domain listlengths 0 3 | difficulty float64 1 9.5 | problem stringlengths 18 1.21k | solution stringlengths 2 9.24k | answer stringlengths 1 136 | source stringclasses 56
values |
|---|---|---|---|---|---|
[
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 1.5 | Anna thinks of an integer that is not a multiple of three, not a perfect square, and the sum of its digits is a prime number. What could the integer be? | Since 12 and 21 are multiples of 3 (12 = 4 \times 3 and 21 = 7 \times 3), the answer is not 12 or 21. 16 is a perfect square (16 = 4 \times 4) so the answer is not 16. The sum of the digits of 26 is 8, which is not a prime number, so the answer is not 26. Since 14 is not a multiple of three, 14 is not a perfect square,... | 14 | fermat |
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 1.5 | It takes Pearl 7 days to dig 4 holes. It takes Miguel 3 days to dig 2 holes. If they work together and each continues digging at these same rates, how many holes in total will they dig in 21 days? | Since Pearl digs 4 holes in 7 days and $\frac{21}{7}=3$, then in 21 days, Pearl digs $3 \cdot 4=12$ holes. Since Miguel digs 2 holes in 3 days and $\frac{21}{3}=7$, then in 21 days, Miguel digs $7 \cdot 2=14$ holes. In total, they dig $12+14=26$ holes in 21 days. | 26 | fermat |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1.5 | At the end of which year did Steve have more money than Wayne for the first time? | Steve's and Wayne's amounts of money double and halve each year, respectively. By 2004, Steve has more money than Wayne. | 2004 | pascal |
[
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 1.5 | A rectangle has width $x$ and length $y$. The rectangle is cut along the horizontal and vertical dotted lines to produce four smaller rectangles. The sum of the perimeters of these four rectangles is 24. What is the value of $x+y$? | The sum of the lengths of the horizontal line segments is $4x$, because the tops of the four small rectangles contribute a total of $2x$ to their combined perimeter and the bottoms contribute a total of $2x$. Similarly, the sum of the lengths of the vertical line segments is $4y$. In other words, the sum of the perimet... | 6 | cayley |
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 1 | Erin walks $\frac{3}{5}$ of the way home in 30 minutes. If she continues to walk at the same rate, how many minutes will it take her to walk the rest of the way home? | Since Erin walks $\frac{3}{5}$ of the way home in 30 minutes, then she walks $\frac{1}{5}$ of the way at the same rate in 10 minutes. She has $1-\frac{3}{5}=\frac{2}{5}$ of the way left to walk. This is twice as far as $\frac{1}{5}$ of the way. Since she continues to walk at the same rate and $\frac{1}{5}$ of the way t... | 20 | pascal |
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 1.5 | Which number from the set $\{1,2,3,4,5,6,7,8,9,10,11\}$ must be removed so that the mean (average) of the numbers remaining in the set is 6.1? | The original set contains 11 elements whose sum is 66. When one number is removed, there will be 10 elements in the set. For the average of these elements to be 6.1, their sum must be $10 \times 6.1=61$. Since the sum of the original 11 elements is 66 and the sum of the remaining 10 elements is 61, then the element tha... | 5 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1 | A numerical value is assigned to each letter of the alphabet. The value of a word is determined by adding up the numerical values of each of its letters. The value of SET is 2, the value of HAT is 7, the value of TASTE is 3, and the value of MAT is 4. What is the value of the word MATH? | From the given information, we know that $S+E+T=2$, $H+A+T=7$, $T+A+S+T+E=3$, and $M+A+T=4$. Since $T+A+S+T+E=3$ and $S+E+T=2$, then $T+A=3-2=1$. Since $H+A+T=7$ and $T+A=1$, then $H=7-1=6$. Since $M+A+T=4$ and $H=7$, then $M+(A+T)+H=4+6=10$. Therefore, the value of the word MATH is 10. | 10 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 1 | Determine which of the following expressions has the largest value: $4^2$, $4 \times 2$, $4 - 2$, $\frac{4}{2}$, or $4 + 2$. | We evaluate each of the five choices: $4^{2}=16$, $4 \times 2=8$, $4-2=2$, $\frac{4}{2}=2$, $4+2=6$. Of these, the largest is $4^{2}=16$. | 16 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1.5 | If points $P, Q, R$, and $S$ are arranged in order on a line segment with $P Q=1, Q R=2 P Q$, and $R S=3 Q R$, what is the length of $P S$? | Since $P Q=1$ and $Q R=2 P Q$, then $Q R=2$. Since $Q R=2$ and $R S=3 Q R$, then $R S=3(2)=6$. Therefore, $P S=P Q+Q R+R S=1+2+6=9$. | 9 | fermat |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1.5 | Ten numbers have an average (mean) of 87. Two of those numbers are 51 and 99. What is the average of the other eight numbers? | Since 10 numbers have an average of 87, their sum is $10 \times 87 = 870$. When the numbers 51 and 99 are removed, the sum of the remaining 8 numbers is $870 - 51 - 99$ or 720. The average of these 8 numbers is $\frac{720}{8} = 90$. | 90 | cayley |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1 | If $2x-3=10$, what is the value of $4x$? | Since $2x-3=10$, then $2x=13$ and so $4x=2(2x)=2(13)=26$.
