but that person cannot shake hands with anyone on his own team. So that person has only 30 people who’s hands he can shake.", "topic": "probability"}
{"id": "GMAT QUANT_p79_c3_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 79, "page_end": 79, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Therefore, the number of combinations is (35*30)/2.", "topic": "probability"}
{"id": "GMAT QUANT_p80_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 80, "page_end": 80, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. COMBINATIONS One concept that you need to know for the exam is that when dealing with combinations and permutations, each result corresponds to a unique set of circumstances.", "topic": "probability"}
{"id": "GMAT QUANT_p80_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 80, "page_end": 80, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: No other value of z would yield exactly 15 different groups of two. So if you know how many subgroups of a certain size you can choose from an unknown original larger group, you can deduce the size of the larger group Seating 5 people in 3 chairs, is the same as seating 3 people in 5 chairs.", "topic": "probability"}
{"id": "GMAT QUANT_p81_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 81, "page_end": 81, "topic_guess": "word_problems", "text": "GMAT word problem note: Separate total outcomes from desired outcomes and count carefully. If you have 4 types of bread, 3 types of cheese and 2 types of ham and wish to make a sandwich, you can make it in 4*3*2 = 24 different ways.", "topic": "probability"}
{"id": "GMAT QUANT_p81_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 81, "page_end": 81, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. How many codes are possible?", "topic": "number_properties"}
{"id": "GMAT QUANT_p81_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 81, "page_end": 81, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: In this problem, you do not want to choose the five digits in order. Instead, you should start by picking the first and last digits (which must be odd), because these digits are the most restricted.", "topic": "number_properties"}
{"id": "GMAT QUANT_p81_c2_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 81, "page_end": 81, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Because there are 5 different odd digits (1, 3, 5, 7, and 9), there are 5 ways of picking the first digit. Since no repetition is allowed, there are only 4 odd digits left for the last digit.", "topic": "number_properties"}
{"id": "GMAT QUANT_p81_c2_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 81, "page_end": 81, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: You can then pick the other three digits in any order, but make sure you account for the lack of repetition. For those three choices, you have only 8, 7, and 6 digits available.", "topic": "number_properties"}
{"id": "GMAT QUANT_p81_c2_r5", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 81, "page_end": 81, "topic_guess": "sequences_patterns", "text": "GMAT sequences and patterns note: Therefore, the total number of lock codes is 4 x 5 x 8 x 7 x 6 = 6,720.", "topic": "number_properties"}
{"id": "GMAT QUANT_p81_c3_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 81, "page_end": 81, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. For those three choices, you have only 8, 7, and 6 digits available.", "topic": "statistics"}
{"id": "GMAT QUANT_p81_c3_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 81, "page_end": 81, "topic_guess": "statistics", "text": "GMAT statistics note: Therefore, the total number of lock codes is 4 x 5 x 8 x 7 x 6 = 6,720 SIMPLE FACTORIALS The number of ways in which n different things can be arranged in a straight line, such that no one thing is allowed to appear more than once in any of the arrangements (i.e. When repetitions are not allowed) is n!.", "topic": "statistics"}
{"id": "GMAT QUANT_p81_c3_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 81, "page_end": 81, "topic_guess": "statistics", "text": "GMAT statistics note: In how many different orders can she arrange the books? Using the Fundamental Counting Principle, we see that we have 6 choices for the book that goes first, 5 choices for the book that goes next, and so forth.", "topic": "statistics"}
{"id": "GMAT QUANT_p81_c3_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 81, "page_end": 81, "topic_guess": "statistics", "text": "GMAT statistics note: Ultimately, we have this total: 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720 different orders.", "topic": "statistics"}
{"id": "GMAT QUANT_p81_c4_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 81, "page_end": 81, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Hat goes first, 5 choices for the book that goes next, and so forth.", "topic": "statistics"}
{"id": "GMAT QUANT_p81_c4_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 81, "page_end": 81, "topic_guess": "statistics", "text": "GMAT statistics note: Ultimately, we have this total: 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720 different orders ANAGRAMS An anagram is a rearrangement of the letters in a word or phrase.", "topic": "statistics"}
{"id": "GMAT QUANT_p81_c4_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 81, "page_end": 81, "topic_guess": "statistics", "text": "GMAT statistics note: For instance, the word DEDUCTIONS is an anagram of DISCOUNTED, and so is the gibberish \"word“ CDDEINOSTU Anagrams with repeated words The number of anagrams of a word is the factorial of the total number of letters, divided by the factorial(s) corresponding to each set of repeated letters.", "topic": "statistics"}
{"id": "GMAT QUANT_p81_c5_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 81, "page_end": 81, "topic_guess": "statistics", "text": "GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Possible combinations from the 7 letters in the word PIZZAZZ.", "topic": "statistics"}
{"id": "GMAT QUANT_p81_c5_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 81, "page_end": 81, "topic_guess": "statistics", "text": "GMAT statistics note: However, there are only 210. Why are there, relatively speaking, so few anagrams?", "topic": "statistics"}
{"id": "GMAT QUANT_p81_c5_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 81, "page_end": 81, "topic_guess": "statistics", "text": "GMAT statistics note: The answer lies in repetition: the four Z's are indistinguishable from each other. If the four Z's were all different letters, then we would have 7!", "topic": "statistics"}
{"id": "GMAT QUANT_p81_c5_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 81, "page_end": 81, "topic_guess": "statistics", "text": "GMAT statistics note: To picture that scenario, imagine labeling the Z's with subscripts:. We could then list the 5,040 anagram.", "topic": "statistics"}
