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) ↓ 0.5r ↓ (4) Solve the equation. (a) Write the equation: (b) Use the distributive property: 0.5r 2(r 15) 150 0.5r 2r 30 150 Using Formulas to Solve Problems 137 (c) Combine like terms: (d) Add 30, the opposite of 30 to each side of the equation: (e) Divide each side by 2.5: 2.5r 2.5r 30 150 30 30 120 2.5 5 120 2.5r 2... |
h, find h when A 3.6 m2 and b 0.90 m. 1 2bh 11. If A, find h when A 24 sq ft and b 8.0 ft. 12. If V lwh, find w when V 72 yd3, l 0.75 yd, and h 12 yd. 13. If d rt, find r when d 120 mi and t 3 hr. 14. If I prt, find the principal, p, when the interest, I, is $135, the yearly rate of interest, r, is 2.5%, and the time, ... |
a rectangle is 3 yards less than its length. The perimeter is 130 yards. Find the length and the width of the rectangle. 25. The length of each side of an equilateral triangle is 5 centimeters more than the length of each side of a square. The perimeters of the two figures are equal. Find the lengths of the sides of t... |
to be used. All numbers may be considered to be exact values. 35. Rahul has 25 coins, all quarters and dimes. Copy the table given below and organize the facts in the table using the answers to a through c. Number of coins in one denomination b a Value of one coin a b Total value of the coins of that denomination Numb... |
when she works Monday through Friday and $9.00 an hour when she works on Saturday. Last week, her salary was $273 for 42 hours of work. How many hours did she work on Saturday? (Make a table similar to that given in exercise 39.) 41. Candice earns $8.25 an hour and is paid every two weeks. Last week she worked 4 hours... |
traveled. 46. Carla and Candice left from the same place at the same time and rode their bicycles in the same direction along a straight road. Candice bicycled at an average speed that was threequarters of Carla’s average speed. After 2 hours they were 28 miles apart. What was the average speed of Carla and Candice? 4... |
A 3b 3b ✔ Check x a b b 1 a 2 a 5? b b b ✔ Solution Compare with 2x 10 3x. 2x 10 3x 3x 3ax 3x 2ax 10a2 3ax 3ax 5x 10 5x 5 5 10 5 x 2 5ax 10a2 5a 5 10a2 5ax 5a x 2a Transforming Formulas 143 Check 2ax 10a2 3ax 2a(2a) 5? 10a2 2 3a(2a) 5? 10a2 2 6a2 4a2 4a2 4a2 ✔ Answer x 2a EXERCISES Writing About Mathematics 1. Write a... |
ve the formula d = rt for t. b. Use the answer obtained in part a to find the value of t when d 200 miles and r 40 miles per hour. Solution a. d rt d r 5 rt r d r 5 t b. t 5 d r 5 200 40 5 Answers a. t d r b. t 5 hours Note that the rate is 40 miles per hour, that is, 40 miles 1 hour. Therefore, 200 miles 40 miles 1 ho... |
stand at a movie theater wants to sell popcorn in containers that are in the shape of a cylinder. The volume of the cylinder is given by the formula V = pr2h, where V is the volume, r is the radius of the base, and h is the height of the container. a. Solve the formula for h. b. If the container is to hold 1,400 cubic... |
3 x < 3 Properties of Inequalities 147 The real number that makes the corresponding equality, x 3, true is a single point on the number line. This point, x 3, is also the boundary between the values of x that make x 3 true and the values of x that make x 3 true. The circle is filled in, indicating that 3 belongs to th... |
PLE 1 Use the inequality 5 9 to write a new inequality: a. by adding 6 to both sides b. by adding 9 to both sides Solution a. 5 6 9 6 11 15 b. 5 (–9) 9 (–9) –4 0 Answers a. 11 15 b. 4 0 The Multiplication Property of Inequality The following table shows the result of multiplying both sides of an inequality by the same ... |
. 21 3 b. by multiplying both sides by. Solution a. 6 9 b. 6 9 6(2)? 9(2) 12 18 6 A Answers a. 12 18 b. 2 3 21 3 21? 9 3 A 2 3 B B 150 First Degree Equations and Inequalities in One Variable EXERCISES Writing About Mathematics 1. Sadie said that if 5 4, then it must be true that 5x 4x. Do you agree with Sadie? Explain ... |