(We did not have to determine the value of $x$.) | 26 | pascal |
[
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 1.5 | How many solid $1 imes 1 imes 1$ cubes are required to make a solid $2 imes 2 imes 2$ cube? | The volume of a $1 imes 1 imes 1$ cube is 1 . The volume of a $2 imes 2 imes 2$ cube is 8 . Thus, 8 of the smaller cubes are needed to make the larger cube. | 8 | fermat |
[
"Mathematics -> Algebra -> Intermediate Algebra -> Inequalities"
] | 1.5 | The value of $\frac{x}{2}$ is less than the value of $x^{2}$. The value of $x^{2}$ is less than the value of $x$. Which of the following could be a value of $x$? | Since $x^{2}<x$ and $x^{2} \geq 0$, then $x>0$ and so it cannot be the case that $x$ is negative. Thus, neither (D) nor (E) is the answer. Since $x^{2}<x$, then we cannot have $x>1$. This is because when $x>1$, we have $x^{2}>x$. Thus, (A) is not the answer and so the answer is (B) or (C). If $x=\frac{1}{3}$, then $x^{... | \frac{3}{4} | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 1 | Calculate the value of the expression $\left(2 \times \frac{1}{3}\right) \times \left(3 \times \frac{1}{2}\right)$. | Re-arranging the order of the numbers being multiplied, $\left(2 \times \frac{1}{3}\right) \times \left(3 \times \frac{1}{2}\right) = 2 \times \frac{1}{2} \times 3 \times \frac{1}{3} = \left(2 \times \frac{1}{2}\right) \times \left(3 \times \frac{1}{3}\right) = 1 \times 1 = 1$. | 1 | cayley |
[
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 1 | In the list 7, 9, 10, 11, 18, which number is the average (mean) of the other four numbers? | The average of the numbers \(7, 9, 10, 11\) is \(\frac{7+9+10+11}{4} = \frac{37}{4} = 9.25\), which is not equal to 18, which is the fifth number. The average of the numbers \(7, 9, 10, 18\) is \(\frac{7+9+10+18}{4} = \frac{44}{4} = 11\), which is equal to 11, the remaining fifth number. | 11 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 1.5 | In the subtraction shown, $K, L, M$, and $N$ are digits. What is the value of $K+L+M+N$?\n$$\begin{array}{r}6 K 0 L \\ -\quad M 9 N 4 \\ \hline 2011\end{array}$$ | We work from right to left as we would if doing this calculation by hand. In the units column, we have $L-4$ giving 1. Thus, $L=5$. (There is no borrowing required.) In the tens column, we have $0-N$ giving 1. Since 1 is larger than 0, we must borrow from the hundreds column. Thus, $10-N$ gives 1, which means $N=9$. In... | 17 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1.5 | If $x \%$ of 60 is 12, what is $15 \%$ of $x$? | Since $x \%$ of 60 is 12, then $\frac{x}{100} \cdot 60=12$ or $x=\frac{12 \cdot 100}{60}=20$. Therefore, $15 \%$ of $x$ is $15 \%$ of 20, or $0.15 \cdot 20=3$. | 3 | fermat |
[
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 1 | Which of the following is equal to $110 \%$ of 500? | Solution 1: $10 \%$ of 500 is $\frac{1}{10}$ of 500, which equals 50. Thus, $110 \%$ of 500 equals $500+50$, which equals 550. Solution 2: $110 \%$ of 500 is equal to $\frac{110}{100} \times 500=110 \times 5=550$. | 550 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1.5 | If $(2)(3)(4) = 6x$, what is the value of $x$? | Since $(2)(3)(4) = 6x$, then $6(4) = 6x$. Dividing both sides by 6, we obtain $x = 4$. | 4 | cayley |
[
"Mathematics -> Algebra -> Algebra -> Algebraic Expressions"
] | 1 | Evaluate the expression $2x^{2}+3x^{2}$ when $x=2$. | When $x=2$, we obtain $2x^{2}+3x^{2}=5x^{2}=5 \cdot 2^{2}=5 \cdot 4=20$. | 20 | fermat |
[
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 1 | There are 20 students in a class. In total, 10 of them have black hair, 5 of them wear glasses, and 3 of them both have black hair and wear glasses. How many of the students have black hair but do not wear glasses? | Since 10 students have black hair and 3 students have black hair and wear glasses, then a total of $10-3=7$ students have black hair but do not wear glasses. | 7 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1 | Calculate the value of the expression $\frac{1+(3 \times 5)}{2}$. | Using the correct order of operations, $\frac{1+(3 \times 5)}{2}=\frac{1+15}{2}=\frac{16}{2}=8$. | 8 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1 | Calculate the value of $(3,1) \nabla (4,2)$ using the operation ' $\nabla$ ' defined by $(a, b) \nabla (c, d)=ac+bd$. | From the definition, $(3,1) \nabla (4,2)=(3)(4)+(1)(2)=12+2=14$. | 14 | cayley |
[
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other"
] | 1 | A box contains 5 black ties, 7 gold ties, and 8 pink ties. What is the probability that Stephen randomly chooses a pink tie? | There are \(5+7+8=20\) ties in the box, 8 of which are pink. When Stephen removes a tie at random, the probability of choosing a pink tie is \(\frac{8}{20}\) which is equivalent to \(\frac{2}{5}\). | \frac{2}{5} | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 1 | At Wednesday's basketball game, the Cayley Comets scored 90 points. At Friday's game, they scored $80\%$ as many points as they scored on Wednesday. How many points did they score on Friday? | On Friday, the Cayley Comets scored $80\%$ of 90 points. This is equal to $\frac{80}{100} \times 90 = \frac{8}{10} \times 90 = 8 \times 9 = 72$ points. Alternatively, since $80\%$ is equivalent to 0.8, then $80\%$ of 90 is equal to $0.8 \times 90 = 72$. | 72 | cayley |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 1.5 | What is the sum of all of the possibilities for Sam's number if Sam thinks of a 5-digit number, Sam's friend Sally tries to guess his number, Sam writes the number of matching digits beside each of Sally's guesses, and a digit is considered "matching" when it is the correct digit in the correct position? | We label the digits of the unknown number as vwxyz.