{"id": "GMAT QUANT_p81_c5_r5", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 81, "page_end": 81, "topic_guess": "statistics", "text": "GMAT statistics note: Now erase the subscripts from those 5,040 anagrams. You will notice that many \"different“ arrangements-like AIPZ1Z2Z3Z4, AIPZ4Z2Z3Z1' and so on-are now the same.", "topic": "statistics"}
{"id": "GMAT QUANT_p81_c5_r6", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 81, "page_end": 81, "topic_guess": "statistics", "text": "GMAT statistics note: In fact, for any genuinely unique anagram-like AIPZZZZ-there are now 4! = 24 identical copies in the list of 5,040 anagrams, because there are 4!", "topic": "statistics"}
{"id": "GMAT QUANT_p81_c5_r7", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 81, "page_end": 81, "topic_guess": "statistics", "text": "GMAT statistics note: = 24 ways to rearrange the four Z's in the word PIZZAZZ without changing anything. Because this 24-fold repetition occurs for every unique anagram of PIZZAZZ, we take 7!", "topic": "statistics"}
{"id": "GMAT QUANT_p81_c5_r8", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 81, "page_end": 81, "topic_guess": "statistics", "text": "GMAT statistics note: (which counts the arrangements as if the letters were all distinct) and divide by 4!", "topic": "statistics"}
{"id": "GMAT QUANT_p82_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 82, "page_end": 82, "topic_guess": "word_problems", "text": "GMAT word problem note: Separate total outcomes from desired outcomes and count carefully. SOLVING COMBINATORICS PROBLEMS – ADVANCED STRATEGIES.", "topic": "probability"}
{"id": "GMAT QUANT_p83_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 83, "page_end": 83, "topic_guess": "word_problems", "text": "GMAT word problem note: Separate total outcomes from desired outcomes and count carefully. SOLVING COMBINATORICS PROBLEMS – ADVANCED STRATEGIES.", "topic": "probability"}
{"id": "GMAT QUANT_p84_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 84, "page_end": 84, "topic_guess": "word_problems", "text": "GMAT word problem note: Separate total outcomes from desired outcomes and count carefully. SOLVING COMBINATORICS PROBLEMS – ADVANCED STRATEGIES.", "topic": "probability"}
{"id": "GMAT QUANT_p85_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 85, "page_end": 85, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. PROBABILITY PROBABILITY = NUMBER OF FAVORABLE OUTCOMES TOTAL NUMBER OF OUTCOMES PROBABILITY OF MULTIPLE EVENTS AND If two events have to occur together, generally an \"and\" is used.", "topic": "probability"}
{"id": "GMAT QUANT_p85_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 85, "page_end": 85, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: If a coin is tossed twice, what is the probability that on the first toss the coin lands heads and on the second toss the coin lands tails? The probability that the coin will land on heads is 1/2.", "topic": "probability"}
{"id": "GMAT QUANT_p85_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 85, "page_end": 85, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: The probability that the coin will land on Tails is also 1/2. Since we want them to happen together, we multiply individual probabilities 1/2 × 1/2 = 1/4 OR If both events do not necessarily have to occur together, an “or” may be used.", "topic": "probability"}
{"id": "GMAT QUANT_p85_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 85, "page_end": 85, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. John will win $100 if, from a deck of 52 standard playing cards, he chooses either a 7 or a 9 when pulling a single card from the deck.", "topic": "probability"}
{"id": "GMAT QUANT_p85_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 85, "page_end": 85, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: What is the probability that John will win $100? He can win by pulling out either a 7 or a 9.", "topic": "probability"}
{"id": "GMAT QUANT_p85_c2_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 85, "page_end": 85, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: His chances of doing that are higher than if he could win only by pulling out a 7. In that case, he’d only have 4 cards that would make him win $100 (because there are 4 7's in a standard deck), now he has 8 cards.", "topic": "probability"}
{"id": "GMAT QUANT_p85_c2_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 85, "page_end": 85, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: To find the total probability, we need to figure out the probability of each event and then add them together. Drawing from a pack of cards without replacing the card drawn MUTUALLY EXCLUSIVE EVENTS.", "topic": "probability"}
{"id": "GMAT QUANT_p85_c3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 85, "page_end": 85, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Watch divisibility, factors, remainders, parity, and prime structure.", "topic": "number_properties"}
{"id": "GMAT QUANT_p86_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 86, "page_end": 86, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. PROBABILITY – 2 The greatest probability- the certainty that an event will occur-is 1.", "topic": "probability"}
{"id": "GMAT QUANT_p86_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 86, "page_end": 86, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Thus, a probability of 1 means that the event must occur The lowest probability-the impossibility that an event will occur is 0. Thus, a probability of 0 means that an event will NOT occur.", "topic": "probability"}
{"id": "GMAT QUANT_p86_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 86, "page_end": 86, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: What is the probability that, on three rolls of a single fair die, at least of the rolls will be a six? What is the probability that NONE of the rolls will yield a 6?", "topic": "probability"}
{"id": "GMAT QUANT_p86_c1_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 86, "page_end": 86, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: On each roll, there is a 5/6 probability that the die will not yield a 6. Thus, the probability that on all 3 rolls the die will not yield a 6 is (5/6)*(5/6)*(5*6) = 125/216 Now, we originally defined success as rolling at least one six.", "topic": "probability"}
{"id": "GMAT QUANT_p86_c1_r5", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 86, "page_end": 86, "topic_guess": "sets_probability_counting", "text": "GMAT sets, probability, and counting note: Since we have found the probability of failure, we answer the original question by subtracting this probability from 1: 1 – (125/216) = 91/256.", "topic": "probability"}