24. If x 5 and 5 y, then x? y. 23. If 2x 6, then?? 3(4) or y? 12. or x? 12. 2(6) 26. If 3 7, then 7? 3. 28. 1f 9 x, then x? 9. or x? 3. 25. If m 7 and 7 a, then m? a. 27. If 4 12, then 12? 4. 29. If 7 a, then a? 7. 30. If x 10 and 10 z, then x? z. 31. If a b and c b, then a? c. 4-8 FINDING AND GRAPHING THE SOLUTION OF... |
alternative method of expressing the solution set is interval notation. When this notation is used, the solution set is written as (5, ). The first number, 5 names the lower boundary. The symbol, often called infinity, indicates that there is no upper boundary, that is, that the set of real numbers continues without e... |
2 3 Graphing the Intersection of Two Sets The inequality 3 x 6 is equivalent to (3 x) and (x 6). This statement is true when both simple statements are true and false when one or both statements are false. The solution set of this inequality consists of all of the numbers that are in the solution set of both simple in... |
) Graphing the Union of Two Sets The inequality (x 3) or (x 6) is true when one or both of the simple statements are true. It is false when both simple statements are false. The solution set of the inequality consists of the union of the solution sets of the two simple statements. The graph of the solution set can be d... |
Inequalities in One Variable Developing Skills In 3–37, find and graph the solution set of each inequality. The domain is the set of real numbers. 3. x 2 4 7. x 3 6 11. y 4 4 15. 15 3y 19. –10x 20 4. z 6 4 8. 19 y 17 12. 25 d 22 16. –10 4h 20. 12 1.2r 24. –10 2.5z 28. 5x 4 4 3x 5. 31 9. 4 13. 3t 6 17. –6y 24 1 3x. 2 2... |
or (x 2). Using Inequalities to Solve Problems 157 4-9 USING INEQUALITIES TO SOLVE PROBLEMS Many problems can be solved by writing an inequality that describes how the numbers in the problem are related and then solving the inequality. An inequality can be expressed in words in different ways. For example: x 12 A numb... |
Therefore, x 3.578595318. Since the domain is the set of whole numbers, the solution set is {x : x is a counting number less than or equal to 3} or {0, 1, 2, 3}. (4) Check the solution in the words of the problem. 0 shirt costs $14.95(0) $0 1 shirt costs $14.95(1) $14.95 2 shirts cost $14.95(2) $29.90 3 shirts cost $1... |
is 30. 9. The maximum value of 4x 6 is 54. 11. The product of 3x and x 1 is less than 35. 8. The sum of 5x and 2x is at least 70. 10. The minimum value of 2x 1 is 13. 12. When x is divided by 3 the quotient is greater than 7. In 13–19, in each case write and solve the inequality that represents the given conditions. U... |
must save in each of the remaining 8 months? 27. Two consecutive even numbers are such that their sum is greater than 98 decreased by twice the larger. Find the smallest possible values for the integers. 28. Minou wants $29 to buy music online. Her father agrees to pay her $6 an hour for gardening in addition to her $... |
Addition property of inequality: When the same number is added to or subtracted from both sides of an inequality, the order of the new inequality is the same as the order of the original one. Multiplication property of inequality: When both sides of an inequality are multiplied or divided by the same positive number, ... |
b 0.9. 17. If P 2l + 2w, find w when P 17 and l 5. 18. If F 9 5C 32, find C when F 68. In 19–26, find and graph the solution set of each inequality. 20. 2x 3 5 19. 6 x 3 23. 3 x 1 2 22. x 4 1 1 3x 21. 24. (x 2 5) and (2x 14) Review Exercises 163 25. (x 2) or (x 0) 26. (x 4 1) and (2x 18) In 27–30, tell whether each st... |
the maximum load that it can carry is 2,000 pounds. The maintenance supervisor wants to move a replacement part for the air-conditioning unit to the roof. The part weighs 1,600 pounds, and the mechanized cart on which it is being moved weighs 250 pounds. When the maintenance supervisor drives the cart onto the elevato... |
x 6. Which of the following sets is closed under division? (1) nonzero whole numbers (2) nonzero integers (3) nonzero even integers (4) nonzero rational numbers 7. The measure of one side of a rectangle is 20.50 feet. This measure is given to how many significant digits? (1) 1 (2) 2 (3) 3 (4) 4 8. In the coordinate pl... |