Since vwxyz and 71794 have 0 matching digits, then $v \neq 7$ and $w \neq 1$ and $x \neq 7$ and $y \neq 9$ and $z \neq 4$.
Since vwxyz and 71744 have 1 matching digit, then the preceding information tells us that $y=4$.
Since $v w x 4 z$ and 51545 have 2 matchin... | 526758 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1.5 | In the subtraction shown, $K, L, M$, and $N$ are digits. What is the value of $K+L+M+N$? | We work from right to left as we would if doing this calculation by hand. In the units column, we have $L-1$ giving 1. Thus, $L=2$. (There is no borrowing required.) In the tens column, we have $3-N$ giving 5. Since 5 is larger than 3, we must borrow from the hundreds column. Thus, $13-N$ gives 5, which means $N=8$. In... | 20 | cayley |
[
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 1 | If \( (2^{a})(2^{b})=64 \), what is the mean (average) of \( a \) and \( b \)? | Since \( (2^{a})(2^{b})=64 \), then \( 2^{a+b}=64 \), using an exponent law. Since \( 64=2^{6} \), then \( 2^{a+b}=2^{6} \) and so \( a+b=6 \). Therefore, the average of \( a \) and \( b \) is \( \frac{1}{2}(a+b)=3 \). | 3 | fermat |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1.5 | If $4x + 14 = 8x - 48$, what is the value of $2x$? | Since $4x + 14 = 8x - 48$, then $14 + 48 = 8x - 4x$ or $62 = 4x$. Dividing both sides of this equation by 2, we obtain $\frac{4x}{2} = \frac{62}{2}$ which gives $2x = 31$. | 31 | pascal |
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 1.5 | The first two hours of Melanie's trip were spent travelling at $100 \mathrm{~km} / \mathrm{h}$. The remaining 200 km of Melanie's trip was spent travelling at $80 \mathrm{~km} / \mathrm{h}$. What was Melanie's average speed during this trip? | In 2 hours travelling at $100 \mathrm{~km} / \mathrm{h}$, Melanie travels $2 \mathrm{~h} \times 100 \mathrm{~km} / \mathrm{h}=200 \mathrm{~km}$. When Melanie travels 200 km at $80 \mathrm{~km} / \mathrm{h}$, it takes $\frac{200 \mathrm{~km}}{80 \mathrm{~km} / \mathrm{h}}=2.5 \mathrm{~h}$. Melanie travels a total of $20... | 89 \mathrm{~km} / \mathrm{h} | pascal |
[
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 1 | What is the perimeter of the figure shown if $x=3$? | Since $x=3$, the side lengths of the figure are $4,3,6$, and 10. Thus, the perimeter of the figure is $4+3+6+10=23$. (Alternatively, the perimeter is $x+6+10+(x+1)=2x+17$. When $x=3$, this equals $2(3)+17$ or 23.) | 23 | pascal |
[
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 1 | What is the perimeter of the shaded region in a \( 3 \times 3 \) grid where some \( 1 \times 1 \) squares are shaded? | The top, left and bottom unit squares each contribute 3 sides of length 1 to the perimeter. The remaining square contributes 1 side of length 1 to the perimeter. Therefore, the perimeter is \( 3 \times 3 + 1 \times 1 = 10 \). | 10 | cayley |
[
"Mathematics -> Number Theory -> Congruences"
] | 1.5 | How many of the four integers $222, 2222, 22222$, and $222222$ are multiples of 3? | We could use a calculator to divide each of the four given numbers by 3 to see which calculations give an integer answer. Alternatively, we could use the fact that a positive integer is divisible by 3 if and only if the sum of its digits is divisible by 3. The sums of the digits of $222, 2222, 22222$, and $222222$ are ... | 2 | cayley |
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 1 | What was the range of temperatures on Monday in Fermatville, given that the minimum temperature was $-11^{\circ} \mathrm{C}$ and the maximum temperature was $14^{\circ} \mathrm{C}$? | Since the maximum temperature was $14^{\circ} \mathrm{C}$ and the minimum temperature was $-11^{\circ} \mathrm{C}$, then the range of temperatures was $14^{\circ} \mathrm{C} - (-11^{\circ} \mathrm{C}) = 25^{\circ} \mathrm{C}$. | 25^{\circ} \mathrm{C} | fermat |
[
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 1.5 | What is \( 110\% \) of 500? | Solution 1: \( 10\% \) of 500 is \( \frac{1}{10} \) or 0.1 of 500, which equals 50. \( 100\% \) of 500 is 500. Thus, \( 110\% \) of 500 equals \( 500 + 50 \), which equals 550. Solution 2: \( 110\% \) of 500 is equal to \( \frac{110}{100} \times 500 = 110 \times 5 = 550 \). | 550 | cayley |
[
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 1.5 | In $\triangle ABC$, points $D$ and $E$ lie on $AB$, as shown. If $AD=DE=EB=CD=CE$, what is the measure of $\angle ABC$? | Since $CD=DE=EC$, then $\triangle CDE$ is equilateral, which means that $\angle DEC=60^{\circ}$. Since $\angle DEB$ is a straight angle, then $\angle CEB=180^{\circ}-\angle DEC=180^{\circ}-60^{\circ}=120^{\circ}$. Since $CE=EB$, then $\triangle CEB$ is isosceles with $\angle ECB=\angle EBC$. Since $\angle ECB+\angle CE... | 30^{\circ} | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Integers",