{"id": "GMAT QUANT_p88_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 88, "page_end": 88, "topic_guess": "geometry", "text": "GMAT geometry note: LINES AND ANGLES Two straight lines which meet at a point form an angle between them Acute angle is between 0 and 90 degrees Right angle is 90 degrees Obtuse angle is between 90 degrees and 180 degrees Reflex angle is between 180 degrees and 360 degrees A straight line is the shortest distance between two points and measures 180 degrees Parallel lines are lines that never intersect, no matter how far they are stretched, and are equidistant Perpendicular lines intersect at 90 degrees (may or may not bisect the line segment) The sum of angles at a point is 360 degrees INTERSECTING LINES EXTERIOR ANGLES OF A TRIANGLE.", "topic": "geometry"}
{"id": "GMAT QUANT_p89_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 89, "page_end": 89, "topic_guess": "geometry", "text": "GMAT geometry note: LINES AND ANGLES PARALLEL LINES CUT BY A TRANSVERSAL GMAT can disguise the lines, by not drawing them completely. Extending them can help you to find out whether they are transversal or parallel.", "topic": "geometry"}
{"id": "GMAT QUANT_p89_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 89, "page_end": 89, "topic_guess": "geometry", "text": "GMAT geometry note: GMAT uses the symbol II to indicate 2 lines are parallel. MN II OP means those segments are parallel Properties Alternate interior angles are equal Angles 3 and 6 are equal.", "topic": "geometry"}
{"id": "GMAT QUANT_p89_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 89, "page_end": 89, "topic_guess": "geometry", "text": "GMAT geometry note: Angles 4 and 5 are equal Alternate exterior angles are equal Angles 1 and 8 are equal.", "topic": "geometry"}
{"id": "GMAT QUANT_p89_c1_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 89, "page_end": 89, "topic_guess": "geometry", "text": "GMAT geometry note: Angles 2 and 7 are equal Corresponding angles are equal (corresponding angles are one interior and one exterior angle that are on the same side of the transversal) Angles 1 and 5 Angles 2 and 6 Angles 3 and 7 Angles 4 and 8 It is important to note that if two lines cut by a transversal have any of the above properties, then the two lines must be parallel.", "topic": "geometry"}
{"id": "GMAT QUANT_p90_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 90, "page_end": 90, "topic_guess": "geometry", "text": "GMAT geometry note: TWO – DIMENSIAL POLYGONS TWO DIMENSIONAL POLYGONS TRIANGLES – 3 SIDED QUADRILATERALS – 4 SIDED OTHERS – N SIDED Equilateral Isosceles Scalene Pentagon – 5 sides Hexagon – 6 sides Comprised of 6 equilateral triangles.", "topic": "geometry"}
{"id": "GMAT QUANT_p90_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 90, "page_end": 90, "topic_guess": "geometry", "text": "GMAT geometry note: Therefore, the area of a hexagon will be 6 * area of an equilateral triangle Octagon – 8 sides Irregular quadrilaterals Parallelogram Rectangle Rhombus Square All quadrilaterals are not necessarily parallelograms.", "topic": "geometry"}
{"id": "GMAT QUANT_p90_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 90, "page_end": 90, "topic_guess": "geometry", "text": "GMAT geometry note: For a quadrilateral to be a parallelogram opposite sides and opposite angles have to be equal and parallel Trapezium SUM OF THE INTERIOR ANGLES OF A POLYGON – This depends on the number of sides (n) a polygon has: (n-2)*180 SUM OF THE EXTERIOR ANGLES OF A POLYGON = 360.", "topic": "geometry"}
{"id": "GMAT QUANT_p91_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 91, "page_end": 91, "topic_guess": "geometry", "text": "GMAT geometry note: TRIANGLES EQUILATERAL ISOSCELES SCALENE TYPES OF TRIANGLES – SIDES TWO SIDES ARE EQUAL AND THE THIRD SIDE IS LESS THAN THE SUM OF THE OTHER TWO SIDES All THREE SIDES ARE EQUAL ALL THREE SIDES ARE DIFFERENT ACUTE OBTUSE RIGHT – ANGLED TYPES OF TRIANGLES – ANGLES ONE ANGLE IS GREATER THAN 90 DEGREES In a triangle ABC with the vertex at B, AB2 + BC2 < AC2 ALL ANGLES ARE LESS THAN 90 DEGREES In a triangle ABC with the vertex at B, AB2 + BC2 > AC2 ONE ANGLE IS 90 DEGREES The 45-45-90 Triangle is known as a right-angled isosceles triangle In a triangle ABC with the vertex at B, AB2 + BC2 = AC2.", "topic": "geometry"}
{"id": "GMAT QUANT_p94_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 94, "page_end": 94, "topic_guess": "geometry", "text": "GMAT geometry note: BASIC PROPERTIES OF TRIANGLES – B SUM OF THE ANGLES OF A TRIANGLE = 180 DEGREES When one side is extended in any direction, an angle is formed with another side. This is called the exterior angle.", "topic": "geometry"}
{"id": "GMAT QUANT_p94_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 94, "page_end": 94, "topic_guess": "geometry", "text": "GMAT geometry note: There are six exterior angles in a triangle ANGLES CORRESPOND TO THEIR OPPOSITE SIDE This means that the largest angle is opposite the longest side, while the smallest angle is opposite the shortest side.", "topic": "geometry"}
{"id": "GMAT QUANT_p94_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 94, "page_end": 94, "topic_guess": "geometry", "text": "GMAT geometry note: Additionally, if two angles are equal, their sides are also equal SUM OF ANY TWO SIDES OF A TRIANGLE MUST BE GREATER THAN THE THIRD SIDE DIFFERENCE BETWEEN ANY TWO SIDES IS LESS THAN THE THIRD SIDE THE LENGTH OF THE THIRD SIDE MUST LIE BETWEEN THE SUM AND THE DIFFERENCE OF THE TWO GIVEN SIDES.", "topic": "geometry"}
{"id": "GMAT QUANT_p95_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 95, "page_end": 95, "topic_guess": "geometry", "text": "GMAT geometry note: SIMILAR AND CONGRUENT TRIANGLES Similarity – if any of the conditions are met – Angle, angle, angle – if the corresponding angles of each triangle have the same measurement. In other words, the above triangles are similar if:.", "topic": "geometry"}
{"id": "GMAT QUANT_p95_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 95, "page_end": 95, "topic_guess": "geometry", "text": "GMAT geometry note: Angle M = Angle P Side, angle, side - An angle in one triangle is the same measurement as an angle in the other triangle and the two sides containing these angles have the same ratio. Side LM/Side OP = Side LN/Side OQ Side, side, side - Each pair of corresponding sides have the same ratio.", "topic": "geometry"}
{"id": "GMAT QUANT_p95_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 95, "page_end": 95, "topic_guess": "geometry", "text": "GMAT geometry note: Side LM/Side OP = Side LN/Side OQ = Side MN/Side PQ.", "topic": "geometry"}
{"id": "GMAT QUANT_p96_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 96, "page_end": 96, "topic_guess": "geometry", "text": "GMAT geometry note: RIGHT- ANGLED TRIANGLE AND THE PYTHAGOREAN THEOREM Pythagorean Theorem: c2 = a2 + b2 a and b are the legs, while c is the hypotenuse THE 45-45-90 ISOSCELES TRIANGLE The 45-45-90 triangle is a special triangle with 2 equal sides and a relation between each side. If you are given one dimension on a 45-45-90 triangle, you can find the others.", "topic": "geometry"}
