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Math Olympics

Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Give two ways to write the algebraic

expression p ÷ 10 in words. a. the product of p and 10

p times 10 b. p subtracted from 10

p less than 10 c. the quotient of 10 and p

10 divided by p d. the quotient of p and 10

p divided by 10

2. Julia wrote 14 letters to friends each month for y months in a row. Write an expression to show how many total letters Julia wrote. a. 14 – y b. 14y c. 14 + y d.

3. Salvador’s class has collected 88 cans in a food drive. They plan to sort the cans into x bags, with an equal number of cans in each bag. Write an expression to show how many cans there will be in each bag. a. 88x b. 88 + x c. d. 88 – x

4. Evaluate the expression m + o for m = 9 and o = 7. a. 15 b. 16 c. 2 d. 63

5. Evaluate the expression q – v for q = 5 and v = 1. a. 3 b. 4 c. 5 d. 6

6. Evaluate the expression xy for x = 6 and y = 3. a. 21 b. 24 c. 9 d. 18

7. Evaluate the expression a ÷ b for a = 24 and b = 8. a. 192 b. 16 c. 3 d. 4

8. Mike scored 40 points in the first half of the basketball game, and he scored y points in the second half of the game. Write an expression to determine the number of points he scored in all Then, find the number of points he scored in all if he scored 2 points in the second half of the game. a. 40y; 42 points b. 40 – y; 38 points c. ; 38 points d. 40 + y; 42 points

9. Aaron has saved 72 sand dollars and wants to give them away equally to y friends. Write an expression to show how many sand dollars each of Aaron’s friends will receive. Then, find the total number of sand dollars each of Aaron’s friends will get if Aaron gives them to 12 friends. a. 72 – y; 60 sand dollars b. 72 + y; 60 sand dollars c. ; 6 sand dollars d. 72y; 6 sand dollars

10. Salvador reads 12 books from the library each month for n months in a row. Write an expression to show how many books Salvador read in all. Then, find the number of books Salvador read if he read for 7 months.

a. 12 – n; 19 books b. ; 84 books c. 12 + n; 19 books d. 12n; 84 books

11. Evaluate the expression for and .

a. 23 b. 25 c. 32 d. 18

12. Subtract using a number line. –5 – (–3)

0 1 2 3 4 5 6 7 80–1–2–3–4–5–6–7–8

–5

– (–3)

a. –3 b. 2 c. –5 d. –2

13. Add. 34 + (–21) a. 55 b. 13 c. –55 d. –13

14. Evaluate x + (–9) for x = 35. a. –44 b. –26 c. 26 d. 44

15. Subtract. –5 – (–8) a. –13 b. 13 c. 3 d. –3

16. Evaluate x – (–10) for x = 12. a. –22 b. 2 c. 22 d. –2

17. The highest temperature recorded in the town of Westgate this summer was 101ºF. Last winter, the lowest temperature recorded was –9ºF. Find the difference between these extremes. a. 92ºF b. –92ºF c. 110ºF d. –110ºF

18. The temperature on the ground during a plane’s takeoff was 4ºF. At 38,000 feet in the air, the temperature outside the plane was –38ºF. Find the difference between these two temperatures.

a. –34ºF b. 42ºF c. 34ºF d. –42ºF

19. The elevator in the a downtown skyscraper goes from the top floor down to the lowest level of the underground parking garage. If the building is 470 feet tall and the elevator descends 530 feet from top to bottom, how far underground does the parking garage go? a. 990 feet b. 60 feet c. 1,000 feet d. 50 feet

20. Multiply. –8 • 9 a. 1 b. –72 c. –17 d. 72

21. Evaluate –5u for u = –4. a. –9 b. 25 c. –20 d. 20

22. Divide. –48 8 a. –384 b. 6 c. –56 d. –6

23. Evaluate k (–11) for k = –33. a. 363 b. 3 c. –22 d. –3

24. Divide.

a. 6 821 b. 457 c. 8

821 d. 6

314

25. Divide. 0 ÷ 5.928 a. –5.928 b. 5.928 c. undefined d. 0

26. Carina hiked at Yosemite National Park for 1.75 hours. Her average speed was 3.5 mi/h. How many miles did she hike? a. 2 mi b. 20 mi c. 61.25 mi d. 6.125 mi

27. Write the power represented by the geometric model.

a. 35 b. 52 c. 53 d. 25

28. Simplify . a. 27 b. 93 c. 729 d. 12

29. Simplify . a. –81 b. 81 c. 1 d. –12

30. Simplify . a. –2 b. 16 c. –8 d. –16

31. Simplify .

a. b. 536 c. 256 d.

2536

32. Write 9 as a power of the base 3. a. b. c. d.

33. Suppose you have developed a scale that indicates the brightness of sunlight. Each category in the table is 6 times brighter than the next lower category. For example, a day that is dazzling is 6 times brighter than a day that is radiant. How many times brighter is a dazzling day than an illuminated day?

Sunlight Intensity Category Brightness

Dim 2 Illuminated 3

Radiant 4 Dazzling 5

a. 36 times brighter b. 2 times brighter c. 6 times brighter d. 216 times brighter

34. If the population of an ant hill doubles every 10 days and there are currently 40 ants living in the ant hill, what will the ant hill population be in 20 days? a. 320 ants b. 160 ants c. 1,600 ants d. 80 ants

35. The design shows the layout of a vegetable garden and the surrounding path. The path is 1.5 feet wide. First, find the total area of the vegetable garden and path. Then, find the area of the vegetable garden and the area of the path. If one bag of gravel covers 10 square feet, how many bags of gravel are needed to cover the path?

12 ft

12 ft

a. The total area is 81 sq ft. The area of the vegetable garden is 144 sq ft, and the area of the path is 63 sq ft.

To cover the path, 7 bags of gravel are needed. b. The total area is 144 sq ft. The area of the vegetable garden is 110.25 sq ft, and the area of the path is 33.75 sq ft. To cover the path, 4 bags of gravel are needed. c. The total area is 144 sq ft. The area of the vegetable garden is 81 sq ft, and the area of the path is 63 sq ft. To cover the path, 7 bags of gravel are needed. d. The total area is 144 sq ft. The area of the vegetable garden is 72 sq ft, and the area of the path is 72 sq ft. To cover the path, 8 bags of gravel are needed.

36. Find the square root.

a. 14 b. 98 c. 38416 d. –14

37. The area of a square garden is 202 square feet. Estimate the side length of the garden. a. 16 ft b. 12 ft c. 17 ft d. 14 ft

38. Write all classifications that apply to the real number . a. rational number, terminating decimal b. rational number, repeating decimal c. irrational number d. rational number

39. Write all classifications that apply to the real number . a. irrational number, integer b. irrational number c. rational number, terminating decimal, integer, whole number, natural number d. rational number, terminating decimal

40. A set of numbers is said to be closed under a certain operation if, when you perform the operation on any two numbers in the set, the result is also a number in the set. Is the set of irrational numbers closed under addition? Explain.

a. Yes, the set of irrational numbers is closed under addition. For example, the sum of 0.121221222.. and 0.131331333.. is 0.252552555... which is an irrational number. b. Yes, the set of irrational numbers is closed under addition. The result of adding any two irrational numbers is an irrational number. c. No, the set of irrational numbers is not

closed under addition. For example, the sum of and is not an irrational number. d. No, the set of irrational numbers is not closed under addition. The result of adding any two irrational numbers is an irrational number.

41. Simplify . a. –1 b. 2 c. 22 d. 14

42. Simplify . a. 21 b. 75 c. 39 d. 93

43. Evaluate for x = 9. a. –68 b. 58 c. –5 d. 72

44. Evaluate 1 + x2 • 6 for x = 4. a. 102 b. 97 c. 94 d. 150

45. Simplify the expression .

a. 14 b. 23 c. 4 d. 22 46. Translate the word phrase, the

product of 8.5 and the difference of –4 and –8, into a numerical expression. a. b. c. d.

47. Tatia has coins in pennies, nickels, dimes, and quarters. The total amount of money she has in dollars can be found using the expression (P + 5N + 10D + 25Q) ÷ 100. Use the table to find how much money Tatia has.

P N D Q 20 16 4 2

a. $140.50 b. $33.30 c. $0.42 d. $1.90

48. Use the numbers 2, 3, 5, and 8 to write an expression that has a value of . You may use any operations, and you must use each of the numbers at least once.

a. b. c.

d.

49. Simplify the expression .

a. 1059 b. 10 c. 959 d. 9

50. Write 11 • 59 using the Distributive Property. Then simplify. a. 11 • 50 + 11 • 9; 649 b. (11 + 50)(11 + 9); 1,220 c. 11 • 59 + 11 • 9; 748 d. 11 • 5 + 11 • 9; 154

51. Write using the Distributive Property. Then simplify.

a. ; 130 b. ; 114 c. ; 60 d. ; 168

52. Simplify by combining like terms.

a. b. c. d.