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1.5 | Yann writes down the first $n$ consecutive positive integers, $1,2,3,4, \ldots, n-1, n$. He removes four different integers $p, q, r, s$ from the list. At least three of $p, q, r, s$ are consecutive and $100<p<q<r<s$. The average of the integers remaining in the list is 89.5625. What is the number of possible values of... | When Yann removes 4 of the $n$ integers from his list, there are $n-4$ integers left. Suppose that the sum of the $n-4$ integers left is $T$. The average of these $n-4$ integers is $89.5625=89.5+0.0625=89+\frac{1}{2}+\frac{1}{16}=89 \frac{9}{16}=\frac{1433}{16}$. Since the sum of the $n-4$ integers is $T$, then $\frac{... | 22 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 1.5 | What is the number halfway between $\frac{1}{12}$ and $\frac{1}{10}$? | The number halfway between two numbers is their average. Therefore, the number halfway between $\frac{1}{10}$ and $\frac{1}{12}$ is $\frac{1}{2}\left(\frac{1}{10}+\frac{1}{12}\right)=\frac{1}{2}\left(\frac{12}{120}+\frac{10}{120}\right)=\frac{1}{2}\left(\frac{22}{120}\right)=\frac{11}{120}$. | \frac{11}{120} | fermat |
[
"Mathematics -> Number Theory -> Factorization"
] | 1.5 | The integer 119 is a multiple of which number? | The ones digit of 119 is not even, so 119 is not a multiple of 2. The ones digit of 119 is not 0 or 5, so 119 is not a multiple of 5. Since $120=3 \times 40$, then 119 is 1 less than a multiple of 3 so is not itself a multiple of 3. Since $110=11 \times 10$ and $121=11 \times 11$, then 119 is between two consecutive mu... | 7 | cayley |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1.5 | In 12 years, Janice will be 8 times as old as she was 2 years ago. How old is Janice now? | Suppose that Janice is \( x \) years old now. Two years ago, Janice was \( x - 2 \) years old. In 12 years, Janice will be \( x + 12 \) years old. From the given information \( x + 12 = 8(x - 2) \) and so \( x + 12 = 8x - 16 \) which gives \( 7x = 28 \) and so \( x = 4 \). | 4 | cayley |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1.5 | What is the largest possible value for $n$ if the average of the two positive integers $m$ and $n$ is 5? | Since the average of $m$ and $n$ is 5, then $\frac{m+n}{2}=5$ which means that $m+n=10$. In order for $n$ to be as large as possible, we need to make $m$ as small as possible. Since $m$ and $n$ are positive integers, then the smallest possible value of $m$ is 1, which means that the largest possible value of $n$ is $n=... | 9 | cayley |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1.5 | The three numbers $5, a, b$ have an average (mean) of 33. What is the average of $a$ and $b$? | Since $5, a, b$ have an average of 33, then $\frac{5+a+b}{3}=33$. Multiplying by 3, we obtain $5+a+b=3 \times 33=99$, which means that $a+b=94$. The average of $a$ and $b$ is thus equal to $\frac{a+b}{2}=\frac{94}{2}=47$. | 47 | cayley |
[
"Mathematics -> Algebra -> Prealgebra -> Percentages -> Other",
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 1.5 | Meg started with the number 100. She increased this number by $20\%$ and then increased the resulting number by $50\%$. What was her final result? | $20\%$ of the number 100 is 20, so when 100 is increased by $20\%$, it becomes $100 + 20 = 120$. $50\%$ of a number is half of that number, so $50\%$ of 120 is 60. Thus, when 120 is increased by $50\%$, it becomes $120 + 60 = 180$. Therefore, Meg's final result is 180. | 180 | cayley |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1.5 | When a number is tripled and then decreased by 5, the result is 16. What is the original number? | To get back to the original number, we undo the given operations. We add 5 to 16 to obtain 21 and then divide by 3 to obtain 7. These are the 'inverse' operations of decreasing by 5 and multiplying by 3. | 7 | cayley |
[
"Mathematics -> Number Theory -> Prime Numbers"
] | 1.5 | Which of the following integers cannot be written as a product of two integers, each greater than 1: 6, 27, 53, 39, 77? | We note that $6=2 imes 3$ and $27=3 imes 9$ and $39=3 imes 13$ and $77=7 imes 11$, which means that each of $6,27,39$, and 77 can be written as the product of two integers, each greater than 1. Thus, 53 must be the integer that cannot be written in this way. We can check that 53 is indeed a prime number. | 53 | fermat |
[
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 1.5 | The mean (average) of 5 consecutive integers is 9. What is the smallest of these 5 integers? | Since the mean of five consecutive integers is 9, then the middle of these five integers is 9. Therefore, the integers are $7,8,9,10,11$, and so the smallest of the five integers is 7. | 7 | cayley |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1.5 | What is the value of $k$ if the side lengths of four squares are shown, and the area of the fifth square is $k$? | Let $s$ be the side length of the square with area $k$.
The sum of the heights of the squares on the right side is $3+8=11$.
The sum of the heights of the squares on the left side is $1+s+4=s+5$.
Since the two sums are equal, then $s+5=11$, and so $s=6$.