{"id": "GMAT QUANT_p96_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 96, "page_end": 96, "topic_guess": "geometry", "text": "GMAT geometry note: Relationship between sides 45° - 45° - 90° Leg - leg - hypotenuse x: x: x√2 Therefore, in a 45-45-90 degree triangle, you only need to know the value of one side to determine the others A 45-45-90 is exactly half of a square. Two 45-45-90 triangles put together make up a square.", "topic": "geometry"}
{"id": "GMAT QUANT_p96_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 96, "page_end": 96, "topic_guess": "geometry", "text": "GMAT geometry note: So, if you are given the diagonal of a square, you can find the side by using the relation above.", "topic": "geometry"}
{"id": "GMAT QUANT_p97_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 97, "page_end": 97, "topic_guess": "geometry", "text": "GMAT geometry note: HYPOTENUSE HYPOTENUSE EQUILATERAL TRIANGLES AND THE 30-60-90 TRIANGLE An equilateral triangle is one in which all three sides and all three angles are equal Equilateral Triangles and the 30-60-90 Triangle:.", "topic": "geometry"}
{"id": "GMAT QUANT_p98_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 98, "page_end": 98, "topic_guess": "geometry", "text": "GMAT geometry note: INSCRIBED AND CIRCUMSCRIBED TRIANGLES AND CIRCLES A right triangle inscribed in a semi- circle must have its hypotenuse as the diameter of the circle. The reverse is also true: if the diameter of the circle is also the triangle’s hypotenuse, then that triangle is a right triangle.", "topic": "geometry"}
{"id": "GMAT QUANT_p99_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 99, "page_end": 99, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT QUANT_p100_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 100, "page_end": 100, "topic_guess": "geometry", "text": "GMAT geometry note: QUADRILATERALS How to prove a quadrilateral is a parallelogram If both pairs of opposite sides of a quadrilateral are equal If both pairs of opposite angles of a quadrilateral are equal If all pairs of consecutive angles of a quadrilateral are supplementary If one pair of opposite sides of a quadrilateral is both equal and parallel If the diagonals of a quadrilateral bisect each other The area of a parallelogram is twice the area of a triangle created by one of its diagonals Of all quadrilaterals with a given perimeter, the square has the largest area.", "topic": "geometry"}
{"id": "GMAT QUANT_p100_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 100, "page_end": 100, "topic_guess": "geometry", "text": "GMAT geometry note: Conversely, of all the quadrilaterals with a given area, the square is the one with the smaller perimeter The diagonal formed by joining vertices of two smaller angles will be greater than the diagonal formed by joining vertices of two greater angles in a rhombus The diagonals of a kite are perpendicular bisectors of each other.", "topic": "geometry"}
{"id": "GMAT QUANT_p100_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 100, "page_end": 100, "topic_guess": "geometry", "text": "GMAT geometry note: The kite also has two pairs of adjacent, congruent sides (i.e., all sides are not equal)– THE KITE IS NOT A PARALLELOGRAM.", "topic": "geometry"}
{"id": "GMAT QUANT_p101_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 101, "page_end": 101, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT QUANT_p102_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 102, "page_end": 102, "topic_guess": "geometry", "text": "GMAT geometry note: QUADRILATERALS TRAPEZIOD One pair of opposite sides is parallel, but not equal Area = 1/2 * (a+b)*h Isosceles Trapezoid What is the area of the trapezoid as show? Angle A = 120 degrees The perimeter of the trapezoid ABCD = 36 The area of a trapezoid is equal to the average of the bases multiplied by the height.", "topic": "geometry"}
{"id": "GMAT QUANT_p102_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 102, "page_end": 102, "topic_guess": "geometry", "text": "GMAT geometry note: In this problem, you are given the top base (AB = 6), but not the bottom base (CD) or the height. (Note: 8 is NOT the height!) In order to find the area, you will need a way to figure out this missing data.", "topic": "geometry"}
{"id": "GMAT QUANT_p102_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 102, "page_end": 102, "topic_guess": "geometry", "text": "GMAT geometry note: Drop 2 perpendicular lines from points A and B to the horizontal base CD, and label the points at which the lines meet the base E and F.", "topic": "geometry"}
{"id": "GMAT QUANT_p102_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 102, "page_end": 102, "topic_guess": "geometry", "text": "GMAT geometry note: Drop 2 perpendicular lines from points A and B to the horizontal base CD, and label the points at which the lines meet the base E and F. The congruent symbols in the drawing tell you that Angle A and Angle B are congruent, and that Angle C and Angle D are congruent.", "topic": "geometry"}
{"id": "GMAT QUANT_p102_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 102, "page_end": 102, "topic_guess": "geometry", "text": "GMAT geometry note: This tells you that AC = BD and CE = FD. Therefore, since the sum of all 4 angles must yield 360 (which is the total number of degrees in any four-sided polygon), we know that Angle B = 120, Angle C = 60, and Angle D = 60.", "topic": "geometry"}
{"id": "GMAT QUANT_p102_c2_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 102, "page_end": 102, "topic_guess": "geometry", "text": "GMAT geometry note: This means that triangle ACE and triangle BDF are both 30-60-90 triangles. The relationship among the sides of a 30-60-90 triangle is in the ratio of x: x √3: 2x, where x is the shortest side.", "topic": "geometry"}
{"id": "GMAT QUANT_p102_c2_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 102, "page_end": 102, "topic_guess": "geometry", "text": "GMAT geometry note: The same measurements hold for triangle BFD. Thus we have the length of the bottom base (4 + 6 + 4) and the height and we can calculate the area of the trapezoid.", "topic": "geometry"}
{"id": "GMAT QUANT_p102_c3_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 102, "page_end": 102, "topic_guess": "geometry", "text": "GMAT geometry note: Asurements hold for triangle BFD. We already know that the lengths of sides AB (6), AC (8), and BD (8) sum to 22.", "topic": "geometry"}