53. The table shows, step-by-step, how to simplify the algebraic expression . Justify Step 4.

Step Procedure Justification

1. 2. Distributive Property 3. 4. 5. 6.

a. Multiply b. Associative Property c. Combine like terms d. Commutative Property

54. Fill in the missing justifications.

Procedure Justification Definition of subtraction

?

?

?

Simplify Definition of subtraction a. Distributive Property; Associative Property; Commutative Property b. Associative Property; Commutative Property; Distributive Property c. Commutative Property; Distributive Property; Associative Property d. Commutative Property; Associative Property; Distributive Property

55. Graph the point (1, 4).

a.

5–5 x

5

–5

y

b.

5–5 x

5

–5

y

c.

5–5 x

5

–5

y

d.

5–5 x

5

–5

y

56. Name the quadrant where the point (–3, 2) is located.

5–5 x

5

–5

y

a. Quadrant III b. Quadrant I c. Quadrant IV d. Quadrant II

57. Name the quadrant where the point (3, 0) is located.

5–5 x

5

–5

y

a. Quadrant III b. No quadrant (y-axis) c. No quadrant (x-axis) d. Quadrant I

58. A phone company advertises a new plan in which the customer pays a fixed amount of $25 per month for unlimited calls in the country, and $0.10 per minute for international calls. Find a rule for the monthly payment a customer pays according to the new plan. Write ordered pairs for the monthly payment when the customer uses 90, 120, 145, and 150 international minutes in a month. a. ; (34, 90), (37, 120), (39.5, 145), (40, 150) b. ; (90, 34), (120, 37), (145, 39.5), (150, 40) c. ; (34, 90), (37, 120), (145, 39.5), (150, 40) d. ; (34, 90), (37, 120), (145, 39.5), (150, 40)

59. Create a table of ordered pairs for the function using the values x = –2, –1, 0, 1, and 2. Graph the ordered pairs and describe the shape of the graph. a.

The points form an S shape. b.

The points form a straight line. c.

The points form a V shape. d.

The points form a U shape.

60. The coordinates of three vertices of a rectangle are , , and . Find the coordinates of the fourth vertex. Then, find the area of the rectangle. a. ; Area = 80 square units b. ; Area = 72 square units c. ; Area = 80 square units d. ; Area = 72 square units

61. Give two ways to write the algebraic expression 6p in words. a. the quotient of 6 and p

6 divided by p b. p subtracted from 6

p less than 6 c. 6 times p

6 groups of p d. p more than 6

p added to 6

62. Add using a number line. 3 + 3

0 1 2 3 4 5 6 7 80–1–2–3–4–5–6–7–8

3

3

a. 0 b. 6 c. –6 d. 3

63. Solve . a. p = 22 b. p = –22 c. p = 10 d. p = –10

64. Solve .

a. s = 52 b. s = 42 c. s = 43 d. s = 54

65. Solve –14 + s = 32.

a. s = 46 b. s = 18 c. s = –46 d. s = –18 66. A toy company's total payment for salaries for the first two months of 2005 is

$21,894. Write and solve an equation to find the salaries for the second month if the first month’s salaries are $10,205. a.

The salaries for the second month are $11,689. b.

The salaries for the second month are $21,894. c.

The salaries for the second month are $10,947. d.

The salaries for the second month are $32,099.

67. The range of a set of scores is 23, and the lowest score is 33. Write and solve an equation to find the highest score. (Hint: In a data set, the range is the difference between the highest and the lowest values.) a.

The highest score is 10. b.

The highest score is 56. c.

The highest score is –10. d.

The highest score is 79.

68. Solve .

a. q = 46 b. q = 205 c. q = 36 d. q = 815

69. Solve 3n = 42. a. n = 39 b. n = 15 c. n = 45 d. n = 14

70. Solve .

a. b. c. d.

71. The time between a flash of lightning and the sound of its thunder can be used to estimate the distance from a lightning strike. The distance from the strike is the number of seconds between seeing the flash and hearing the thunder divided by 5. Suppose you are 17 miles from a lightning strike. Write and solve an equation to find how many seconds there would be between the flash and thunder. a. , so t is about 85 seconds.

b. , so t is about 3.4 seconds.

c. , so t is about 22 seconds. d. , so t is about 0.3 seconds.

72. If , find the value of . a. 3 b. –5 c. 5 d. –3

73. Solve . a. a = –29 b. a = 29 c. a = 15 d. a = –15

74. Solve .

a. b. c. d.

75. Solve . a. b. c. d.

76. Sara needs to take a taxi to get to the movies. The taxi charges $4.00 for the first mile, and then $2.75 for each mile after that. If the total charge is $20.50, then how far was Sara’s taxi ride to the movie? a. 6 miles b. 7 miles c. 5.1 miles d. 7.5 miles

77. If 8y – 8 = 24, find the value of 2y. a. 8 b. 11 c. 2 d. 24

78. The formula gives the profit p when a number of items n are each sold at a cost c and expenses e are subtracted. If

, , and , what is the value of c? a. 0.80 b. 1.55 c. 1.25 d. 0.95

79. Solve . a. b. c. d.

80. Solve . a. n = 112 b. n = −4

12 c. n = 3

12 d. n = −1

16

81. Solve . Tell whether the equation has infinitely many solutions or no solutions. a. Two solutions b. No solutions c. Infinitely many solutions d. Only one solution

82. A video store charges a monthly membership fee of $7.50, but the charge to rent each movie is only $1.00 per movie. Another store has no membership fee, but it costs $2.50 to rent each movie. How many movies need to be rented each month for the total fees to be the same from either company? a. 3 movies b. 5 movies c. 7 movies d. 9 movies

83. Find three consecutive integers such that twice the greatest integer is 2 less than 3 times the least integer. a. 2, 3, 4 b. 4, 5, 6 c. 6, 7, 8 d. 8, 9, 10

84. A professional cyclist is training for the Tour de France. What was his average speed in kilometers per hour if he rode the 194 kilometers from Laval to Blois in 4.7 hours? Use the formula , and round your answer to the nearest tenth. a. 189.3 kph b. 911.8 kph c. 115.3 kph d. 41.3 kph

85. The formula for the resistance of a conductor with voltage V and current I is . Solve for V.

a. I = Vr b. c. V = Ir

d.

86. Solve for x.

a. b.

c. d.

87. Solve for y.

a. b.

c. d.

88. The fuel for a chain saw is a mix of oil and gasoline. The ratio of ounces of oil to gallons of gasoline is 7:19. There are 38 gallons of gasoline. How many ounces of oil are there? a. 103.1 ounces b. 20 ounces c. 14 ounces d. 3.5 ounces

89. Ramon drives his car 150 miles in 3 hours. Find the unit rate. a. Ramon drives 50 miles per hour. b. Ramon drives 1 mile per 50 hours. c. Ramon drives 30 miles per hour. d. Ramon drives 150 miles per 3 hours.

90. The local school sponsored a mini-marathon and supplied 84 gallons of water per hour for the runners. What is the amount of water in quarts per hour? a. 672 qt/h b. 336 qt/h c. 168 qt/h d. 21 qt/h

91. Solve the proportion .

a. x = 36 b. x = 26 c. x = 0.03 d. x = 25

92. An architect built a scale model of a shopping mall. On the model, a circular fountain is 20 inches tall and 22.5 inches in diameter. If the actual fountain is to be 8 feet tall, what is its diameter? a. 7 ft b. 7.1 ft c. 9 ft d. 10.5 ft

93. Complementary angles are two angles whose measures add to 90°. The ratio of the measures of two complementary angles is 4:11. What are the measures of the angles? a. 51.4°, 38.6° b. 26°, 64° c. 24°, 66° d. 24°, 114°

94. Find the value of MN if cm, cm, and cm. ABCD LMNO

a. 22.4 cm b. 12.6 cm c. 22.8 cm d. 23.8 cm

95. On a sunny day, a 5-foot red kangaroo casts a shadow that is 7 feet long. The shadow of a nearby eucalyptus tree is 35 feet long. Write and solve a proportion to find the height of the tree.

a. ; 25 feet

b. ; 49 feet

c. ; 245 feet

d. ; 175 feet

96. A right triangle has legs 15 inches and 12 inches. Every dimension is multiplied by to form a new right triangle with legs 5 inches and 4 inches. How is the ratio of the areas related to the ratio of corresponding sides? a. The ratio of the areas is the square of the ratio of the corresponding sides. b. The ratio of the areas is equal to the ratio of the corresponding sides. c. The ratio of the areas is the cube of the ratio of the corresponding sides. d. None of the above

97. Triangles C and D are similar. The area of triangle C is 47.6 . The base of triangle D is 6.72 in. Each dimension of D is the corresponding dimension of C. What is the height of D ? a. 20.4 in b. 17 in c. 5.6 in d. 57.12 in

98. Find 55% of 125. a. 227.27 b. 68.75 c. 70.25 d. 6875

99. What percent of 74 is 481? If necessary, round your answer to the nearest tenth of a percent. a. 6.5% b. 650% c. 550% d. 15.38%