is greater than 2 is divided by 4? Explain why or why not. 14. A plum and a pineapple cost the same as three peaches. Two plums cost the same as a peach. How many plums cost the same as a pineapple? Part IV Answer all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary st... |
Scientific Notation 5-8 Dividing by a Monomial 5-9 Dividing by a Binomial Chapter Summary Vocabulary Review Exercises Cumulative Review 167 168 Operations with Algebraic Expressions 5-1 ADDING AND SUBTRACTING ALGEBRAIC EXPRESSIONS Recall that an algebraic expression that is a number, a variable, or a product or quotie... |
. 15abc 6abc d. 8x2 y x2y e. 9y 9y f. 2(a b) 6(a b) EXAMPLE 2 Adding and Subtracting Algebraic Expressions 169 Answers [3 (8)]a 5a [12 (5)]b2 [12 5]b2 7b2 (15 6)abc 9abc (8 1)x2y 7x2y (9 9)y 0y 0 (2 6)(a b) 8(a b) An isosceles triangle has two sides that are equal in length. The length of each of the two equal sides of... |
2x 4. A polynomial has been simplified or is in simplest form when it contains no like terms. For example, 5x3 8x2 5x3 7, when expressed in simplest form, becomes 8x2 7. A polynomial is said to be in descending order when the exponents of a particular variable decrease as we move from left to right. The polynomial x3 ... |
5 3(4)2 5 53 6x2 8 6(4)2 8 104 9x2 13 9(4)2 13 157 ✔ EXAMPLE 5 Simplify: 6a [5a (6 3a)] Solution When one grouping symbol appears within another, first simplify the expres- sion within the innermost grouping symbol. How to Proceed (1) Write the expression (2) Use the commutative property: (3) Use the associative prope... |
) 19. 9y [7 (6y 7)] 21. 5a [3b (2a 4b)] 23. 3y2 [6y2 (3y 4)] 25. d2 [9d (2 4d2)] 27. (x3 9x 5) (4x2 12x 5) 4. (4a) (6a) 6. (7w) (7w) 8. (6x) (4x) (5x) (10x) 10. 4m 9m 12m m 12. 4a (9a 3) 14. 8c (7 9c) 16. r (s 2r) 18. (5x 3) (6x 5) 20. (5 6y) (9y 2) 22. (5x2 4) (3x2 9) 24. (x3 3x2) (2x2 9) 26. (x2 5x 24) (x2 4x 9) In 2... |
deposited in the 3 months. 35. On Tuesday, Melita read 3 times as many pages as she read on Monday. On Wednesday she read 1.5 times as many pages as on Monday, and on Thursday she read half as many pages as on Monday. If Melita read p pages on Monday, represent in terms of p, the total number of pages she read in the ... |
The exponent in each product is the sum of the exponents in the factors, as shown in these examples. In general, when x is a real number and a and b are positive integers: xa xb xa b EXAMPLE 1 Simplify each of the following products: a. x5 x2 b. a7 a c. 32 34 Answers a. x5 x2 x52 x7 b. a7 a a71 a8 c. 32 34 324 36 Note... |
. (ab2)4 c. (32 42)3 Solution a. (a2)3 a2 a2 a2 a222 a6 or (a2)3 a2(3) a6 Answer a6 b. (ab2)4 ab2 ab2 ab2 ab2 (a a a a)(b2 b2 b2 b2) a4b8 or (ab2)4 a1(4)b2(4) a4b8 Answer a4b8 c. (32 42)3 (32)3 (42)3 36 46 (3 4)6 126 or (32 42)3 ((3 4)2)3 (122)3 126 Answer 126 176 Operations with Algebraic Expressions To evaluate the e... |
107 36. 33 22 66 33. 24 22 28 37. 54 5 55 34. 33 22 65 38. (22)3 25 35. 1480 1410 1490 39. (63)4 (64)3 Multiplying by a Monomial 177 Applying Skills 40. Two students attended the first meeting of the Chess Club. At that meeting, they decided that each person would bring one additional person to the next meeting, doubl... |
with Algebraic Expressions EXAMPLE 1 Multiply: Answers 24xyz a. (8xy)(3z) c. (6y3)(y) 6y4 e. (5x2y3)(2xy2) 10x3y5 g. (3x2)3 b. (4a3)(5a5) Answers 20a8 12a5b7 d. (3a2b3)(4a3b4) f. (6c2d4)(0.5d) 3c2d5 (3x2)(3x2)(3x2) 27x6 or (3)3(x2)3 27x6 EXAMPLE 2 Solution Represent the area of a rectangle whose length is 3x and whose... |
: Next, multiply: Finally, combine like terms by addition 8y 2(7y 4y) 5 8y 2(3y) 5 8y 6y 5 2y 5 or subtraction: In many expressions, however, the terms within parentheses cannot be combined because they are unlike terms. When this happens, we use the distributive property to clear parentheses and then follow the order ... |