{"id": "GMAT QUANT_p102_c3_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 102, "page_end": 102, "topic_guess": "geometry", "text": "GMAT geometry note: Further, since EF = 6, we can determine that CE = FD = 4. From this information, we can work with either Triangle ACE or Triangle BDF, and use the Pythagorean theorem to figure out the height of the trapezoid.", "topic": "geometry"}
{"id": "GMAT QUANT_p102_c3_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 102, "page_end": 102, "topic_guess": "geometry", "text": "GMAT geometry note: Now, knowing the lengths of both bases, and the height, we can calculate the area of the trapezoid EACH STATEMENT ALONE IS SUFFICIENT.", "topic": "geometry"}
{"id": "GMAT QUANT_p103_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 103, "page_end": 103, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT QUANT_p104_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 104, "page_end": 104, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT QUANT_p105_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 105, "page_end": 105, "topic_guess": "geometry", "text": "GMAT geometry note: CIRCLES Circumference of a circle = 2πr Circumference of a semi-circle = πr Perimeter of a semi-circle = πr + D (Diameter of the circle) Area of a circle = πr2 Area of a semi-circle = πr2 / 2 Length of the arc = (θ/360) × (2 π r) Area of the sector = (θ/360) × πr2 Distance travelled by a wheel in n revolutions = n x circumference Perimeter of a sector = (θ/360) × (2 π r) + 2r Circles, when graphed on the coordinate plane, have an equation of x2 + y2 = r2 where r is the radius (standard form) when the center of the circle is the origin.", "topic": "geometry"}
{"id": "GMAT QUANT_p105_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 105, "page_end": 105, "topic_guess": "geometry", "text": "GMAT geometry note: When the center of the circle is (h, k) and the radius is of length r, the equation of a circle (standard form) is (x - h)2 + (y - k)2 = r2.", "topic": "geometry"}
{"id": "GMAT QUANT_p106_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 106, "page_end": 106, "topic_guess": "geometry", "text": "GMAT geometry note: CIRCLES, ARCS AND SECTORS A circle is a set of points that are equidistant from a fixed point called the centre. RADIUS: distance from the centre of the circle to any point on the circle.", "topic": "geometry"}
{"id": "GMAT QUANT_p106_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 106, "page_end": 106, "topic_guess": "geometry", "text": "GMAT geometry note: Can be shown as a line segment connecting the centre to a point on the circle – OC DIAMETER: is a line segment that connects two points on the circle and goes through the centre of the circle - AE Diameter = 2r CHORD: is any line segment whose endpoints are any two points on the circle – BD The tangent line and the radius of the circle that has an endpoint at the point of tangency are perpendicular to each other – AO IS PERPENDICULAR TO XY ARC – a portion/distance on the circle A C B E O D Y X A B O C.", "topic": "geometry"}
{"id": "GMAT QUANT_p107_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 107, "page_end": 107, "topic_guess": "geometry", "text": "GMAT geometry note: INSCRIBED CIRCLES When a circle is inscribed inside a square, the side equals the diameter.", "topic": "geometry"}
{"id": "GMAT QUANT_p109_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 109, "page_end": 109, "topic_guess": "geometry", "text": "GMAT geometry note: CO-ORDINATE GEOMETRY – GENERAL CONCEPTS CO-ORDINATE POINTS Points in the plane are identified by using an ordered pair of numbers, (x, y), where x is the x co-ordinate and y is the y co-ordinate.", "topic": "geometry"}
{"id": "GMAT QUANT_p109_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 109, "page_end": 109, "topic_guess": "geometry", "text": "GMAT geometry note: The point where a line intersects a co-ordinate axis is called an intercept X intercept: the point at which the line intersects the x axis The x-intercept is expressed using the ordered pair (x, 0), where x is the point where the line intersects the x-axis.", "topic": "geometry"}
{"id": "GMAT QUANT_p109_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 109, "page_end": 109, "topic_guess": "geometry", "text": "GMAT geometry note: The x-intercept is the point on the line at which y = O Y intercept: the point at which the line intersects the y axis The y-intercept is expressed using the ordered pair (0, y), where y is the point where the line intersects the y-axis. The y-intercept is the point on the line at which x = O.", "topic": "geometry"}
{"id": "GMAT QUANT_p109_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 109, "page_end": 109, "topic_guess": "algebra", "text": "GMAT algebra note: Ssed using the ordered pair (0, y), where y is the point where the line intersects the y-axis.", "topic": "geometry"}
{"id": "GMAT QUANT_p109_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 109, "page_end": 109, "topic_guess": "algebra", "text": "GMAT algebra note: The y-intercept is the point on the line at which x = O LINE A line can be formed in any of the three ways By knowing the co-ordinates of any two points on the line By knowing the co-ordinates of one point on the line and the slope of the line By knowing the equation of the line SLOPE OF A LINE:.", "topic": "geometry"}
{"id": "GMAT QUANT_p109_c2_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 109, "page_end": 109, "topic_guess": "algebra", "text": "GMAT algebra note: Y2 – Y1 / X2 – X1 A line in the plane is formed by the connection of two or more points. Therefore, in the equation y = 4x + 8.", "topic": "geometry"}
{"id": "GMAT QUANT_p109_c2_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 109, "page_end": 109, "topic_guess": "algebra", "text": "GMAT algebra note: The slope is 8 If the equation of the line is given in the general form ax + by + c = 0. Then the slope is –a/b and the intercept is –c/b.", "topic": "geometry"}
{"id": "GMAT QUANT_p109_c3_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 109, "page_end": 109, "topic_guess": "algebra", "text": "GMAT algebra note: The slope is 8 If the equation of the line is given in the general form ax + by + c = 0. Then the slope is –a/b and the intercept is –c/b The slope of all points/ co-ordinates on a line will ALWAYS be the same.", "topic": "geometry"}
{"id": "GMAT QUANT_p109_c3_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 109, "page_end": 109, "topic_guess": "algebra", "text": "GMAT algebra note: However, just because two points have the same slope, doesn’t necessarily mean that the lie on the same line and vice versa. For instance, parallel lines have the same slope.", "topic": "geometry"}