100. 66 is 56% of what number? If necessary, round your answer to the nearest hundredth. a. 0.85 b. 117.86 c. 1.18 d. 36.96

101. A compound is made up of various elements totaling 80 ounces. If the total amount of lead in the compound weighs 15 ounces, what percent of the compound is made up of lead? If necessary, round your answer to the nearest hundredth of a percent. a. 81.25% b. 18.75% c. 5.33% d. 0.19%

102. According to the United States Census Bureau, the United States population was projected to be 293,655,404 people on July 1, 2004. The two most populous states were California, with a population of 35,893,799, and Texas, with a population of 22,490,022. About what percent of the United States population lived in California or Texas? Round your answer to the nearest percent. a. 8% b. 12% c. 20% d. 37%

103. Aaron works part time as a salesperson for an electronics store. He earns $6.75 per hour plus a percent commission on all of his sales. Last week Aaron worked 17 hours and earned a gross income of $290.63. Find Aaron’s percent commission if his total sales for the week were $3,350. If necessary, round your answer to the nearest hundredth of a percent. a. 1.03% b. 0.05% c. 5.25% d. 6%

104. After 6 months the simple interest earned annually on an investment of $8000 was $975. Find the interest rate to the nearest tenth of a percent. a. 0.2 % b. 22.4% c. 0.244% d. 24.4%

105. Hidemi is a waiter. He waits on a table of 4 whose bill comes to $69.98. If Hidemi receives a 20% tip, approximately how much will he receive?

a. $14.00 b. $84.00 c. $13.55 d. $3.50

106. Hannah had dinner at her favorite restaurant. If the sales tax rate is 4% and the sales tax on the meal came to $1.25, what was the total cost of the meal, including sales tax and a 20% tip? a. $52.50 b. $45.63 c. $31.25 d. $38.75

107. Find the percent change from 52 to 390. Tell whether it is a percent increase or decrease. If necessary, round your answer to the nearest percent. a. 650% decrease b. 87% decrease c. 650% increase d. 87% increase

108. Find the result when 28 is decreased by 25%. a. 21 b. 35 c. 7 d. 3

109. The price of a train ticket from Atlanta to Oklahoma City is normally $117.00. However, children under the age of 16 receive a 70% discount. Find the sale price for someone under the age of 16. a. $35.10 b. $198.90 c. $81.90 d. $49.14

110. A bookstore buys Algebra 1 books at a wholesale price of $16 each. It then marks up the price by 83%, and sells the Algebra 1 books. What is the amount of the markup? What is the selling price?

a. The amount of the markup is $29.28, and the selling price is $13.28. b. The amount of the markup is $13.28, and the selling price is $29.28. c. The amount of the markup is $13.28, and the selling price is $2.72. d. The amount of the markup is $83, and the selling price is $99.00.

111. Mr. Chang sells holiday greeting cards in his gift shop. Before the holidays, he sells the cards at a 225% markup on the price he paid his supplier. After the holidays, he discounts the cards 60%. What is the post-holiday price of two cards he originally bought from his supplier for $1.50 and $2.00, respectively? a. $2.03; $2.70 b. $1.35; $1.80 c. $2.93; $3.9 d. $1.95; $2.60

112. Solve . a. x = 0 or x = –14 b. x = 7 c. x = 0 d. x = 7 or x = –21

113. Solve . a. x = 1 b. x = 116 c. No solution d. x =

83

114. Describe the solutions of in words. a. The value of y is a number less than or equal to 3. b. The value of y is a number greater than 4. c. The value of y is a number equal to 3 d. The value of y is a number less than 4.

115. Graph the inequality m < –3.4.

a.

0 1 2 3 4 50–1–2–3–4–5

b.

0 1 2 3 4 50–1–2–3–4–5

c.

0 1 2 3 4 50–1–2–3–4–5

d.

0 1 2 3 4 50–1–2–3–4–5

116. Write the inequality shown by the graph.

0 1 2 3 4 5 6 70–1–2–3–4–5–6–7 m

a. m ≤ –3 b. m > –3 c. m ≥ –3 d. m < –3

117. To join the school swim team, swimmers must be able to swim at least 500 yards without stopping. Let n represent the number of yards a swimmer can swim without stopping. Write an inequality describing which values of n will result in a swimmer making the team. Graph the solution. a.

0 100 200 300 400 500 600 700 800 900 10000 n

b.

0 100 200 300 400 500 600 700 800 900 10000 n

c.

0 100 200 300 400 500 600 700 800 900 10000 n

d.

0 100 200 300 400 500 600 700 800 900 10000 n

118. Sam earned $450 during winter vacation. He needs to save $180 for a camping trip over spring break. He can spend the remainder of the money on music. Write an inequality to show how much he can spend on music. Then, graph the inequality. a. ;

0 100 200 300 400 5000–100–200–300–400–500

s

b. ;

0 100 200 300 400 5000–100–200–300–400–500

s

c. ;

0 100 200 300 400 5000–100–200–300–400–500

s

d. ;

0 100 200 300 400 5000–100–200–300–400–500

s

119. Solve the inequality n + 6 < –1.5 and graph the solutions. a. n < 4.5

0 2 4 6 8 100–2–4–6–8–10

b. n < –7.5

0 2 4 6 8 100–2–4–6–8–10

c. n < –7.5

0 2 4 6 8 100–2–4–6–8–10

d. n < 4.5

0 2 4 6 8 100–2–4–6–8–10

120. Carlotta subscribes to the HotBurn music service. She can download no more than 11 song files per week. Carlotta has already downloaded 8 song files this week. Write, solve, and graph an inequality to show how many more songs Carlotta can download. a. s > 3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 170

b. s ≥ 3

0 1 2 3 4 5 6 7 8 9 10 110

c. s ≤ 3

0 1 2 3 4 5 6 7 8 9 10 110

d. s < 3

0 1 2 3 4 5 6 7 8 9 10 110

121. Denise has $365 in her saving account. She wants to save at least $635. Write and solve an inequality to determine how much more money Denise must save to reach her goal. Let d represent the amount of money in dollars Denise must save to reach her goal. a. ; b. ;

c. ; d. ;

122. Solve the inequality and graph the solution.

a. 815

0 3 6 9 12 15 18 210–3–6–9

b. 525

0 3 6 9 12 15 18 210–3–6–9

c. 525

0 3 6 9 12 15 18 210–3–6–9

d. 815

0 3 6 9 12 15 18 210–3–6–9

123. Solve the inequality > 3 and graph the solutions.

a. x > 24

0 5 10 15 20 25 30 35 40 45 500

b. x > 24

0 5 10 15 20 25 30 35 40 45 500

c. x > 38

0 1 2 3 4 5 6 7 8 9 10 11 120

d. x > 24

0 5 10 15 20 25 30 35 40 45 500

124. Solve the inequality 2m ≤ 18 and graph the solutions. a. m ≤ 9

0 1 2 3 4 5 6 7 8 9 10 110–1

b. m ≤ 36

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 950

c. m ≤ 36

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 950

d. m ≤ 9

0 1 2 3 4 5 6 7 8 9 10 110–1

125. Solve the inequality ≤ 2 and graph the

solutions. a. z ≤ –8

0 2 4 6 8 100–2–4–6–8–10

b. z ≥ 8

0 2 4 6 8 100–2–4–6–8–10

c. z ≤ 8

0 2 4 6 8 100–2–4–6–8–10

d. z ≥ –8

0 2 4 6 8 100–2–4–6–8–10

126. Solve the inequality 2f ≥ –8 and graph the solutions. a. f ≥ –4

0 2 4 6 8 100–2–4–6–8–10

b. f ≤ –4

0 2 4 6 8 100–2–4–6–8–10

c. f ≤ 4

0 2 4 6 8 100–2–4–6–8–10

d. f ≥ 4

0 2 4 6 8 100–2–4–6–8–10

127. Marco’s Drama class is performing a play. He wants to buy as many tickets as he can afford. If tickets cost $2.50 each and he has $14.75 to spend, how many tickets can he buy? a. 0 tickets b. 5 tickets c. 6 tickets d. 4 tickets

128. What is the greatest possible integer solution of the inequality ? a. 5.33 b. 4 c. 6 d. 5

129. Solve the inequality −n – 4 < 3 and graph the solutions. a. n < –7

0 2 4 6 8 100–2–4–6–8–10

b. n > –7

0 2 4 6 8 100–2–4–6–8–10

c. n > 1

0 2 4 6 8 100–2–4–6–8–10

d. n < 1

0 2 4 6 8 100–2–4–6–8–10

130. Solve the inequality z + 8 + 3z ≤ –4 and graph the solutions. a. z ≥ –3

0 2 4 6 8 100–2–4–6–8–10

b. z ≤ 1

0 2 4 6 8 100–2–4–6–8–10

c. z ≤ –3

0 2 4 6 8 100–2–4–6–8–10

d. z ≥ 1

0 2 4 6 8 100–2–4–6–8–10

131. A family travels to Bryce Canyon for three days. On the first day, they drove 150 miles. On the second day, they drove 190 miles. What is the least number of miles they drove on the third day if their average number of miles per day was at least 180? a. 540 mi b. 180 mi c. 201 mi d. 200 mi

132. Solve and graph .

a. x > 5

0 1 2 3 4 5 6 7 8 9 10 11 12 130–1–2–3–4–5–6–7–8–9–10–11–12–13

b. x < 5

0 1 2 3 4 5 6 7 8 9 10 11 12 130–1–2–3–4–5–6–7–8–9–10–11–12–13

c. x > 3

0 1 2 3 4 5 6 7 8 9 10 11 12 130–1–2–3–4–5–6–7–8–9–10–11–12–13

d. x < –5

0 1 2 3 4 5 6 7 8 9 10 11 12 130–1–2–3–4–5–6–7–8–9–10–11–12–13

133. Mrs. Williams is deciding between two field trips for her class. The Science Center charges $135 plus $3 per student. The Dino Discovery Museum simply charges $6 per student. For how many students will the Science Center charge less than the Dino Discovery Museum? a. 132 or more students b. 132 or fewer students c. More than 45 students d. Fewer than 45 students

134. Solve the inequality and graph the solution.

a.