ES Writing About Mathematics 1. In an algebraic term, how do you show the product of a constant times a variable or the product of different variables? 2. In the expression 2 3(7y), which operation is performed first? Explain your answer. 3. In the expression (2 3)(7y), which operation is performed first? Explain your ... |
y y2) 36. B A 39. 5c2(15c 4c) 42. 3ab(5a2 7b2) 45. 8(2x2 3x 5) 216 12 B 32. 2(8a 6b) B 28 4r 2 1 4s 35. A 38. 5d(d2 3d) 41. ab(a b) 44. 10d(2a 3c 4b) 47. 5r2s2(2r2 3rs 4s2) In 48–50, represent the area of each rectangle whose length l and width w are given. 48. l 5y, w 3y 51. The dimensions of the outer rectangle pictu... |
as a polynomial in simplest form. 74. If 1 pound of grass seed costs 25x cents, represent in terms of x the cost of 7 pounds of seed. 75. If a bus travels at the rate of 10z miles per hour for 4 hours, represent in terms of z the dis- tance traveled. 76. If Lois has 2n nickels, represent in terms of n the number of ce... |
(x 3) x2 3x 4x 12 x2 7x 12 This result can also be illustrated geometricallyx + 4)(x + 3) = 3 + x x(x + 3) 4(x + 3) = 3 + x x 3x x2 4 12 4x 3 x (x + 4)(x + 3) x(x + 3) + 4(x + 3) x2 + 3x + 4x + 12 In general, for all a, b, c, and d: (a b)(c d) a(c d) b(c d) ac ad bc bd Notice that each term of the first polynomial mult... |
3y) x3 3x2y 3x2y 9xy2 9xy2 27y3 x3 0x2y 0xy2 27y3 ➤ ➤ Answer x3 27y3 EXAMPLE 3 Solution Simplify: (2x 5)2 (x 3) (2x 5)2 (x 3) (2x 5)(2x 5) (x 3) 2x(2x) 2x(5) 5(2x) 5(5) (x) (+3) 4x2 10x 10x 25 x 3 4x2 21x 28 Answer 4x2 21x 28 Multiplying Polynomials 185 EXERCISES Writing About Mathematics 1. The product of two binomia... |
) 14. (2y 7)(2y 7) 17. (5y 2)(3y 1) 20. (x y)(x y) 23. (a b)2 26. (9x 5y)(2x 3y) 29. (x 2)(x2 3x 5) 32. (2x 1)(3x 4)(x 3) 35. (x y)3 In 36–43, simplify each expression. 36. (x 7)(x 2) x2 38. r(r 2) (r 5) 40. (x 4)(x 3) (x 2)(x 5) 42. (y 4)2 (y 3)2 37. 2(3x 1)(2x 3) 14x 39. 8x2 (4x 3)(2x 1) 41. (3y 5)(2y 3) (y 7)(5y 1) ... |
b a. • Since x2 x3 x5, then x5 x3 x2. • Since y5 y4 y9, then y9 y4 y5. • Since c4 c c5, then c5 c1 c4. Observe that the exponent in each quotient is the difference between the exponent of the dividend and the exponent of the divisor. In general, when x 0 and a and b are positive integers with a b: xa xb xab Procedure ... |
5a x2a 22. sx s2 (x 2) 20. y10b y2b 23. ab ab 21. rc rd (c d) 24. 2a 2b (a b) In 25–32: a. Simplify each expression by using the rules for multiplying and dividing powers with like bases. b. Evaluate the expression using a calculator. Compare your answers to parts a and b. 25. 29. 23? 24 22 106 102? 104 26. 30. 58 54? ... |
1 (x 0) permits us to say that the zero power of any number except 0 equals 1. 40 1 (4)0 1 (4x)0 1 (4x)0 1 A calculator will return this value. For example, to evaluate 40: ENTER: 4 ^ 0 ENTER DISPLAY: 4 ^ 0 1 Note that 4x0 41 x0 4 1 4 but (4x)0 40 x0 1 1 1. The Negative Integral Exponent We know that, for x 0, x3 x5 5... |
B A B 16 5 1 1 16 5 6 27 5 2 9 Use the laws of exponents to perform the indicated operations. Answers Answers a. 27 23 27(3) 24 b. 36 32 36(2) 362 34 c. (x4)3 x4(3) x12 d. (y2)4 y2(4) y8 Scientific Notation 191 EXERCISES Writing About Mathematics 1. Sasha said that for all x 0, x2 is a positive number less than 1. Do ... |
Scientific Notation To write a number in scientific notation, first write it as the product of a number between 1 and 10 times a power of 10. Then express the power of 10 in exponential form. The table at the right shows some integral powers of 10. When the exponent is a positive integer, the power can be written as 1... |
the exponent will be negative. Place a caret after the first nonzero digit to indicate the position of the dec- imal point in scientific notation. Answer 0.0000029 0.000002 9 2.9 106 ^ 6 Graphing calculators can be placed in scientific notation mode and will return the results shown in Examples 1 and 2 when the given ... |
the mass of 2.70 1015 hydrogen atoms if the mass of one hydrogen atom is 1.67 1024 grams. Round the answer to three significant digits. Solution Multiply the mass of one hydrogen atom by the number of hydrogen atoms. (1.67 1024) (2.70 1015) (1.67 2.70) (1024 1015) (1.67 2.70) (1024 15) 4.509 109 Round 4.509 to 4.51, w... |