{"id": "GMAT QUANT_p109_c3_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 109, "page_end": 109, "topic_guess": "algebra", "text": "GMAT algebra note: Conversely, if two lines have the same slope they could either be parallel or lie on each other If the slope is 1, the angle formed by the slope is 45 degrees Parallel lines have equal slopes Perpendicular lines have negative reciprocal slopes SLOPE DIRECTION - There are four types of slope an equation can have Positive – rises upward from left to right Negative – falls downward from left to right Zero – parallel to the x axis Undefined – parallel to the y axis Every line (but the one that crosses the origin OR the one that is parallel to X or Y axis OR the X and Y axis themselves) crosses three quadrants.", "topic": "geometry"}
{"id": "GMAT QUANT_p110_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 110, "page_end": 110, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. CO-ORDINATE GEOMETRY – EQUATION OF A LINE/ SLOPE-INTERCEPT LINE What is the slope-intercept form for a line with the equation 6x + 3y = 18?", "topic": "algebra"}
{"id": "GMAT QUANT_p110_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 110, "page_end": 110, "topic_guess": "algebra", "text": "GMAT algebra note: Instead, they are expressed as simple, one-variable equations. Horizontal lines are expressed in the form: y = some number, such as y = 3 or y = 5 (i.e., equation of the line parallel to the x axis is 3 or 5).", "topic": "algebra"}
{"id": "GMAT QUANT_p110_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 110, "page_end": 110, "topic_guess": "algebra", "text": "GMAT algebra note: Vertical lines are expressed in the form: x = some number, such as x = 4 or x = 7 (i.e., equation of the line parallel to the y axis is 4 or 7) Equation of the X axis is Y = 0 Equation of the Y axis is X = 0.", "topic": "algebra"}
{"id": "GMAT QUANT_p111_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 111, "page_end": 111, "topic_guess": "word_problems", "text": "GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating.", "topic": "word_problems"}
{"id": "GMAT QUANT_p111_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 111, "page_end": 111, "topic_guess": "word_problems", "text": "GMAT word problem note: Draw a right triangle connecting the points Find the lengths of the two legs of the triangle by calculating the rise and the run The y-coordinate changes from 3 to -5, a difference of 8 (the vertical leg) The x-coordinate changes from 1 to 7, a difference of 6 (the horizontal leg) Use the Pythagorean Theorem to calculate the length of the diagonal, which is the distance between the points 62 + 82 = c2 36 + 64 = c2 100 = c2 c = 10 POSITIVE AND NEGATIVE QUADRANTS Quadrant I: x and y are both positive Quadrant II: x is negative, y is positive Quadrant III: x is negative, y is negative Quadrant IV: x is positive, y is negative.", "topic": "word_problems"}
{"id": "GMAT QUANT_p111_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 111, "page_end": 111, "topic_guess": "word_problems", "text": "GMAT word problem note: Nd y are both positive Quadrant II: x is negative, y is positive Quadrant III: x is negative, y is negative Quadrant IV: x is positive, y is negative The GMAT sometimes asks you to determine which quadrants a given line passes through. Which quadrants does the line 2x + Y = 5 pass through?", "topic": "geometry"}
{"id": "GMAT QUANT_p111_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 111, "page_end": 111, "topic_guess": "word_problems", "text": "GMAT word problem note: 2x + y = 5 y = 5 – 2x Since b = 5, the y-intercept is the point (0, 5). The slope is -2, so the line slopes downward steeply to the right from the y-intercept.", "topic": "geometry"}
{"id": "GMAT QUANT_p111_c2_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 111, "page_end": 111, "topic_guess": "word_problems", "text": "GMAT word problem note: If Point A is (5,20) and Point B is (30,7), define a line through point C (12,10) that is parallel to the line that passes through both A and B Slope of AB = (7-20)/(30-5)  -13/25  -0.52 Substituting in equation y-y1 = m(x-x1)  y = -0.52(x – 12) + 10 y = -0.52x + 16.24 Distance between two parallel lines y = mx + b and y = mx + c can be found by the formula D = | b – c |/ √m2 + 1.", "topic": "geometry"}
{"id": "GMAT QUANT_p112_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 112, "page_end": 112, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: CO-ORDINATE GEOMETRY – 4 PERPENDICULAR BISECTORS The perpendicular bisector of a line segment forms a 90° angle with the segment and divides the segment exactly in HALF. The key to solving perpendicular bisector problems is remembering this property: the perpendicular bisector has the negative reciprocal slope of the line segment it bisects.", "topic": "geometry"}
{"id": "GMAT QUANT_p112_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 112, "page_end": 112, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: If the coordinates of point A are (2, 2) and the coordinates of point B are (0, -2), what is the equation of the perpendicular bisector of line segment AB? Slope of AB is 2 Slope of the perpendicular bisector is – (1/2) Now we know that the equation of the perpendicular bisector has the following form: y = (-1/2)x + b.", "topic": "geometry"}
{"id": "GMAT QUANT_p112_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 112, "page_end": 112, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: However, we still need to find the value of b To do this, we will need to find one point on the perpendicular bisector The perpendicular bisector passes through the midpoint of AB. Thus, if we find the midpoint of AB, we will have found a point on the perpendicular bisector.", "topic": "geometry"}
{"id": "GMAT QUANT_p112_c1_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 112, "page_end": 112, "topic_guess": "coordinate_geometry", "text": "GMAT coordinate geometry note: The x co-ordinate of the perpendicular bisector will be the average of the x co-ordinates of A and B. Similarly, the y co-ordinate of the perpendicular bisector of AB.", "topic": "geometry"}
{"id": "GMAT QUANT_p112_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 112, "page_end": 112, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Dinate of the perpendicular bisector of AB will be the average of the y co-ordinates of A and B.", "topic": "algebra"}
{"id": "GMAT QUANT_p112_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 112, "page_end": 112, "topic_guess": "algebra", "text": "GMAT algebra note: Plug in the values of x (1) and y (0) to get b = 1/2 INTERSECTION OF TWO LINES Recall that a line in the coordinate plane is defined by a linear equation relating x and y. That is, if a point (x, y) lies on the line, then those values of x and y satisfy the equation.", "topic": "algebra"}