Therefore, the square with area $k$ has side length 6, an... | 36 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 1 | The value of \( \frac{1}{2} + \frac{2}{4} + \frac{4}{8} + \frac{8}{16} \) is what? | In the given sum, each of the four fractions is equivalent to \( \frac{1}{2} \). Therefore, the given sum is equal to \( \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} = 2 \). | 2 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Exponents -> Other"
] | 1 | If $4^{n}=64^{2}$, what is the value of $n$? | We note that $64=4 \times 4 \times 4$. Thus, $64^{2}=64 \times 64=4 \times 4 \times 4 \times 4 \times 4 \times 4$. Since $4^{n}=64^{2}$, then $4^{n}=4 \times 4 \times 4 \times 4 \times 4 \times 4$ and so $n=6$. | 6 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1 | If $2n + 5 = 16$, what is the value of the expression $2n - 3$? | Since $2n + 5 = 16$, then $2n - 3 = (2n + 5) - 8 = 16 - 8 = 8$. Alternatively, we could solve the equation $2n + 5 = 16$ to obtain $2n = 11$ or $n = \frac{11}{2}$. From this, we see that $2n - 3 = 2\left(\frac{11}{2}\right) - 3 = 11 - 3 = 8$. | 8 | cayley |
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 1.5 | Pascal High School organized three different trips. Fifty percent of the students went on the first trip, $80 \%$ went on the second trip, and $90 \%$ went on the third trip. A total of 160 students went on all three trips, and all of the other students went on exactly two trips. How many students are at Pascal High Sc... | Let $x$ be the total number of students at Pascal H.S. Let $a$ be the total number of students who went on both the first trip and the second trip, but did not go on the third trip. Let $b$ be the total number of students who went on both the first trip and the third trip, but did not go on the second trip. Let $c$ be ... | 800 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1.5 | Lauren plays basketball with her friends. She makes 10 baskets. Each of these baskets is worth either 2 or 3 points. Lauren scores a total of 26 points. How many 3 point baskets did she make? | Suppose that Lauren makes $x$ baskets worth 3 points each. Since she makes 10 baskets, then $10-x$ baskets that she made are worth 2 points each. Since Lauren scores 26 points, then $3 x+2(10-x)=26$ and so $3 x+20-x=26$ which gives $x=6$. Therefore, Lauren makes 6 baskets worth 3 points. | 6 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1.5 | If the number of zeros in the integer equal to $(10^{100}) imes (100^{10})$ is sought, what is this number? | Since $100=10^{2}$, then $100^{10}=(10^{2})^{10}=10^{20}$. Therefore, $(10^{100}) imes (100^{10})=(10^{100}) imes (10^{20})=10^{120}$. When written out, this integer consists of a 1 followed by 120 zeros. | 120 | cayley |
[
"Mathematics -> Precalculus -> Functions"
] | 1.5 | Numbers $m$ and $n$ are on the number line. What is the value of $n-m$? | On a number line, the markings are evenly spaced. Since there are 6 spaces between 0 and 30, each space represents a change of $\frac{30}{6}=5$. Since $n$ is 2 spaces to the right of 60, then $n=60+2 \times 5=70$. Since $m$ is 3 spaces to the left of 30, then $m=30-3 \times 5=15$. Therefore, $n-m=70-15=55$. | 55 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 1 | The operation \( \otimes \) is defined by \( a \otimes b = \frac{a}{b} + \frac{b}{a} \). What is the value of \( 4 \otimes 8 \)? | From the given definition, \( 4 \otimes 8 = \frac{4}{8} + \frac{8}{4} = \frac{1}{2} + 2 = \frac{5}{2} \). | \frac{5}{2} | pascal |
[
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations",
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1.5 | Alvin, Bingyi, and Cheska play a two-player game that never ends in a tie. In a recent tournament between the three players, a total of 60 games were played and each pair of players played the same number of games. When Alvin and Bingyi played, Alvin won \(20\%\) of the games. When Bingyi and Cheska played, Bingyi won ... | Since 60 games are played and each of the 3 pairs plays the same number of games, each pair plays \(60 \div 3 = 20\) games. Alvin wins \(20\%\) of the 20 games that Alvin and Bingyi play, so Alvin wins \(\frac{20}{100} \times 20 = \frac{1}{5} \times 20 = 4\) of these 20 games and Bingyi wins \(20 - 4 = 16\) of these 20... | 28 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 1.5 | Three integers from the list $1,2,4,8,16,20$ have a product of 80. What is the sum of these three integers? | The three integers from the list whose product is 80 are 1, 4, and 20, since $1 \times 4 \times 20=80$. The sum of these integers is $1+4+20=25$.
(Since 80 is a multiple of 5 and 20 is the only integer in the list that is a multiple of 5, then 20 must be included in the product. This leaves two integers to choose, an... | 25 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1 | If $2x + 6 = 16$, what is the value of $x + 4$? | Solution 1: Since $2x + 6 = 16$, then $\frac{2x + 6}{2} = \frac{16}{2}$ and so $x + 3 = 8$. Since $x + 3 = 8$, then $x + 4 = (x + 3) + 1 = 8 + 1 = 9$. Solution 2: Since $2x + 6 = 16$, then $2x = 16 - 6 = 10$. Since $2x = 10$, then $\frac{2x}{2} = \frac{10}{2}$ and so $x = 5$. Since $x = 5$, then $x + 4 = 5 + 4 = 9$. | 9 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 1 | What is the value of \( \frac{5-2}{2+1} \)? | Simplifying, \( \frac{5-2}{2+1}=\frac{3}{3}=1 \). | 1 | cayley |
[
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 1.5 | In how many different ways can André form exactly \( \$10 \) using \( \$1 \) coins, \( \$2 \) coins, and \( \$5 \) bills? | Using combinations of \( \$5 \) bills, \( \$2 \) coins, and \( \$1 \) coins, there are 10 ways to form \( \$10 \). | 10 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1.5 | A sign has 31 spaces on a single line. The word RHOMBUS is written from left to right in 7 consecutive spaces. There is an equal number of empty spaces on each side of the word. Counting from the left, in what space number should the letter $R$ be put? | Since the letters of RHOMBUS take up 7 of the 31 spaces on the line, there are $31-7=24$ spaces that are empty. Since the numbers of empty spaces on each side of RHOMBUS are the same, there are $24 \div 2=12$ empty spaces on each side. Therefore, the letter R is placed in space number $12+1=13$, counting from the left. | 13 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1.5 | Which of the following is closest in value to 7? | We note that $7=\sqrt{49}$ and that $\sqrt{40}<\sqrt{49}<\sqrt{50}<\sqrt{60}<\sqrt{70}<\sqrt{80}$.