0 1 2 3 4 5 6 7 8 9 100–1–2–3–4–5–6–7–8–9–10

b.

0 1 2 3 4 5 6 7 8 9 100–1–2–3–4–5–6–7–8–9–10

c.

0 1 2 3 4 5 6 7 8 9 100–1–2–3–4–5–6–7–8–9–10

d.

0 1 2 3 4 5 6 7 8 9 100–1–2–3–4–5–6–7–8–9–10

135. Solve the inequality . a. z ≤ −2 916 b. z ≤ 3

716 c. all real numbers

d. no solutions

136. Solve . a. b. c. d.

137. Fly with Us owns a D.C.10 airplane that has seats for 240 people. The company flies this airplane only if there are at least 100 people on the plane. Write a compound inequality to show the possible number of people in a flight on a D.C.10 with Fly with Us. Let n represent the possible number of people in the flight. Graph the solutions. a.

0 50 100 150 200 2500–50–100–150–200–250

b.

0 50 100 150 200 2500–50–100–150–200–250

c.

0 50 100 150 200 2500–50–100–150–200–250

d.

0 50 100 150 200 2500–50–100–150–200–250

138. Solve and graph the solutions of the compound inequality . a. AND

0 1 2 3 4 50

b. AND

0 1 2 3 4 50

c. AND

0 1 2 3 4 50

d. AND

0 1 2 3 4 50

139. Solve and graph the compound inequality. OR

a. OR

0 1 2 3 4 5 6 7 8 9 100–1–2–3–4–5–6–7–8–9–10 s

b. OR

0 1 2 3 4 5 6 7 8 9 100–1–2–3–4–5–6–7–8–9–10 s

c. OR

0 1 2 3 4 5 6 7 8 9 100–1–2–3–4–5–6–7–8–9–10 s

d. OR

0 1 2 3 4 5 6 7 8 9 100–1–2–3–4–5–6–7–8–9–10 s

140. Write the compound inequality shown by the graph.

0 1 2 3 4 5 6 7 8 9 100–1–2–3–4–5–6–7–8–9–10 x

a. AND b. AND c. OR d. OR

141. Which of the following is a solution of AND ? a. 2 b. 14 c. 12 d. –6

142. Solve the inequality and graph the solutions. Then write the solutions as a compound inequality. a.

0 3 6 9 12 15 18 21 24 27 300–3–6–9–12–15–18–21–24–27

b.

0 3 6 9 12 15 18 21 24 27 300–3–6–9–12–15–18–21–24–27–30

c.

0 3 6 9 12 15 18 21 24 27 300–3–6–9–12–15–18–21–24–27

d.

0 3 6 9 12 15 18 21 24 27 300–3–6–9–12–15–18–21–24–27–30

143. Solve and graph the solutions of . Write the solutions as a compound inequality. a. –9 < x < 21

0 2 4 6 8 10 12 14 16 18 20 22 240–2–4–6–8–10–12–14–16–18–20–22–24

b. x < – 15 OR x > 15

0 2 4 6 8 10 12 14 16 18 20 22 240–2–4–6–8–10–12–14–16–18–20–22–24

c. x > 21

0 2 4 6 8 10 12 14 16 18 20 22 240–2–4–6–8–10–12–14–16–18–20–22–24

d. x < – 9 OR x > 21

0 2 4 6 8 10 12 14 16 18 20 22 240–2–4–6–8–10–12–14–16–18–20–22–24

Numeric Response 144. An architect charges $1800 for a first

draft of a three-bedroom house. If the work takes longer than 8 hours, the architect charges $105 for each additional hour. What would be the total cost for a first draft that took 14 hours to complete?

145. The maximum speed of a greyhound is 153 miles per hour less than 3 times the maximum speed of a cheetah. If a greyhound’s maximum speed is 42 miles per hour, what is the maximum speed of a greyhound? Check to make sure your answer is reasonable.

146. A car rental company increases daily rental fees 15% in the summer to cover increased fuel costs. They then have a “25% off” promotion for the fall. If a car rented for $36.00 per day before the summer, what would the per-day rental cost be during the fall promotion?

147. What is the least possible integer solution of the inequality ?

148. A volleyball team scored 14 more points in its first game than in its third game. In the second game, the team scored 28 points. The total number of points scored was less than 80. What is the greatest number of points the team could have scored in its first game?

Matching Match each vocabulary term with its definition. a. algebraic expression b. numerical expression c. like terms d. absolute value e. evaluate f. variable g. constant

149. a symbol used to represent a quantity that can change

150. a value that does not change

151. an expression that contains at least one variable

152. a mathematical phrase that contains operations and numbers

153. to find the value of an algebraic expression by substituting a number for each variable and simplifying by using the order of operations

Match each vocabulary term with its definition. a. real numbers b. additive inverse c. opposites d. multiplicative inverse e. natural numbers f. reciprocal g. absolute value

154. numbers that are the same distance from zero on opposite sides of the number line

155. the opposite of a number

156. for any real number

157. the distance from zero on a number line

158. the reciprocal of the number Match each vocabulary term with its definition. a. coefficient b. variable

c. power d. perfect square e. square root f. exponent g. base

159. the number in a power that is used as a factor

160. a number that is multiplied to itself to form a product

161. a number whose positive square root is a whole number

162. an expression written with a base and an exponent or the value of such an expression

163. the number that indicates how many times the base in a power is used as a factor

Match each vocabulary term with its definition. a. real numbers b. positive numbers c. negative numbers d. integers e. irrational numbers f. rational numbers g. natural numbers h. whole numbers

164. the set of numbers that can be written in the form , where a and b are integers and

165. the set of counting numbers

166. the set of natural numbers and zero

167. the set of rational and irrational numbers

168. the set of whole numbers and their opposites

169. the set of real numbers that cannot be written as a ratio of integers

Match each vocabulary term with its definition. a. repeating decimal b. terminating decimal c. reciprocal d. absolute value e. term f. coefficient g. like terms h. order of operations

170. a rational number in decimal form that has a block of one or more digits that repeats continuously

171. a number multiplied by a variable

172. a part of an expression to be added or subtracted

173. terms with the same variables raised to the same exponents

174. A rule for evaluating expressions: First, perform operations in parentheses or other grouping symbols. Second, evaluate powers and roots. Third, perform all multiplication and division from left to right.

Fourth, perform all addition and subtraction from left to right.

175. a rational number in decimal form that has a finite number of digits after the decimal point

Match each vocabulary term with its definition. a. coordinate plane b. ordered pair c. origin d. quadrant e. y-axis f. x-axis g. axes

176. the intersection of the x- and y-axes in a coordinate plane

177. the vertical axis in a coordinate plane

178. the horizontal axis in a coordinate plane

179. the two perpendicular number lines, also known as the x-axis and the y-axis, used to define the location of a point in a coordinate plane

180. a pair of numbers that can be used to locate a point on a coordinate plane

Match each vocabulary term with its definition. a. x-axis b. x-coordinate c. y-coordinate d. input e. output f. y-axis g. quadrant h. coordinate plane

181. the second number in an ordered pair, which indicates the vertical distance of a point from the origin on the coordinate plane

182. the first number in an ordered pair, which indicates the horizontal distance of a point from the origin on the coordinate plane

183. the result of substituting a value for a variable in a function

184. a value that is substituted for the independent variable in a relation or function

185. one of the four regions into which the x- and y-axis divide the coordinate plane

186. a plane that is divided into four regions by a horizontal line called

the x-axis and a vertical line called the y-axis

Match each vocabulary term with its definition. a. expression b. solution of an equation c. contradiction d. identity e. formula f. inequality g. equation h. literal equation