{"id": "GMAT QUANT_p112_c2_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 112, "page_end": 112, "topic_guess": "algebra", "text": "GMAT algebra note: For instance, the point (3, 2) lies on the line defined by the equation y = 4x - 10, since the equation is true when we plug in x = 3 and y = 2. On the other hand, the point (7, 5) does not lie on that line, because the equation is false when we plug in x = 7 and y = 5 So, what does it mean when two lines intersect in the coordinate plane?", "topic": "algebra"}
{"id": "GMAT QUANT_p112_c2_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 112, "page_end": 112, "topic_guess": "algebra", "text": "GMAT algebra note: It means that at the point of intersection, BOTH equations representing the lines are true. That is, the pair of numbers (x, y) that represents the point of intersection solves BOTH equations.", "topic": "algebra"}
{"id": "GMAT QUANT_p112_c2_r5", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 112, "page_end": 112, "topic_guess": "algebra", "text": "GMAT algebra note: At what point does the line represented by y = 4x - 10 intersect the line represented by 2x + 3y = 26? Solving the equations simultaneously, we get x = 4 and y = 6 Therefore, the point of intersection is (4,6) If two lines in a plane do not intersect, then the lines are parallel.", "topic": "algebra"}
{"id": "GMAT QUANT_p112_c3_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 112, "page_end": 112, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation.", "topic": "algebra"}
{"id": "GMAT QUANT_p112_c3_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 112, "page_end": 112, "topic_guess": "algebra", "text": "GMAT algebra note: In this case, infinitely many points (x, y) along the line satisfy the two equations (which must actually be the same equation in two disguises) The coordinates of the point P(x, y) which divides the line segment joining the points A(x1, y1) and B(x2, y2), internally, in the ratio m:n are (nx1 + mx2)/m+n and (ny1 + my2)/m+n.", "topic": "algebra"}
{"id": "GMAT QUANT_p113_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 113, "page_end": 113, "topic_guess": "geometry", "text": "GMAT geometry note: ADVANCED CONCEPTS EQUATION OF A CIRCLE The equation of a circle with centre (h, k) and radius r is (x – h)2 + (y – k)2 = r2 The equation of a circle with centre (0, 0) and radius r is x2 + y2 = r2 In the coordinate plane, a circle has center (2, -3) and passes through the point (5, 0). What is the area of the circle?", "topic": "geometry"}
{"id": "GMAT QUANT_p113_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 113, "page_end": 113, "topic_guess": "geometry", "text": "GMAT geometry note: (5 – 2)2 + (0 – (-3))2 = r2 (3)2 + (3)2 = r2 18 = r2 Therefore, area = 18 π Find the equation of a circle whose center is at (4, 2) and is tangent to y-axis Since the circle is tangent to y-axis, the radius of the circle is perpendicular to y-axis.", "topic": "geometry"}
{"id": "GMAT QUANT_p113_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 113, "page_end": 113, "topic_guess": "geometry", "text": "GMAT geometry note: It also means that the length of the radius is also the length of the perpendicular segment from the center of the circle to y –axis. The point of tangency is at (0, 2).", "topic": "geometry"}
{"id": "GMAT QUANT_p113_c1_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 113, "page_end": 113, "topic_guess": "geometry", "text": "GMAT geometry note: To find the length of the radius, use the distance formula: (0 – 4)2 + (2 – 2)2 = r2 r = 4.", "topic": "geometry"}
{"id": "GMAT QUANT_p113_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 113, "page_end": 113, "topic_guess": "geometry", "text": "GMAT geometry note: The point of tangency is at (0, 2). To find the length of the radius, use the distance formula: (0 – 4)2 + (2 – 2)2 = r2 r = 4 Write the equation of the circle with the given condition: (10, 8) and (4, -2) are the endpoints of the diameter In a circle, the radius is one-half of the diameter.", "topic": "geometry"}
{"id": "GMAT QUANT_p113_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 113, "page_end": 113, "topic_guess": "geometry", "text": "GMAT geometry note: Since the given are the endpoints of the diameter, the center of the circle is the midpoint of the diameter The next step is to get the length of the radius. Since radius is one-half of the circle, it will equal the distance from the center to any one end point of the diameter: (10 – 7)2 + (8 – 3)2 = r2  r = root 34 Equation = (x – 7)2 + (y – 3)2 = 34.", "topic": "geometry"}
{"id": "GMAT QUANT_p113_c3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 113, "page_end": 113, "topic_guess": "word_problems", "text": "GMAT word problem note: X and y are co-ordinates of the point Therefore, A = 3, B = -4 and C = -24, x = 1, y = 0 Radius = 21/5.", "topic": "geometry"}
{"id": "GMAT QUANT_p114_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 114, "page_end": 114, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. ADVANCED CONCEPTS PARABOLAS The equation y = Ax2 + Bx + C is the equation of the quadratic graph which is a parabola with axis parallel to y axis C is the y intercept (i.e., the value of y when x is 0) If A > 0, the parabola opens upwards.", "topic": "algebra"}
{"id": "GMAT QUANT_p114_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 114, "page_end": 114, "topic_guess": "algebra", "text": "GMAT algebra note: If A < 0, the parabola opens downwards If B2 > 4AC, the parabola cuts the x-axis at 2 different points If B2 = 4AC, the parabola touches the x-axis at one point (the two points become co-incident) If B2 < 4AC, the parabola does not cut the x-axis at all The vertex of a parabola represents the maximum or minimum value of the function.", "topic": "algebra"}
{"id": "GMAT QUANT_p114_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 114, "page_end": 114, "topic_guess": "algebra", "text": "GMAT algebra note: The vertex is located at point (-b/2a, c – (b2/4a)).", "topic": "algebra"}
{"id": "GMAT QUANT_p116_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 116, "page_end": 116, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}
{"id": "GMAT QUANT_p117_c1_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 117, "page_end": 117, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. (1997 is not a leap year When Mark purchased Product X in January 1997, the warranty did not expire until March 1997 When Santos purchased Product X in May 1997, the warranty expired in May 1997 Rephrase the two statements in terms of extreme possibilities:.", "topic": "percent"}
{"id": "GMAT QUANT_p117_c1_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 117, "page_end": 117, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: Statement 1 Shortest possible warranty period:. 1 (29 days later) Longest possible warranty period:.", "topic": "percent"}
{"id": "GMAT QUANT_p117_c1_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 117, "page_end": 117, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: 31 (89 days later) Statement 2 Shortest possible warranty period:. May 1 to May 2, or similar (1 day later) Longest possible warranty period:.", "topic": "percent"}