This means that $\sqrt{40}$ or $\sqrt{50}$ is the closest to 7 of the given choices.
Since $\sqrt{40} \approx 6.32$ and $\sqrt{50} \approx 7.07$, then $\sqrt{50}$ is closest to 7. | \sqrt{50} | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 1 | Calculate the expression $8 \times 10^{5}+4 \times 10^{3}+9 \times 10+5$. | First, we write out the powers of 10 in full to obtain $8 \times 100000+4 \times 1000+9 \times 10+5$. Simplifying, we obtain $800000+4000+90+5$ or 804095. | 804095 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 1 | Evaluate the expression $8- rac{6}{4-2}$. | Evaluating, $8- rac{6}{4-2}=8- rac{6}{2}=8-3=5$. | 5 | fermat |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1 | If \( 8 + 6 = n + 8 \), what is the value of \( n \)? | Since \( 8+6=n+8 \), then subtracting 8 from both sides, we obtain \( 6=n \) and so \( n \) equals 6. | 6 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 1.5 | In the addition problem shown, $m, n, p$, and $q$ represent positive digits. What is the value of $m+n+p+q$? | From the ones column, we see that $3 + 2 + q$ must have a ones digit of 2. Since $q$ is between 1 and 9, inclusive, then $3 + 2 + q$ is between 6 and 14. Since its ones digit is 2, then $3 + 2 + q = 12$ and so $q = 7$. This also means that there is a carry of 1 into the tens column. From the tens column, we see that $1... | 24 | cayley |
[
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 1.5 | Ewan writes out a sequence where he counts by 11s starting at 3. Which number will appear in Ewan's sequence? | Ewan's sequence starts with 3 and each following number is 11 larger than the previous number. Since every number in the sequence is some number of 11s more than 3, this means that each number in the sequence is 3 more than a multiple of 11. Furthermore, every such positive integer is in Ewan's sequence. Since $110 = 1... | 113 | fermat |
[
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 1 | If \( 3-5+7=6-x \), what is the value of \( x \)? | Simplifying the left side of the equation, we obtain \( 5=6-x \). Therefore, \( x=6-5=1 \). | 1 | fermat |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1 | If $x=2$, what is the value of $4x^2 - 3x^2$? | Simplifying, $4 x^{2}-3 x^{2}=x^{2}$. When $x=2$, this expression equals 4 . Alternatively, when $x=2$, we have $4 x^{2}-3 x^{2}=4 \cdot 2^{2}-3 \cdot 2^{2}=16-12=4$. | 4 | fermat |
[
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 1 | Calculate the value of the expression $2+3 imes 5+2$. | Calculating, $2+3 imes 5+2=2+15+2=19$. | 19 | fermat |
[
"Mathematics -> Algebra -> Prealgebra -> Integers",
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 1.5 | The smallest of nine consecutive integers is 2012. These nine integers are placed in the circles to the right. The sum of the three integers along each of the four lines is the same. If this sum is as small as possible, what is the value of $u$? | If we have a configuration of the numbers that has the required property, then we can add or subtract the same number from each of the numbers in the circles and maintain the property. (This is because there are the same number of circles in each line.) Therefore, we can subtract 2012 from all of the numbers and try to... | 2015 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1.5 | If \( x=2 \) and \( v=3x \), what is the value of \((2v-5)-(2x-5)\)? | Since \( v=3x \) and \( x=2 \), then \( v=3 \cdot 2=6 \). Therefore, \((2v-5)-(2x-5)=(2 \cdot 6-5)-(2 \cdot 2-5)=7-(-1)=8\). | 8 | fermat |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1.5 | John ate a total of 120 peanuts over four consecutive nights. Each night he ate 6 more peanuts than the night before. How many peanuts did he eat on the fourth night? | Suppose that John ate \( x \) peanuts on the fourth night. Since he ate 6 more peanuts each night than on the previous night, then he ate \( x-6 \) peanuts on the third night, \((x-6)-6=x-12\) peanuts on the second night, and \((x-12)-6=x-18\) peanuts on the first night. Since John ate 120 peanuts in total, then \( x+(... | 39 | fermat |
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 1.5 | Arturo has an equal number of $\$5$ bills, of $\$10$ bills, and of $\$20$ bills. The total value of these bills is $\$700$. How many $\$5$ bills does Arturo have? | Since Arturo has an equal number of $\$5$ bills, of $\$10$ bills, and of $\$20$ bills, then we can divide Arturo's bills into groups, each of which contains one $\$5$ bill, one $\$10$ bill, and one $\$20$ bill. The value of the bills in each group is $\$5 + \$10 + \$20 = \$35$. Since the total value of Arturo's bills i... | 20 | cayley |
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 1.5 | Last summer, Pat worked at a summer camp. For each day that he worked, he earned \$100 and he was not charged for food. For each day that he did not work, he was not paid and he was charged \$20 for food. After 70 days, the money that he earned minus his food costs equalled \$5440. On how many of these 70 days did Pat ... | Let \( x \) be the number of days on which Pat worked. On each of these days, he earned \$100 and had no food costs, so he earned a total of \( 100x \) dollars. Since Pat worked for \( x \) of the 70 days, then he did not work on \( 70-x \) days. On each of these days, he earned no money and was charged \$20 for food, ... | 57 | fermat |