187. a mathematical sentence that shows that two expressions are equivalent

188. an equation that contains two or more variables

189. an equation that is true for all values of the variables

190. a value or values that make the equation or inequality true

191. a literal equation that states a rule for a relationship among quantities

192. an equation that is not true for any value of the variable

Match each vocabulary term with its definition. a. proportion b. formula c. ratio d. unit rate e. identity f. conversion factor g. rate

193. a rate in which the second quantity in the comparison is one unit

194. the ratio of two equal quantities, each measured in different units

195. a ratio that compares two quantities measured in different units

196. an equation that states that two ratios are equal

197. a comparison of two numbers by division

Match each vocabulary term with its definition. a. conversion factor b. scale c. scale drawing d. scale factor e. scale model f. proportion g. similar

198. a drawing that uses a scale to represent an object as smaller or larger than the original object

199. the ratio of any length in a drawing to the corresponding actual length

200. in a dilation, the ratio of a linear measurement of the image to the corresponding measurement of the preimage

201. a three-dimensional model that uses a scale to represent an object as smaller or larger than the actual object

202. the ratio of two equal quantities, each measured in different units

Match each vocabulary term with its definition. a. proportion b. corresponding sides c. indirect measurement d. like terms e. cross products f. similar g. corresponding angles

203. the product of the means bc and the product of the extremes ad in the statement

204. sides in the same relative position in two different polygons that have the same number of sides

205. two figures that have the same shape, but not necessarily the same size

206. angles in the same relative position in two different polygons that have the same number of angles

207. a method of measuring an object by using formulas, similar figures, and/or proportions

Match each vocabulary term with its definition. a. commission b. interest c. rate d. sales tax e. markup f. principal g. tip

208. an amount of money added to a bill for service

209. the amount of money charged for borrowing money or the amount of money earned when saving or investing money

210. money paid to a person or company for making a sale

211. a percent of the cost of an item that is charged by governments to raise money

212. an amount of money borrowed or invested

Match each vocabulary term with its definition. a. rate b. markup c. percent change d. percent decrease e. ratio f. percent g. percent increase h. discount

213. a decrease given as a percent of the original amount

214. an increase given as a percent of the original amount

215. an amount by which an original price is reduced

216. an increase or decrease given as a percent of the original amount

217. a ratio that compares a number to 100

218. the amount by which a wholesale cost is increased

Match each vocabulary term with its definition. a. compound inequality b. inequality c. intersection d. solution of an inequality e. union f. Venn diagram g. equation

219. the set of all elements that are common to both sets, denoted by

220. the set of all elements that are in either set, denoted by

221. a statement that compares two expressions by using one of the following signs: , , , or

222. a value or values that make the inequality true

223. two inequalities that are combined into one statement by the word and or or

Math Olympics Answer Section

MULTIPLE CHOICE

1. ANS: D

The operation means “divided by” or “quotient”. p ÷ 10: the quotient of p and 10 p divided by 10 Feedback A Check the operation in the algebraic expression. B Check the operation in the algebraic expression. C Check the order of the variable and constant. D Correct!

TOP: 1-1 Variables and Expressions

2. ANS: B y represents the number of letters Julia wrote. Think: y groups of 14 letters. 14y Feedback A Think: how many groups of letters are there? B Correct! C Think: how many groups of letters are there? D To translate words into an algebraic expression, look for words

that indicate the action.


3. ANS: C x represents the number of bags. Think: How many groups of 88 are in x?

Feedback A Think: how many groups of cans are in the number of bags? B Think: how many groups of cans are in the number of bags?

C Correct! D To translate words into an algebraic expression, look for words

that indicate the action.


4. ANS: B m + o 9 + 7 Substitute 9 for m and 7 for o. 16 Simplify. Feedback A Check your addition. B Correct! C This expression involves addition, not subtraction. D This expression involves addition, not multiplication.


5. ANS: B Substitute the values for q and v into the expression, and then subtract. Feedback A Check your subtraction. B Correct! C This expression involves subtraction, not division. D This expression involves subtraction, not addition.


6. ANS: D Substitute the values for x and y into the expression, and then multiply. Feedback A Check your multiplication. B Check your multiplication. C This expression involves multiplication, not addition. D Correct!


7. ANS: C Substitute the values for a and b into the expression, and then divide. Feedback

A This expression involves division, not multiplication. B This expression involves division, not subtraction. C Correct! D Check your division.


8. ANS: D The expression 40 + y models the number of points Mike scored in all Evaluate 40 + y for y = 2. 40 + 2 = 42 If Mike scored 2 points in the second half of the game, then he scored 42 points in all. Feedback A Use a different operation. B Use a different operation. C Use a different operation instead of division. D Correct!


9. ANS: C

The expression models the number of sand dollars each of Aaron’s friends will receive. Evaluate for y = 12.

= 6 If Aaron gives 72 sand dollars to 12 friends, each friend will get 6 sand dollars. Feedback A Use a different operation. B Use a different operation. C Correct! D Use a different operation instead of multiplication.


10. ANS: D The expression 12n models the number books Salvador read in all. Evaluate 12n for n = 7.

12(7) = 84 If Salvador read for 7 months, then that means Salvador read 84 books. Feedback A Use a different operation. B Use a different operation. C Use a different operation. D Correct!


11. ANS: A Substitute 7 for m and 9 for n.

Simplify. Remember: means 2 times m.

Feedback A Correct! B You switched the values of the variables. C First, substitute the given values. Then, simplify the

expression. D When there is no operation sign between a number and a

variable, it means it is multiplication.


12. ANS: D The lower vector shows the minuend and the upper vector shows the subtrahend. The number at which the upper vector stops is the difference of the two integers. Feedback A Move left on a number line to subtract a positive integer; move

right to subtract a negative integer. B Move left on a number line to subtract a positive integer; move

right to subtract a negative integer. C Move left on a number line to subtract a positive integer; move

right to subtract a negative integer. D Correct!

TOP: 1-2 Adding and Subtracting Real Numbers

13. ANS: B To add two integers with the same sign, find the sum of their absolute values and use the sign of the two integers. To add two integers with different signs, find the difference of their absolute values and use the sign of the integer with the greater absolute value. Feedback A When adding two integers with the same sign, find the sum of

their absolute values. When adding two integers with different signs, find the difference of their absolute values.

B Correct! C When adding two integers with the same sign, find the sum of


D Check the sign of your answer.


14. ANS: C Substitute 35 for x, and then add the integers. To add two integers with the same sign, find the sum of their absolute values and use the sign of the two integers. To add two integers with different signs, find the difference of their absolute values and use the sign of the integer with the greater absolute value. Feedback A Substitute for x, and then add the integers. B Check the sign of your answer. C Correct! D When adding two integers with the same sign, find the sum of



15. ANS: C Change the subtraction sign to an addition sign, and change the sign of the second number. Feedback A Change the subtraction sign to an addition sign, and change

the sign of the second number. B Change the subtraction sign to an addition sign, and change

the sign of the second number. C Correct! D Pay attention to the sign.


16. ANS: C Substitute 12 for x, and then subtract the integers. To subtract, change the subtraction sign to an addition sign, and change the sign of the second number. Feedback A Pay attention to the sign. B Change the subtraction sign to an addition sign, and change

the sign of the second number. C Correct! D Change the subtraction sign to an addition sign, and change

the sign of the second number.


17. ANS: C Subtract the negative temperature from the positive temperature to calculate the difference in the two readings. Feedback A Check the signs. B Check the signs. C Correct! D Subtract the lower temperature from the higher one.


18. ANS: B Subtract the lower temperature from the higher temperature to calculate the difference in the two readings. Feedback A Subtract the lower temperature from the higher temperature. B Correct! C Subtract the lower temperature from the higher temperature. D Check the signs.


19. ANS: B

Subtract the height of the building from the height of the elevator. The difference represents how far underground the parking garage goes. Feedback A Check your subtraction. B Correct! C Subtract the numbers instead of adding them. D Check your subtraction.


20. ANS: B Multiply the two integers. If the signs are the same, the product is positive; if the signs are different, the product is negative. Feedback A Multiply the integers, not add. B Correct! C Be sure to multiply the integers. D If the signs of the two integers are the same, the product will

be positive. If the signs are different, the product will be negative.

TOP: 1-3 Multiplying and Dividing Real Numbers

21. ANS: D Substitute –4 for u. Then multiply. Feedback A Substitute the value in the variable, and then multiply. B Check your multiplication. C If the signs of the two integers are the same, the product will

be positive; if they are different, the product will be negative. D Correct!


22. ANS: D Divide the two integers. If the signs are the same, the quotient is positive; if the signs are different, the quotient is negative. Feedback A This expression involves division, not multiplication. B If the signs of the two integers are the same, the quotient will

be positive. If the signs are different, the quotient will be

negative. C This expression involves division, not subtraction. D Correct!


23. ANS: B Substitute –33 for k in the expression. Then divide the integers. If the signs are the same, the quotient is positive; if the signs are different, the quotient is negative. Feedback A This expression involves division, not multiplication. B Correct! C This expression involves division, not subtraction. D If the signs of the two integers are the same, the product will

be positive. If the signs are different, the product will be negative.