{"id": "GMAT QUANT_p117_c1_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 117, "page_end": 117, "topic_guess": "fractions_decimals_percents", "text": "GMAT fractions, decimals, and percents note: May 1 to May 31 (30 days later).", "topic": "percent"}
{"id": "GMAT QUANT_p117_c2_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 117, "page_end": 117, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Ement 2 Shortest possible warranty period:.", "topic": "algebra"}
{"id": "GMAT QUANT_p117_c2_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 117, "page_end": 117, "topic_guess": "algebra", "text": "GMAT algebra note: May 1 to May 2, or similar (1 day later) Longest possible warranty period:. May 1 to May 31 (30 days later) Even taking both statements together, there are still two possibilities-29 days and 30 days -so both statements together are still insufficient Note that, had the given year been a leap year, the two statements together would have become sufficient!", "topic": "algebra"}
{"id": "GMAT QUANT_p117_c2_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 117, "page_end": 117, "topic_guess": "algebra", "text": "GMAT algebra note: In general optimization problems, the objective is to maximize or minimize some quantity, given constraints on other quantities. These quantities are all related through some equation.", "topic": "algebra"}
{"id": "GMAT QUANT_p117_c3_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 117, "page_end": 117, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. The objective is to maximize or minimize some quantity, given constraints on other quantities.", "topic": "algebra"}
{"id": "GMAT QUANT_p117_c3_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 117, "page_end": 117, "topic_guess": "algebra", "text": "GMAT algebra note: If no individual guest consumed more than 2.5 pounds of food, what is the minimum number of guests that could have attended the banquet? Begin by considering the extreme case in which each guest eats as much food as possible, or 2.5 pounds apiece.", "topic": "algebra"}
{"id": "GMAT QUANT_p117_c3_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 117, "page_end": 117, "topic_guess": "algebra", "text": "GMAT algebra note: The corresponding number of guests at the banquet works out to 401/2.5 = 160.4 people You obviously cannot have a fractional number of guests at the banquet. Thus the answer must be rounded.", "topic": "algebra"}
{"id": "GMAT QUANT_p117_c3_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 117, "page_end": 117, "topic_guess": "algebra", "text": "GMAT algebra note: To determine whether to round up or down, consider the explicit constraint: the amount of food per guest is a maximum of 2.5 pounds per guest. Therefore, the minimum number of guests is 160.4 (if guests could be fractional), and we must round up to make the number of guests an integer: 161 Note the careful reasoning required!", "topic": "algebra"}
{"id": "GMAT QUANT_p117_c3_r5", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 117, "page_end": 117, "topic_guess": "algebra", "text": "GMAT algebra note: Although the phrase \"minimum number of guests\" may tempt you to round down, you will get an incorrect answer if you do so. In general, as you solve this sort of problem, put the extreme case into the underlying equation, and solve.", "topic": "algebra"}
{"id": "GMAT QUANT_p117_c4_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 117, "page_end": 117, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. In general, as you solve this sort of problem, put the extreme case into the underlying equation, and solve.", "topic": "algebra"}
{"id": "GMAT QUANT_p117_c4_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 117, "page_end": 117, "topic_guess": "algebra", "text": "GMAT algebra note: Then round appropriately In grouping problems, you make complete groups of items, drawing these items out of a larger pool. The goal is to maximize or minimize some quantity, such as the number of complete groups or the number of leftover items that do not fit into complete groups.", "topic": "algebra"}
{"id": "GMAT QUANT_p117_c4_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 117, "page_end": 117, "topic_guess": "algebra", "text": "GMAT algebra note: As such, these problems are really a special case of optimization. One approach is to determine the limiting factor on the number of complete groups.", "topic": "algebra"}
{"id": "GMAT QUANT_p117_c5_r1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 117, "page_end": 117, "topic_guess": "algebra", "text": "GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. As such, these problems are really a special case of optimization.", "topic": "algebra"}
{"id": "GMAT QUANT_p117_c5_r2", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 117, "page_end": 117, "topic_guess": "algebra", "text": "GMAT algebra note: Each group must have exactly one person from Division A, two people from Division B, and three people from Division C. There are 20 people from Division A, 30 people from Division B, and 40 people from Division C at the conference.", "topic": "algebra"}
{"id": "GMAT QUANT_p117_c5_r3", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 117, "page_end": 117, "topic_guess": "algebra", "text": "GMAT algebra note: What is the smallest number of people who will not be able to be assigned to a group? The first step is to find out how many groups you can make with the people from each division separately, ignoring the other divisions.", "topic": "algebra"}
{"id": "GMAT QUANT_p117_c5_r4", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 117, "page_end": 117, "topic_guess": "algebra", "text": "GMAT algebra note: There are enough Division A people for 20 groups, but only enough Division B people for 15 groups and only enough Division C people for 13 groups. So the limiting factor is Division C.", "topic": "algebra"}
{"id": "GMAT QUANT_p117_c5_r5", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 117, "page_end": 117, "topic_guess": "algebra", "text": "GMAT algebra note: Therefore the total number of people left over will be 7 from A + 4 from B + 1 from C = 12 COMPUTATION Very occasionally, the GMAT features problems centered on computation problems that contain no variables at all. If the wire thus made is exactly 1 meter long, how long is each of the identical pieces?", "topic": "algebra"}
{"id": "GMAT QUANT_p118_c1", "source_name": "GMAT Quantitative Review PPT", "source_file": "GMAT QUANT.pptx", "source_type": "pptx", "page_start": 118, "page_end": 118, "topic_guess": "general", "text": "GMAT quant note: Identify the concept first, then apply the matching rule or setup.", "topic": "general"}