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 1.5 | A bicycle trip is 30 km long. Ari rides at an average speed of 20 km/h. Bri rides at an average speed of 15 km/h. If Ari and Bri begin at the same time, how many minutes after Ari finishes the trip will Bri finish? | Riding at 15 km/h, Bri finishes the 30 km in $\frac{30 \text{ km}}{15 \text{ km/h}} = 2 \text{ h}$. Riding at 20 km/h, Ari finishes the 30 km in $\frac{30 \text{ km}}{20 \text{ km/h}} = 1.5 \text{ h}$. Therefore, Bri finishes 0.5 h after Ari, which is 30 minutes. | 30 | fermat |
[
"Mathematics -> Algebra -> Algebra -> Algebraic Expressions"
] | 1 | Which of the following expressions is not equivalent to $3x + 6$? | We look at each of the five choices: (A) $3(x + 2) = 3x + 6$ (B) $\frac{-9x - 18}{-3} = \frac{-9x}{-3} + \frac{-18}{-3} = 3x + 6$ (C) $\frac{1}{3}(3x) + \frac{2}{3}(9) = x + 6$ (D) $\frac{1}{3}(9x + 18) = 3x + 6$ (E) $3x - 2(-3) = 3x + (-2)(-3) = 3x + 6$ The expression that is not equivalent to $3x + 6$ is the expressi... | \frac{1}{3}(3x) + \frac{2}{3}(9) | pascal |
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 1.5 | Hagrid has 100 animals. Among these animals, each is either striped or spotted but not both, each has either wings or horns but not both, there are 28 striped animals with wings, there are 62 spotted animals, and there are 36 animals with horns. How many of Hagrid's spotted animals have horns? | Each of the animals is either striped or spotted, but not both. Since there are 100 animals and 62 are spotted, then there are $100 - 62 = 38$ striped animals. Each striped animal must have wings or a horn, but not both. Since there are 28 striped animals with wings, then there are $38 - 28 = 10$ striped animals with h... | 26 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1 | If $x=3$, $y=2x$, and $z=3y$, what is the average of $x$, $y$, and $z$? | Since $x=3$ and $y=2x$, then $y=2 \times 3=6$. Since $y=6$ and $z=3y$, then $z=3 \times 6=18$. Therefore, the average of $x, y$ and $z$ is $\frac{x+y+z}{3}=\frac{3+6+18}{3}=9$. | 9 | pascal |
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 1.5 | A mass of 15 kg is halfway between 10 kg and 20 kg on the vertical axis. What is the age of the cod when its mass is 15 kg? | A mass of 15 kg is halfway between 10 kg and 20 kg on the vertical axis. The point where the graph reaches 15 kg is halfway between 6 and 8 on the horizontal axis. Therefore, the cod is 7 years old when its mass is 15 kg. | 7 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1.5 | The average (mean) of a list of 10 numbers is 17. When one number is removed from the list, the new average is 16. What number was removed? | When 10 numbers have an average of 17, their sum is $10 \times 17=170$. When 9 numbers have an average of 16, their sum is $9 \times 16=144$. Therefore, the number that was removed was $170-144=26$. | 26 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 1 | When 542 is multiplied by 3, what is the ones (units) digit of the result? | Since \( 542 \times 3 = 1626 \), the ones digit of the result is 6. | 6 | cayley |
[
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 1.5 | A string has been cut into 4 pieces, all of different lengths. The length of each piece is 2 times the length of the next smaller piece. What fraction of the original string is the longest piece? | Let \(L\) be the length of the string. If \(x\) is the length of the shortest piece, then since each of the other pieces is twice the length of the next smaller piece, then the lengths of the remaining pieces are \(2x, 4x\), and \(8x\). Since these four pieces make up the full length of the string, then \(x+2x+4x+8x=L\... | \frac{8}{15} | cayley |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1.5 | In a cafeteria line, the number of people ahead of Kaukab is equal to two times the number of people behind her. There are $n$ people in the line. What is a possible value of $n$? | Suppose that there are $p$ people behind Kaukab. This means that there are $2p$ people ahead of her. Including Kaukab, the total number of people in line is $n = p + 2p + 1 = 3p + 1$, which is one more than a multiple of 3. Of the given choices $(23, 20, 24, 21, 25)$, the only one that is one more than a multiple of 3 ... | 25 | cayley |
[
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 1 | What number should go in the $\square$ to make the equation $\frac{3}{4}+\frac{4}{\square}=1$ true? | For $\frac{3}{4}+\frac{4}{\square}=1$ to be true, we must have $\frac{4}{\square}=1-\frac{3}{4}=\frac{1}{4}$. Since $\frac{1}{4}=\frac{4}{16}$, we rewrite the right side using the same numerator to obtain $\frac{4}{\square}=\frac{4}{16}$. Therefore, $\square=16$ makes the equation true. | 16 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 1 | What is the result of subtracting eighty-seven from nine hundred forty-three? | Converting to a numerical expression, we obtain $943-87$ which equals 856. | 856 | pascal |
[
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 1.5 | Five students play chess matches against each other. Each student plays three matches against each of the other students. How many matches are played in total? | We label the players as A, B, C, D, and E. The total number of matches played will be equal to the number of pairs of players that can be formed times the number of matches that each pair plays. The possible pairs of players are AB, AC, AD, AE, BC, BD, BE, CD, CE, and DE. There are 10 such pairs. Thus, the total number... | 30 | cayley |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1 | If $x=3$, what is the value of $-(5x - 6x)$? | When $x=3$, we have $-(5x - 6x) = -(-x) = x = 3$. Alternatively, when $x=3$, we have $-(5x - 6x) = -(15 - 18) = -(-3) = 3$. | 3 | cayley |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1 | Evaluate the expression $2^{3}-2+3$. | Evaluating, $2^{3}-2+3=2 imes 2 imes 2-2+3=8-2+3=9$. | 9 | pascal |