10n 31. 0.00456 4.56 10n 23. 5,280 5.28 10n 26. 52,000 5.2 10n 29. 0.800 8.00 10n 32. 7,123,000 7.123 10n In 33–44, express each number in scientific notation. 33. 8,400 37. 0.00061 41. 453,000 34. 27,000 38. 0.0000039 42. 0.00381 35. 54,000,000 36. 320,000,000 39. 0.0000000140 40. 0.156 43. 375,000,000 44. 0.0000763 ... |
write: 2x4 5 230 2 a5 5 221 a4?? 23 y2 3y0z1 3 1 z 3z y2? 7a1b3 7ab3 221a5b4 23a4b 15x2 12y2z2 4y2z x6 x4 b4 b 230x6?? 5 12 z2 4 z Procedure To divide a monomial by a monomial: 1. Divide the numerical coefficients. 2. When variable factors are powers of the same base, divide by subtracting exponents. 3. Multiply the q... |
ab 5 7a 2 1 Usually, the two middle steps are done mentally. Procedure To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. Dividing by a Monomial 199 EXAMPLE 3 Divide: a. (8a5 6a4) 2a2 b. 24x3y4 2 18x2y2 2 6xy 26xy Answers 4a3 3a2 4x2y3 3xy 1 EXERCISES Writing About Mathematics 1. ... |
. 30. If 40ab chairs are arranged in 5a rows with equal numbers of chairs in each row, represent the number of chairs in one row. 200 Operations with Algebraic Expressions 5-9 DIVIDING BY A BINOMIAL When we divide 736 by 32, we use repeated subtraction of multiples of 32 to determine how many times 32 is contained in 7... |
3s 2)(2s 3) 3s(2s 3) 2(2s 3) 6s2 9s 4s 6 6s2 5s 6 ✔ Note that we subtracted 9s from 5s by adding 9s to 5s. Answer 3s 2 EXERCISES Writing about Mathematics 1. Nate said that 2. Mason wrote x3 1 as x3 0x2 0x 1 before dividing by x 1. 1 5 x2 2 1 x 1 1 5 x3 x3 2 1 x 1 21. Is Nate correct? Explain why or why not. a. Does x3... |
ials. To subtract one polynomial from another, add the opposite of the polynomial to be subtracted (the subtrahend) to the polynomial from which it is to be subtracted (the minuend). When x is a nonzero real number and a and b are integers: xa xb xa b (xa)b xab xa xb xa b x0 1 x2a 5 1 xa A number is in scientific notat... |
)(2x 1) 11. (2a 5)(2a 5) 14. 17. 40b3c6 28b2c x2 1 x 2 30 x 2 5 In 18–21, use the laws of exponents to perform the operations, and simplify. 18. 35 34 21. 120 122 12 20. [2(102)]3 19. (73)2 In 22–25, express each number in scientific notation. 22. 5,800 23. 14,200,000 24. 0.00006 25. 0.00000277 In 26–29, find the decim... |
(2) 1.67 (3) 1.67 (4) 12 7 1 2. For which of the following values of x is x2 x? x (1) 1 (2) 0 (3) 3 (4) 3. Which of the numbers given below is not a rational number? (2) 11 2 (3) 1.3 (4) 2 3 7 3 (1) 2 " 4. Which of the following inequalities is false? (1) 1.5 11 2 (2) 1.5 11 2 (3) 1.5 1.5 (4) 1.5 1 5. Which of the fol... |
answer with no work shown will receive only 1 credit. 11. The formula for the volume V of a cone is V 5 1 3Bh where B is the area of the base and h is the height. Solve the formula for h in terms of V and B. 12. Each of the numbers given below is different from the others. Explain in what way each is different. 2 7 77... |
offer discounts and other price reductions.These discounts are often expressed as a percent off of the regular price. When the Acme Grocery offers a 25% discount on frozen vegetables and the Shop Rite Grocery advertises “Buy four, get one free,” the price-conscious shopper must decide which is the better offer if she ... |
ratio is a : b because b 3 x 5 ax x bx a b 5 a 24 16 b 3 1 5 a is a fraction, we can divide the numerator and the Also, since a ratio such as denominator of the fraction by the same nonzero number to find equivalent ratios. For example: 24 16 5 24 4 2 16 4 4 5 6 4 A ratio is expressed in simplest form when both terms ... |
4 16 3 10 10 5 64 160 5 64 4 32 160 4 32 5 2 5 210 Ratio and Proportion Calculator Solution On a calculator, divide 6.4 ounces by 16 ounces. ENTER: 6.4 16 ENTER DISPLAY: 6. 4 / 1 6. 4 Change the decimal in the display to a fraction. ENTER: DISPLAY: 2nd ANS MATH ENTER ENTER Answer The ratio is 2 : 5. EXAMPLE 3 Express t... |