24. ANS: C

Write as an improper fraction.

To divide by multiply by .

Multiply.

8 821 Simplify. Feedback A First convert the mixed number to an improper fraction. B Multiply by the reciprocal. C Correct! D First convert the mixed number to an improper fraction. Then

multiply by the reciprocal.


25. ANS: D The quotient of 0 and any nonzero number is 0. Feedback

A Multiply or divide by 0. B Multiply or divide by 0. C Only division by 0 is undefined. D Correct!


26. ANS: D Distance = rate time

Substitute 3.5 for rate and 1.75 for time.

Distance =

Multiply to find the distance.

Feedback A To find distance, multiply rate by time. B To find distance, multiply rate by time. Then estimate to check

if your answer is reasonable. C The decimal point is not in the correct place. Use estimation to

check if your answer is reasonable. D Correct!


27. ANS: C The figure is 5 cubes tall, 5 cubes wide, and 5 cubes long. The factor 5 is used 3 times. Feedback A The length, width, and height of the figure is 5. B Is the figure 2-dimensional or 3-dimensional? C Correct! D The length, width, and height of the figure is 5.

TOP: 1-4 Powers and Exponents

28. ANS: C The exponent tells the number of times to multiply the base number by itself. Multiply 9 by itself 3 times. Feedback A Multiply the base number by itself as many times as the

exponent tells you. B Multiply using the base. The exponent just tells how many

times to multiply the base by itself. C Correct! D Multiply the number by itself rather than adding two different

numbers.


29. ANS: A The exponent tells the number of times to multiply the base number by itself. The negative sign in front of the expression multiplies the expression by –1. Multiply 3 by itself 4 times, and then multiply your answer by –1. Feedback A Correct! B Think of the negative sign in front as multiplying the expression

by -1. C Multiply the base number by itself rather than adding. D Multiply the base number by itself. The exponent tells how

many times to multiply the base by itself.


30. ANS: B The exponent tells the number of times to multiply the base number by itself. Multiply –4 by itself 2 times. Feedback A Multiply the base number by itself rather than adding. B Correct! C This is the product of the base and the exponent. The

exponent tells how many times to multiply the base by itself. D Check the sign of your answer. The product of an even

number of negative factors is positive; the product of an odd number of negative factors is negative.


31. ANS: D The exponent tells how many times to multiply the fraction by itself. Multiply by itself 2 times. Feedback A The exponent tells how many times to multiply the fraction by

itself.

B Raise both the numerator and denominator to the exponent. C Raise both the numerator and denominator to the exponent. D Correct!


32. ANS: B The number given as a base should be multiplied by itself a certain number of times in order to represent the value of the whole number given. The product of two 3’s is 9. Feedback A An exponent is written as a small number raised slightly above

the base number. B Correct! C The exponent tells how many times to multiply the base by

itself. D The number given as a base should be multiplied by itself a

certain number of times in order to represent the value of the whole number given.


33. ANS: A If each category represents sunlight that is 6 times brighter than the category before, then a dazzling day would be 36 times brighter than an illuminated day because: a dazzling day is 6 times brighter than a radiant day, a radiant day is 6 times brighter than an illuminated day, and an illuminated day is 6 times brighter than a dim day. Feedback A Correct! B The brightness number is just for identifying the category. You

need to use the number of times brighter as a factor. C You need to use the number of times brighter as a factor one

more time. D Check to see whether you used the number of times brighter

as a factor too many times.


34. ANS: C

If the population of the ant hill is 40 ants and it doubles every 10 days, then to find its population in 20 days, make a chart to see what the population is after a certain number of days. In 10 days, the population is 40 ants. In 2 • 10 days, the population is 402 ants. In 3 • 10 days, the population is 403 ants. In 4 • 10 days, the population is 404 ants. Feedback A Make a chart to see what the population is after a certain

number of days. B Make a chart to see what the population is after a certain

number of days. C Correct! D Make sure that the ant population doubles.


35. ANS: C Step 1 Find the total area of the vegetable garden and path.

Step 2 Find the area of the vegetable garden and the area of the path. Find the side length of the vegetable garden. Find the area of the vegetable garden. To find the area of the path, subtract the area of the vegetable garden from the total area. Step 3 Find the number of gravel bags needed to cover the path.

So, 7 bags of gravel are needed to cover the path. Feedback A You switched the area of the vegetable garden and the area of

the path. B To find the area of the path, subtract the area of the vegetable

garden from the total area. C Correct! D To find the area of the path, subtract the area of the vegetable

garden from the total area.


36. ANS: A

196 =

What number squared equals 196?

= 14 The sign to the left of the radical determines whether the square root is positive or negative.

Feedback A Correct! B This is half of the number. The square root of a number,

multiplied by itself, equals that number. C Find the square root of the number under the radical sign, not

the square of that number. D The + or - sign to the left of the radical is the sign of the square

root.

TOP: 1-5 Square Roots and Real Numbers

37. ANS: D 202 is between 196 and 225. Since 202 is closer to 196, the best estimate for the side length is 14 ft. Feedback A Find the two perfect squares that the area is between. B Find the two perfect squares that the area is between. C Of the two perfect squares that the area is between, which is

closer to the area? D Correct!


38. ANS: B Any number that can be written as a fraction is a rational number. Rational numbers include terminating decimals and repeating decimals. If a rational number simplifies to a whole number or its opposite, it is also an integer. If a rational number simplifies to a nonzero whole number, it is also a natural number. Feedback

A To check whether the number is a terminating or repeating decimal, divide the numerator by the denominator.

B Correct! C Since this number can be written as a fraction, it is not an

irrational number. D There are more ways to classify the number. Check to see

whether it is a terminating or repeating decimal.


39. ANS: D A rational number can be written as a fraction. Rational numbers include integers, fractions, terminating decimals, and repeating decimals. An irrational number cannot be expressed as either a terminating decimal or repeating decimal. Feedback A A rational number will either terminate or repeat, but an

irrational number will not. If the fraction simplifies to a nonzero whole number, the number is also an integer and a natural number.

B A rational number will either terminate or repeat, but an irrational number will not. If the fraction simplifies to a nonzero whole number, the number is also an integer and a natural number.

C A rational number will either terminate or repeat, but an irrational number will not. If the fraction simplifies to a nonzero whole number, the number is also an integer and a natural number.

D Correct!


40. ANS: C If there is an example of two irrational numbers whose sum is not an irrational number, then the set of irrational numbers is not closed under addition. Add the following irrational numbers: 0.121121112... 0.212212221... The result is 0.33333... which is equal to , and is a rational number.

Another example is and . The sum is 0 which is a rational number.

Feedback A Find an example of two irrational numbers whose sum is not

an irrational number. B If there is an example of two irrational numbers whose sum is

not an irrational number, then the set of irrational numbers is not closed under addition.

C Correct! D The set of irrational numbers being closed under addition

means that when you add any two irrational numbers, the sum is also an irrational number.


41. ANS: D Use the order of operations: 1. Perform operations in parentheses. 2. Evaluate powers. 2. Multiply or divide from left to right. 3. Add or subtract from left to right. Feedback A The exponent tells how many times to use the base as a factor

with itself. B The order of operations is correct, but check your signs. C After evaluating the exponents and evaluating within

parentheses, multiplication must be performed before addition or subtraction.

D Correct!

TOP: 1-6 Order of Operations

42. ANS: D Use the order of operations: 1. Perform operations in parentheses. 2. Evaluate powers. 3. Multiply or divide from left to right. 4. Add or subtract from left to right. Feedback A Use the order of operations. B Perform operations in parentheses first. C Divide before you add. D Correct!


43. ANS: C Substitute 9 for x in the expression. Then use the order of operations to evaluate the expression. 1. Perform operations in parentheses. 2. Evaluate powers. 3. Multiply or divide from left to right. 4. Add or subtract from left to right. Feedback A Use the order of operations. Multiply before adding or

subtracting. B Use the order of operations. Multiply before adding or

subtracting. C Correct! D Use the order of operations. Multiply before adding or

subtracting.


44. ANS: B Substitute 4 for x in the expression. Then use the order of operations to evaluate the expression. 1. Perform operations in parentheses. 2. Evaluate powers. 3. Multiply or divide from left to right. 4. Add or subtract from left to right. Feedback A Use the order of operations. Evaluate powers before

multiplying or adding. B Correct! C Use the order of operations. Evaluate powers before

multiplying or adding. D Use the order of operations. Evaluate powers before

multiplying or adding.


45. ANS: A

First, simplify the numerator of the fraction, and then divide the numerator by the denominator. Next, subtract the terms in the absolute value, and then find the absolute value.

=

Finally, add the two terms. = 14

Feedback A Correct! B Only square the value that has an exponent, not both numbers

in the numerator. C Subtract within the absolute value bars before taking the

absolute value. D Simplify the numerator before dividing by the denominator.


46. ANS: D Use parentheses so that the difference is

evaluated first. Product means multiplication.