[
"Mathematics -> Applied Mathematics -> Math Word Problems",
"Mathematics -> Discrete Mathematics -> Algorithms"
] | 1.5 | A robotic grasshopper jumps 1 cm to the east, then 2 cm to the north, then 3 cm to the west, then 4 cm to the south. After every fourth jump, the grasshopper restarts the sequence of jumps: 1 cm to the east, then 2 cm to the north, then 3 cm to the west, then 4 cm to the south. After a total of $n$ jumps, the position ... | Each group of four jumps takes the grasshopper 1 cm to the east and 3 cm to the west, which is a net movement of 2 cm to the west, and 2 cm to the north and 4 cm to the south, which is a net movement of 2 cm to the south. In other words, we can consider each group of four jumps, starting with the first, as resulting in... | 22 | cayley |
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 1.5 | The Athenas are playing a 44 game season. They have 20 wins and 15 losses so far. What is the smallest number of their remaining games that they must win to make the playoffs, given they must win at least 60% of all of their games? | In order to make the playoffs, the Athenas must win at least 60% of their 44 games. That is, they must win at least $0.6 \times 44=26.4$ games. Since they must win an integer number of games, then the smallest number of games that they can win to make the playoffs is the smallest integer larger than 26.4, or 27. Since ... | 7 | cayley |
[
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 1.5 | In triangle $BCD$, $\angle CBD=\angle CDB$ because $BC=CD$. If $\angle BCD=80+50+30=160$, find $\angle CBD=\angle CDB$. | $\angle CBD=\angle CDB=10$ | 10 | HMMT_11 |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1.5 | Many of the students in M. Gamache's class brought a skateboard or a bicycle to school yesterday. The ratio of the number of skateboards to the number of bicycles was $7:4$. There were 12 more skateboards than bicycles. How many skateboards and bicycles were there in total? | Since the ratio of the number of skateboards to the number of bicycles was $7:4$, then the numbers of skateboards and bicycles can be written in the form $7k$ and $4k$ for some positive integer $k$. Since the difference between the numbers of skateboards and bicycles is 12, then $7k - 4k = 12$ and so $3k = 12$ or $k = ... | 44 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1 | Robyn has 4 tasks to do and Sasha has 14 tasks to do. How many of Sasha's tasks should Robyn do in order for them to have the same number of tasks? | Between them, Robyn and Sasha have \(4 + 14 = 18\) tasks to do. If each does the same number of tasks, each must do \(18 \div 2 = 9\) tasks. This means that Robyn must do \(9 - 4 = 5\) of Sasha's tasks. | 5 | pascal |
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 1.5 | Which number from the set $\{1,2,3,4,5,6,7,8,9,10,11\}$ must be removed so that the mean (average) of the numbers remaining in the set is 6.1? | The original set contains 11 elements whose sum is 66. When one number is removed, there will be 10 elements in the set. For the average of these elements to be 6.1, their sum must be $10 \times 6.1=61$. Since the sum of the original 11 elements is 66 and the sum of the remaining 10 elements is 61, then the element tha... | 5 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 1.5 | Simplify the expression $(\sqrt{100}+\sqrt{9}) \times(\sqrt{100}-\sqrt{9})$. | Simplifying, $(\sqrt{100}+\sqrt{9}) \times(\sqrt{100}-\sqrt{9})=(10+3) \times(10-3)=13 \times 7=91$. | 91 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 1 | On the number line, points $M$ and $N$ divide $L P$ into three equal parts. What is the value at $M$? | The difference between $\frac{1}{6}$ and $\frac{1}{12}$ is $\frac{1}{6}-\frac{1}{12}=\frac{2}{12}-\frac{1}{12}=\frac{1}{12}$, so $L P=\frac{1}{12}$. Since $L P$ is divided into three equal parts, then this distance is divided into three equal parts, each equal to $\frac{1}{12} \div 3=\frac{1}{12} \times \frac{1}{3}=\fr... | \frac{1}{9} | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 1.5 | The expression $(5 \times 5)+(5 \times 5)+(5 \times 5)+(5 \times 5)+(5 \times 5)$ is equal to what? | The given sum includes 5 terms each equal to $(5 \times 5)$. Thus, the given sum is equal to $5 \times(5 \times 5)$ which equals $5 \times 25$ or 125. | 125 | cayley |
[
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 1 | When two positive integers are multiplied, the result is 24. When these two integers are added, the result is 11. What is the result when the smaller integer is subtracted from the larger integer? | The positive divisor pairs of 24 are: 1 and 24, 2 and 12, 3 and 8, 4 and 6. Of these, the pair whose sum is 11 is 3 and 8. The difference between these two integers is $8 - 3 = 5$. | 5 | pascal |
[
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 1 | What fraction of the pizza is left for Wally if Jovin takes $\frac{1}{3}$ of the pizza, Anna takes $\frac{1}{6}$ of the pizza, and Olivia takes $\frac{1}{4}$ of the pizza? | Since Jovin, Anna and Olivia take $\frac{1}{3}, \frac{1}{6}$ and $\frac{1}{4}$ of the pizza, respectively, then the fraction of the pizza with which Wally is left is
$$ 1-\frac{1}{3}-\frac{1}{6}-\frac{1}{4}=\frac{12}{12}-\frac{4}{12}-\frac{2}{12}-\frac{3}{12}=\frac{3}{12}=\frac{1}{4} $$ | \frac{1}{4} | pascal |
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