cents 32. A baseball team played 162 games and won 90. a. What is the ratio of the number of games won to the number of games played? b. For every nine games played, how many games were won? 33. A student did six of ten problems correctly. a. What is the ratio of the number right to the number wrong? b. For every two ... |
in lowest terms when the numbers in its ratio are whole numbers with no common factor other than 1. However, a rate is most frequently written as a ratio with 1 as its second term. As shown in the example above, the second term may be omitted when it is 1. A rate that has a denominator of 1 is called a unit rate. A ra... |
can of shaving cream costs 88 cents, what is the unit cost of the shaving cream in the can? 15. In a supermarket, the regular size of CleanRight cleanser contains 14 ounces and costs 49 cents. The giant size of CleanRight cleanser, which contains 20 ounces, costs 66 cents. a. Find, correct to the nearest tenth of a ce... |
25 60 ✔ Answer The lengths of the sides are 15 feet, 20 feet, and 25 feet. EXAMPLE 2 Two numbers have the ratio 2 : 3. The larger is 30 more than of the smaller. Find the numbers. 1 2 Solution Let 2x the smaller number, 3x the larger number. Verbal Problems Involving Ratio 215 Check The ratio 30 : 45 in lowest terms i... |
the three angles of a triangle is 2 : 2 : 5. Find the measures of each angle. 13. In a triangle, two sides have the same length. The ratio of each of these sides to the third side is 5 : 3. If the perimeter of the triangle is 65 inches, find the length of each side of the triangle. 14. Two positive numbers are in the ... |
mean ↓ ↓ a b 5 c d ↑ ↑ mean extreme In the proportion, 4 : 20 1 : 5, the product of the means, 20(1), is equal to the product of the extremes, 4(5)., the product of the means, 15(10), is equal to the In the proportion, 15 5 10 5 30 product of the extremes, 5(30). a b 5 c, we can show that the product of the means is e... |
20 80 80 Therefore, 16 5 5 4 20 is a proportion. METHOD 3 Use a calculator. Enter the proportion. If the ratios are equal, then the calculator will display 1. If the ratios are not equal, the calculator will display 0. ENTER: 4 16 2nd TEST ENTER 5 20 ENTER DISPLAY Since the calculator displays 1, the statement is true... |
5 5 15 5 1 3 ✔ EXERCISES Writing About Mathematics 1. Jeremy said that if the means and the extremes of a proportion are interchanged, the result- ing ratios form a proportion. Do you agree with Jeremy? Explain why or why not. 2. Mike said that if the same number is added to each term of a proportion, the resulting ra... |
3 times the denominator. If the numerator is decreased by 1 and the denominator is increased by 2, the value of the resulting fraction is. Find the original fraction. 5 2 33. What number must be added to both the numerator and denominator of the fraction make the resulting fraction equal to? 3 4 7 19 to 34. The numera... |
the formula for In a direct variation, the value of each term of the ratio increases when we multiply each variable by a factor greater than 1; the value of each term of the ratio decreases when we divide each variable by a factor greater than 1, as shown below 12 4 3 EXAMPLE 1 If x varies directly as y, and x 1.2 whe... |
trip, Natasha drives at an average speed of 65 miles per hour. She says that each day, her driving time and the distance that she travels are directly proportional. Do you agree with Natasha? Explain why or why not. 2. The cost of parking at the Center City Parking Garage is $5.50 for the first hour or part of an hour... |
vary. b. How will the area of a rectangle whose length is 8 inches compare with the area of a rectangle whose length is 4 inches? c. If l is tripled, what change takes place in A? 27. The variable d varies directly as t. If d 520 when t 13, find d when t 9. 28. Y varies directly as x. If Y 35 when x 5, find Y when x 2... |
16,500? 43. The scale on a map is given as 5 centimeters to 3.5 kilometers. How far apart are two towns if the distance between these two towns on the map is 8 centimeters? 44. David received $8.75 in dividends on 25 shares of a stock. How much should Marie receive in dividends on 60 shares of the same stock? 45. A pic... |