Feedback A "Product" indicates multiplication. B Use parentheses so the difference is evaluated first. C When finding a difference, subtract the second number from

the first. D Correct!


47. ANS: D Use the formula (P + 5N + 10D + 25Q) ÷ 100. Substitute the values from the table. Total 100 100 100 Tatia has $1.90. Feedback

A First perform operations inside parentheses, and then divide. B Multiply before you add. C Multiply the number of coins of each type by its coin value

before performing the addition. D Correct!


48. ANS: C You must use each of the numbers at least once, and you may use any operations. Pay attention to the order of operations. Feedback A Evaluate powers before performing subtraction. B Perform multiplication before subtraction. C Correct! D The number 8 must be used also.


49. ANS: B

Use the Commutative Property.

Use the Associative Property to make groups of compatible numbers. Simplify.

Feedback A The sum of two mixed numbers is the sum of the whole parts

plus the sum of the fractional parts. B Correct! C To add two fractions, first find a common denominator and

then add the numerators. D The sum of the fractional parts is greater than 1.

TOP: 1-7 Simplifying Expressions

50. ANS: A Rewrite 59 as 50 + 9. Then multiply each term by 11 and add the products.

Feedback A Correct! B Multiply the first number by each digit in the second number,

then add the two products. C Multiply the first number by each digit in the second number,

then add the two products. D Multiply the first number by each digit in the second number,

then add the two products.


51. ANS: B Notice that 19 is very close to 20. Rewrite 19 as 20 + . Then use the Distributive Property. Feedback A You've reversed multiplication and addition. Look at the

Distributive Property again. B Correct! C Use mental math. Notice that the two-digit factor is close to a

multiple of 10. D Use mental math. Notice that the two-digit factor is close to a

multiple of 10.


52. ANS: A

Group like terms.

Add or subtract the coefficients. Feedback A Correct! B Combine only like terms. C Check the signs of all the coefficients. D First, group like terms. Then, add or subtract the coefficients.


53. ANS: D The Commutative Property allows for you to add or subtract terms in any order. Feedback

A Multiplication is used in Step 3. B The Associative Property is used in Step 5. C Like terms are combined in Step 6. D Correct!


54. ANS: D Procedure Justification Definition of subtraction

Commutative Property

Associative Property

Distributive Property

Simplify Definition of subtraction Feedback A What is the difference between the Commutative Property and

the Distributive Property? B The Associative Property involves grouping of numbers. What

does the Commutative Property state? C What is the difference between the Associative Property and

the Distributive Property? D Correct!


55. ANS: C The x-coordinate of the ordered pair tells how many units to move left or right from the origin. The y-coordinate of the ordered pair tells how many units to move up or down from the origin. Feedback A The first number in the ordered pair tells whether to move left

or right from (0, 0). The second number tells whether to move up or down.

B The first number in the ordered pair tells whether to move left or right from (0, 0). The second number tells whether to move

up or down. C Correct! D The first number in the ordered pair tells whether to move left

or right from (0, 0). The second number tells whether to move up or down.

TOP: 1-8 Introduction to Functions

56. ANS: D If both x and y are positive, the point is in Quadrant I. If x is negative and y is positive, the point is in Quadrant II. If both x and y are negative, the point is in Quadrant III. If x is positive and y is negative, the point is in Quadrant IV.

Quadrant IQuadrant II

Quadrant III Quadrant IV

5–5 x

5

–5

y

Feedback A The coordinate plane is divided by the x-axis and the y-axis

into four quadrants. The signs of x and y determine which quadrant the point is in.

B The coordinate plane is divided by the x-axis and the y-axis into four quadrants. The signs of x and y determine which quadrant the point is in.

C The coordinate plane is divided by the x-axis and the y-axis into four quadrants. The signs of x and y determine which quadrant the point is in.

D Correct!


57. ANS: C If x = 0, the point is on the y-axis.

If y = 0, the point is on the x-axis. Feedback A The coordinate plane is divided by the x-axis and the y-axis

into four quadrants. If x = 0, the point is on the y-axis. If y = 0, the point is on the x-axis.

B The coordinate plane is divided by the x-axis and the y-axis into four quadrants. If x = 0, the point is on the y-axis. If y = 0, the point is on the x-axis.

C Correct! D The coordinate plane is divided by the x-axis and the y-axis

into four quadrants. If x = 0, the point is on the y-axis. If y = 0, the point is on the x-axis.


58. ANS: B Let y represent the monthly payment and x represent the number of minutes of international calls.

monthly payment

is $25 plus $0.10 for each

international minute

y = 25 + 0.10 x

Number of

international minutes

Rule Monthly payment Ordered

pair

x (input) y (output) (x, y) 90 $34.00 (90, 34)

120 $37.00 (120, 37) 145 $39.50 (145, 39.5) 150 $40.00 (150, 40)

Feedback A The monthly payment is determined by the number of

international minutes, so the number of international minutes is the input and the monthly payment is the output.

B Correct! C The monthly payment is determined by the number of

international minutes, so the number of international minutes is

the input and the monthly payment is the output. D The monthly payment is $25 plus $0.10 for each international

minute.


59. ANS: D

Make a table to find values of (x, y) for . x y (x, y)

–2 (–2, 6) –1 (–1, 0) 0 (0, –2) 1 (1, 0) 2 (2, 6)

The points form a U shape.

Feedback A This is a cubic function. Use the given values for x to get the

values for y. B This is a linear function. Use the given values for x to get the

values for y. C This is an absolute value function. Use the given values for x

to get the values for y. D Correct!


60. ANS: B Step 1 Plot the points.

A

B C

1 2 3 4 5 6 7 8 9 10 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

Step 2 Find the fourth vertex. The fourth vertex will have the same x-coordinate as C(10,3) and the same y-coordinate as A(1, –5). x-coordinate: 10 y-coordinate: –5 The fourth vertex is D(10, –5).

A

B C

D

1 2 3 4 5 6 7 8 9 10 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

Step 3 Find the area of the rectangle.

square units Feedback A The fourth vertex will have the same x-coordinate as point C

and the same y-coordinate as point A. B Correct! C First, plot the points. Then, use the formula for the area of the

rectangle. D The fourth vertex will have the same x-coordinate as point C

and the same y-coordinate as point A.


61. ANS: C The operation means “times”, “multiplied by”, “product”, or “each groups of”. 6p: 6 times p 6 groups of p Feedback A The algebraic expression does not show division. B Check the operation in the algebraic expression. C Correct! D Check the operation in the algebraic expression.


62. ANS: B The lower vector shows the first addend, and the upper vector shows the second addend. The number at which the upper vector stops is the sum of the two integers. Feedback A Move right on a number line to add a positive integer; move

left to add a negative integer. B Correct! C Move right on a number line to add a positive integer; move

left to add a negative integer. D Move right on a number line to add a positive integer; move

left to add a negative integer.


63. ANS: A

Since 6 is subtracted from p, add 6 to both sides to undo the subtraction.

Check:

To check your solution, substitute 22 for p in

the original equation.

Feedback A Correct! B Check the ones place. C Is subtraction the correct operation for solving this equation? D Check the tens place.

TOP: 2-1 Solving Equations by Adding or Subtracting

64. ANS: B

Since 6 is added to s, subtract 6 from both sides to undo the addition.

Check:

To check your solution, substitute 42 for s in the original equation.

Feedback A Check the tens place. B Correct! C Check the ones place. D Is addition the correct operation for solving this equation?


65. ANS: A When something is added to the variable, add its opposite to both sides of the equation to isolate the variable. Here, –14 is added to the variable, so add 14 to both sides of the equation to isolate s. Feedback A Correct! B Add the number that will isolate the variable.

C Add the opposite to isolate the variable. D Add the number that will isolate the variable.


66. ANS: A

First month salaries

Added to

Second month

salaries is 21,894

b + x = 21,894 b + x = 21,894 Write an equation to represent the

relationship. 10,205 + x = 21,894 –10,205 –10,205

Substitute 10,205 for b. Since 10,205 is added to x, subtract 10,205 from both sides to undo the addition.

The salaries for the second month are $11,689. Feedback A Correct! B Subtract the same number from both sides of the equation. C Check your answer. D Use the same operation on both sides of the equation.


67. ANS: B highest score

minus lowest score

equals score range

h – l = 23

Write an equation to represent the relationship.

Substitute 33 for l. Solve the equation.

Feedback A The sum of the lowest value and the range is the highest

value. B Correct! C The range is the difference between the highest and the lowest

values, not the sum. D The range is the difference between the highest and the lowest

values, not the average.


68. ANS: B

Since q is divided by 5, multiply both sides by 5 to undo the division.

q = 205 Check:

To check your solution, substitute 205 for q in the original equation.

Feedback A If the variable is connected to the number by division, then use

multiplication to solve for it. B Correct! C Instead of subtracting, multiply both sides by the denominator. D Multiply on both sides of the equation to isolate the variable.