For example: 50 4 or 50 0.08 4 8 100 Just as we have seen two ways to look at this problem involving sales tax, we will see more than one approach to every percentage problem. Note that when we calculate using percent, we always use the fraction or decimal form of the percent. Percent of Error When we use a measuring ... |
rate of the discount is 25%. Therefore the customer paid (100 25)% or 75% of the regular price. The percentage is given as $73.50, and the base is not known. Let n the regular price, or base. METHOD 1 Use the proportion. percentage base 5 rate n 5 75 73.50 100 75n 7,350 n 98 Check If 25% of 98 is subtracted from 98 do... |
| $30. Percent of increase 30 600 0.05 1 20 Change 0.05 to a percent: 0.05 5% Answer The percent of increase is 5%. EXAMPLE 4 Solution A store reduced the price of a television from $840 to $504. What was the percent of decrease in the price of the television? Original price $840 New price $504 Amount of decrease |$840... |
class? 232 Ratio and Proportion 30. Marie bought a dress that was marked $24. The sales tax is 8%. a. Find the sales tax. b. Find the total amount Marie had to pay. 31. There were 120 planes on an airfield. If 75% of the planes took off for a flight, how many planes took off? 32. One year, the Ace Manufacturing Compan... |
the first day of a sale, a camera was reduced by $8. This represented 10% of the original price. On the last day of the sale, the camera was sold for 75% of the original price. What was the final selling price of the camera? Percent and Percentage Problems 233 46. The regular ticketed prices of four items at Grumbell’... |
if his weight displayed on this scale is 144 pounds? 52. Isaiah answered 80% of the questions correctly on the math midterm, and 90% of the questions correctly on the math final. Can you conclude that he answered 85% of all the questions correctly (the average of 80% and 90%)? Justify your answer or give a counterexam... |
8 cm This answer, rounded to the nearest tenth, can be expressed as 99.4 centimeters. EXAMPLE 1 Solution If there are 5,280 feet in a mile, find, to the nearest hundredth, the number of miles in 1,200 feet. How to Proceed (1) Write a fraction equal to 1 with the required unit in the numerator and the given unit in the ... |
given rate to the required unit of measure. (1) Write 60 miles per hour as a fraction: (2) Change miles to feet. Multiply by a ratio with miles in the denominator to cancel miles in the numerator: (3) Change hours to minutes. Multiply by a ratio with hours in the numerator to cancel hours in the denominator: (4) Chang... |
quarts. 17. Miranda needs boards 0.8 meters long for a building project. The boards available at the local lumberyard are 2 feet, 3 feet, and 4 feet long. a. Express the length, to the nearest hundredth of a foot, of the boards that Miranda needs to buy. b. Which size board should Miranda buy? Explain your answer. 18.... |
extremes, and the inner terms are the means. Proportion: means a : b c : d extremes or extreme mean ↓ ↓ a b 5 c d ↑ ↑ mean extreme In a proportion, the product of the means is equal to the product of the extremes, or alternatively, the cross products are equal. This process is also called cross-multiplication. A direc... |
ratio in simplest form. 3. 30 : 35 4. 8w to 12w 6. 75 millimeters : 15 centimeters 5. 3 8 to 5 8 In 7–9, in each case solve for x and check. 7. 8 2x 5 12 9 8. x x 1 5 5 1 2 9. x 5 6 4 x 1 3 10. The ratio of two numbers is 1 : 4, and the sum of these numbers is 40. Find the numbers. 240 Ratio and Proportion In 11–13, i... |
four carpenters can build four tables in 4 days, how long will it take one carpenter to build one table? 24. How many girls would have to leave a room in which there are 99 girls and 1 boy in order that 98% of the remaining persons would be girls? 25. On an Australian highway, the speed limit was 110 kilometers per ho... |
movie is shown in two versions, the original and the director’s cut. However, movie theatres can play only one of the versions. A journalist for XYZ News, reports that since 30% of theatres are showing the director’s cut and 60% are showing the original, the movie is playing in 90% of all movie theatres. Is the report... |
Part II (4) $70.40 (4) 6 Answer all questions in this part. Each correct answer will receive 2 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 cre... |
the edge. 64 ft 150 ft a. Scott skates once around the rink. Find, to the nearest ten feet, the dis- tance that he skated. b. Scott wants to skate at least 5 miles. What is the smallest number of complete trips around the rink that he must make? GEOMETRIC FIGURES, AREAS, AND VOLUMES A carpenter is building a deck on t... |
plane are undefined words, we must have a clear understanding of what they mean. Knowing the properties and characteristics they possess helps us to achieve this understanding. A point indicates a place or position. It has no length, width, or thickness. A point is usually indicated by a dot and named with a capital l... |
or the length of a line segment is the distance between its endpoints. We use a number line to associate a number with each endpoint. Since the coordinate of A is 0 and the coordinate of B is 5, the length of AB is 5 0 or AB 5 Note: The segment is written as endpoints. The length of the segment is written as AB, with ... |
direction about O to the position rotating, that have the common endpoint O, is TOS. union of the two rays, Note that when three letters are used to name an angle, the letter that names the vertex is always in the middle. Since are the only rays in the diagram that have the common endpoint O, the angle could also have... |
L Obtuse angle 180° T s S Straight angle R 1. The measure of an angle depends only on the amount of rotation, not on the pictured lengths of the rays forming the angle. 2. Since every right angle measures 90°, all right angles are equal in measure. 3. Since every straight angle measures 180°, all straight angles are e... |
. Pairs of Angles 251 In the figures shown below, because mCAB mFDE 25 65 90, CAB and FDE are complementary angles. Also, because mHGI mIGJ 53 37 90, HGI and IGJ are complementary angles. C 25° B A E F 65° D J I 37° 53° G H If the measure of an angle is 50°, the measure of its complement is (90 50)°, or 40°. In general... |
angles: AED and DEB DEB and BEC BEC and CEA CEA and AED The angles of each linear pair are supplementary. • If mAED 130, then mDEB 180 130 50. • If mDEB 50, then mBEC 180 50 130. Therefore, mAED mBEC. A C 130° E D 50° 130° B DEFINITION When two angles have equal measures, they are congruent. We use the symbol to repre... |
the measure of the angle. Find the measure of the angle. Solution Let x measure of angle. Then 4x measure of complement of angle. The sum of the measures of an angle and its complement is 90°. x 4x 90 5x 90 x 18 Check The measure of the first angle is 18°. The measure of the second angle is 4(18°) 72°. The sum of the ... |
. What is the measure of the complement of the angle? b. What is the measure of the supplement of the angle? c. The measure of the supplement of the angle is how much larger than the measure of its complement? 3. 15° 4. 37° 5. 67° 6. x° In 7–10, A and B are complementary. Find the measure of each angle if the measures ... |
of degrees in an angle that measures 8° less than its complement. 25. The supplement of the complement of an acute angle is always: (1) an acute angle (2) a right angle (3) an obtuse angle (4) a straight angle In 26–28, g MN and g RS intersect at T. 26. If mRTM 5x and mNTS 3x 10, find mRTM. 27. If mMTS 4x 60 and mNTR ... |
QS are complementary; mPQR 30; a. mRQS b. mSQT c. mCEA g RQT c. mPQT is a line. Find: 258 Geometric Figures, Areas, and Volumes at R. The measure of LRQ is 80 more than mLRP. Find: b. mLRQ c. mPRM at E. Point F is in the interior of CEB. The measure of CEF is 8 times the 37. 38. g PQ intersects g LM a. mLRP g g'AB CD m... |
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