TOP: 2-2 Solving Equations by Multiplying or Dividing

69. ANS: D 3n = 42

Since n is multiplied by 3, divide both sides by 3 to undo the multiplication.

Check: 3n = 42

To check your solution, substitute 14 for n in the

original equation.

Feedback A Since the variable is multiplied, divide on both sides to undo

the multiplication. B Check your solution by substituting the variable in the original

equation. C To undo multiplication, use division. D Correct!


70. ANS: A

The reciprocal of is . Since is

multiplied by , multiply both sides by .

Feedback A Correct! B Multiply both sides of the equation by the reciprocal of the

fraction. C Multiply both sides of the equation by the reciprocal of the

fraction. D Multiply both sides of the equation by the reciprocal of the

fraction.


71. ANS: A Seconds divided by 5 equals distance.

Write an equation. Let d = distance from the lightning strike in miles and t = number of seconds between flash and thunder.

Substitute 17 for d, the distance from the lightning strike.

Multiply both sides of the equation by 5 to undo the division.

The number of seconds between flash and thunder is about 85 seconds.

Feedback A Correct! B Divide the distance from the lightning strike by 5. C Divide the distance from the lightning strike by 5. D Divide the distance from the lightning strike by 5.


72. ANS: B

Solve the equation.

Substitute 8 for x and simplify. Feedback A Find the value of x by solving the equation. Then substitute it

for x in the given expression and simplify. B Correct! C Subtract the terms in the right order. D Find the value of x by solving the equation. Then substitute it

for x in the given expression and simplify.


73. ANS: D First x is multiplied by –2. Then 14 is added.

Work backward: Subtract 14 from both sides. Since x is multiplied by –2, divide both sides by –2 to

undo the multiplication.

Feedback A To solve for the variable, work backward. B Substitute the solution in the original equation to check your

answer.

C Check the signs. D Correct!

TOP: 2-3 Solving Two-Step and Multi-Step Equations

74. ANS: C

Since is subtracted from , add to both sides to undo the subtraction.

Since f is divided by 45, multiply both sides by 45 to undo the division. Simplify.

Feedback A First, add to undo the subtraction. Then, multiply to undo the

division. B Check your signs. C Correct! D First, add to undo the subtraction. Then, multiply to undo the

division.


75. ANS: A Use the Commutative Property of Addition.

Combine like terms.

Since 10 is added to 17a, subtract 10 from both sides to undo the addition. Since a is multiplied by 17, divide both sides

by 17 to undo the multiplication.

Feedback A Correct! B Check your signs. C Combine like terms, and then solve.

D Combine like terms, and then solve.


76. ANS: B Let d be the distance (in miles) to the movies, then is the number of miles after the first mile. So a formula for the total charge could be first mile charge

+ rate after first mile

= total charge

4.00 + 2.75 = 20.50 Subtract 4.00 from each side.

2.75 = 20.50 4.00

2.75 = 16.5 Divide both sides by 2.75.

=

= 6 Add 1 to both sides.

d = 6 + 1 d = 7

Feedback A Add one for the first mile. B Correct! C The mileage rate is the charge for each mile after the first mile. D Subtract the charge for the first mile.


77. ANS: A 8y – 8 = 24 Add 8 to both sides of the equation.

+ 8 + 8 8y =

32

8y = 32 Divide both sides by 8. 8 8

y = 4 2(4) = 8 Apply 4 to 2y.

Feedback A Correct! B Add before multiplying. C To undo subtraction, add to both sides. D To undo multiplication, divide.


78. ANS: B

Substitute 3750 for p, 3000 for n, and 900 for e.

Add 900 to both sides of the equation. Divide both sides by 3000.

Feedback A Divide both sides of the equation by the coefficient of c. B Correct! C Add the same number to both sides of the equation. D Add the same number to both sides of the equation.


79. ANS: D

To collect the variable terms on one side, subtract 50q from both sides.

Since 81 is subtracted from 2q, add 81 to both sides to undo the subtraction.

Since q is multiplied by 2, divide both sides by 2 to undo the multiplication.

Feedback A Check your signs. B After adding to undo the subtraction, divide to undo the

multiplication. C First, collect the variable terms on one side. Then, add to undo

the subtraction. D Correct!

TOP: 2-4 Solving Equations with Variables on Both Sides

80. ANS: A

Combine like terms.

Add to undo the subtraction. Or subtract to undo the addition. Then, divide to undo the multiplication.

n = 112 Feedback A Correct! B Add to undo the subtraction. Or subtract to undo the addition.

Then, divide to undo the multiplication. C Add to undo the subtraction. Or subtract to undo the addition.

Then, divide to undo the multiplication. D Combine like terms, and then solve.


81. ANS: C Combine like terms on each side of the equation before collecting variable terms on one side. If you get an equation that is always true, the original equation is an identity, and it has infinitely many solutions. If you get a false equation, the original equation is a contradiction, and it has no solutions. Feedback A First, combine like terms on each side of the equation. Then

collect variable terms on one side. Now, if you get an equation that is always true, it means that the original equation has infinitely many solutions. If you get a false equation, the original equation has no solutions.

B If you get an equation that is always true, the original equation is an identity, and it has infinitely many solutions. If you get a

false equation, the original equation is a contradiction and it has no solutions.

C Correct! D If you get an equation that is always true, the original equation

is an identity, and it has infinitely many solutions. If you get a false equation, the original equation is a contradiction and it has no solutions.


82. ANS: B Let m represent the number of movies rented each month. Here are the costs for each company (in dollars).

7.5 + m = 2.5m To collect the variable terms on one side, subtract m from both sides.

7.5 – m = 2.5m – m 7.5 = 1.5 m

Divide both sides by 1.5. = m

5 = m

Feedback A You divided $7.50 by $2.50. Before dividing by 2.50, subtract

the $1.00 charge from the $2.50. B Correct! C You divided $7.50 by $1.00. Subtract the $1.00 charge from

the $2.50 and then divide. D Set up this equation 7.5 + m = 2.5m, where m is the number of

movies.


83. ANS: C Let g represent the greatest integer. The expressions for the three consecutive integers from least to greatest: ,

, g. twice the greatest

integer 3 times the least

integer g ( )

To create an equation, use the additional data that 2g is 2 less than .

Solve the equation.

g 8

The three consecutive numbers are 6, 7, and 8. Feedback A If an expression is x less than a second expression, add x to

the first expression to make it equal to the second one. B If the variable represents the least number, add 2 to the

variable value to find the greatest number. Add 1 to find the second number.

C Correct! D If the variable represents the greatest number, subtract 2 from

the variable value to find the least number. Subtract 1 to find the second number.


84. ANS: D Divide both sides by t.

Substitute the known values.

Simplify. Round to the nearest tenth. Feedback A Solve the equation . B Solve the equation . C Solve the equation . D Correct!

TOP: 2-5 Solving for a Variable

85. ANS: C

Locate V in the equation.

Since V is divided by I, multiply both sides by I to undo the division.

Feedback A Multiply both sides by I to isolate r. B Multiply both sides by I to isolate r. C Correct! D Multiply both sides by I to isolate r.


86. ANS: D

Add z to both sides. Divide both sides by 4.

Feedback A To undo subtraction, add to both sides. B Both terms need to be divided by the coefficient of x. C To undo multiplication, divide. D Correct!


87. ANS: B

Subtract 2 from both sides.

Subtract the term from both sides.

Rewrite 3.8 as .

Multiply by .

Distribute.

Simplify. Feedback

A Divide all terms by 3.8. B Correct! C Multiply by the reciprocal of 3.8. D Keep the x term in the equation.


88. ANS: C

Write a ratio comparing ounces of oil to gallons of gasoline.

Write a proportion. Let x be the amount of oil in ounces.

Since x is divided by 38, multiply both sides of the equation by 38.

There are 14 ounces of oil.

Feedback A Set up a proportion, and use cross products. B First, set up two ratios that compare the ounces of oil to the

gallons of gasoline. Then, solve for the unknown amount of oil. C Correct! D Write a proportion, and use cross products.

TOP: 2-6 Rates Ratios and Proportions

89. ANS: A

Write a proportion to find an equivalent ratio with a second quantity of 1.

Divide on the left side to find x.

The unit rate is .

Ramon drives 50 miles per hour.

Feedback A Correct!

B Divide miles by hours, not hours by miles. C Check that you divided correctly when finding the unit rate. D This is not a unit rate.


90. ANS: B

• To convert the first quantity in a rate, multiply by a conversion factor with that unit in the first quantity.

336 quarts per hour

Feedback A There are 4 quarts in a gallon. B Correct! C There are 4 quarts in a gallon. D To convert the first quantity in a rate, multiply by a conversion

factor with that unit in the first quantity.


91. ANS: D

Use cross products.

Divide both sides by 6.

Feedback A Use cross products to solve. B Cross multiply. C Multiply the numerator of one fraction by the denominator of

the other fraction. D Correct!


92. ANS: C

Write the scale as a fraction.

Let x be the actual diameter.

Use cross products to solve.

Feedback A Use

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