Review Exercises

Real Numbers and the Number Line

Choose an appropriate scale and graph the following sets of real numbers on a number line.

1. {−4, 0, 4}

2. {−30, 10, 40}

3. {−12, −3, 9}

4. {−10, 8, 10}

Fill in the blank with <, $=$, or >.

5. $0 \_\_\_-9$

6. $-75 \_\_\_-5$

7. $-12 \_\_\_-\left(-3\right)$

8. $-\left(-23\right) \_\_\_ 23$

9. $\left|-20\right| \_\_\_\_-\left|-30\right|$

10. $-\left|6\right| \_\_\_\_-\left|-\left(-8\right)\right|$

Determine the unknown.

11. $\left|?\right|=2$

12. $\left|?\right|=1$

13. $\left|?\right|=-7$

14. $\left|?\right|=0$

Translate the following into a mathematical statement.

15. Negative eight is less than or equal to zero.

16. Seventy-eight is not equal to twelve.

17. Negative nine is greater than negative ten.

18. Zero is equal to zero.

Adding and Subtracting Integers

Simplify.

19. $12+\left(-7\right)$

20. $20+\left(-32\right)$

21. $-23-\left(-7\right)$

22. $-8-\left(-8\right)$

23. $-3-\left(-13\right)+\left(-1\right)$

24. $9+\left(-15\right)-\left(-8\right)$

25. $\left(7-10\right)-3$

26. $\left(-19+6\right)-2$

Find the distance between the given numbers on a number line.

27. −8 and 14

28. −35 and −6

29. What is 2 less than 17?

30. What is 3 less than −20?

31. Subtract 30 from the sum of 8 and 12.

32. Subtract 7 from the difference of −5 and 7.

33. An airplane flying at 22,000 feet descended 8,500 feet and then ascended 5,000 feet. What is the new altitude of the airplane?

34. The width of a rectangle is 5 inches less than its length. If the length measures 22 inches, then determine the width.

Multiplying and Dividing Integers

Simplify.

35. $10÷5\cdot 2$

36. $36÷6\cdot 2$

37. $-6\left(4\right)÷2\left(-3\right)$

38. $120÷\left(-5\right)\left(-3\right)\left(-2\right)$

39. $-8\left(-5\right)÷0$

40. $-24\left(0\right)÷8$

41. Find the product of −6 and 9.

42. Find the quotient of −54 and −3.

43. James was able to drive 234 miles on 9 gallons of gasoline. How many miles per gallon did he get?

44. If a bus travels at an average speed of 54 miles per hour for 3 hours, then how far does the bus travel?

Fractions

Reduce each fraction to lowest terms.

45. $\frac{180}{300}$

46. $\frac{252}{324}$

47. Convert to a mixed number: $\frac{23}{8}$.

48. Convert to an improper fraction: $3\frac{5}{9}$.

Simplify.

49. $\frac{3}{5}\left(-\frac{2}{7}\right)$

50. $-\frac{5}{8}\left(-\frac{1}{3}\right)$

51. $-\frac{3}{4}÷\frac{6}{7}$

52. $\frac{4}{15}÷\frac{28}{3}$

53. $4\frac{4}{5}÷6$

54. $5÷8\frac{1}{3}$

55. $\frac{5}{4}÷\frac{15}{2}\cdot 6$

56. $\frac{5}{24}÷\frac{3}{2}÷\frac{5}{12}$

57. $\frac{1}{12}-\frac{1}{4}$

58. $\frac{5}{6}-\frac{3}{14}$

59. $\frac{3}{4}+\frac{2}{3}-\frac{1}{12}$

60. $\frac{3}{10}+\frac{5}{12}-\frac{1}{6}$

61. Subtract $\frac{2}{3}$ from the sum of $-\frac{1}{2}$ and $\frac{2}{9}$.

62. Subtract $\frac{5}{6}$ from the difference of $\frac{1}{3}$ and $\frac{7}{2}$.

63. If a bus travels at an average speed of 54 miles per hour for $2\frac{1}{3}$ hours, then how far does the bus travel?

64. Determine the length of fencing needed to enclose a rectangular pen with dimensions $12\frac{1}{2}$ feet by $8\frac{3}{4}$ feet.

Decimals and Percents

65. Write as a mixed number: 5.32.

66. Write as a decimal: $7\frac{3}{25}$.

Perform the operations.

67. $6.032+2.19$

68. $12.106-9.21$

69. $4.23\times 5.13$

70. $9.246÷4.02$

Convert to a decimal.

71. $7.2\%$

72. $5\frac{3}{8}\%$

73. $147\%$

74. $27\frac{1}{2}\%$

Convert to a percent.

75. 0.055

76. 1.75

77. $\frac{9}{10}$

78. $\frac{5}{6}$

79. Mary purchased 3 boxes of t-shirts for a total of $126. If each box contains 24 t-shirts, then what is the cost of each t-shirt?

80. A retail outlet is offering 12% off the original $39.99 price of tennis shoes. What is the price after the discount?

81. If an item costs $129.99, then what is the total after adding $7\frac{1}{4}\%$ sales tax?

82. It is estimated that 8.3% of the total student population carpools to campus each day. If there are 13,000 students, then estimate the number of students that carpool to campus.

Exponents and Square Roots

Simplify.

83. $8^2$

84. ${\left(-5\right)}^2$

85. $-4^2$

86. $-{\left(-3\right)}^2$

87. ${\left(\frac{2}{9}\right)}^2$

88. ${\left(1\frac{2}{3}\right)}^2$

89. $3^3$

90. ${\left(-4\right)}^3$

91. ${\left(\frac{2}{5}\right)}^3$

92. ${\left(-\frac{1}{6}\right)}^3$

93. $-{\left(-2\right)}^4$

94. $-{\left(-1\right)}^5$

95. $\sqrt{49}$

96. $\sqrt{225}$

97. $2\sqrt{25}$

98. $-\sqrt{121}$

99. $3\sqrt{50}$

100. $-4\sqrt{12}$

101. $\sqrt{\frac{4}{9}}$

102. $\sqrt{\frac{8}{25}}$

103. Calculate the area of a square with sides measuring 3 centimeters. ($A=s^2$)

104. Calculate the volume of a cube with sides measuring 3 centimeters. ($V=s^3$)

105. Determine the length of the diagonal of a square with sides measuring 3 centimeters.

106. Determine the length of the diagonal of a rectangle with dimensions 2 inches by 4 inches.

Order of Operations

Simplify.

107. $-5\left(2\right)-7^2$

108. $1-4^2+2{\left(-3\right)}^2$

109. $2+3{\left(6-2\cdot 4\right)}^3$

110. $5-3{\left(8-3\cdot 4\right)}^2$

111. $-2^3+6\left(3^2-4\right)+{\left(-3\right)}^2$

112. $5^2-40÷5{\left(-2\right)}^2-\left(-4\right)$

113. $\frac{3}{4}{\left[\frac{2}{9}{\left(-3\right)}^2-4\right]}^2$

114. ${\left(\frac{1}{2}\right)}^2-\frac{3}{4}÷\frac{9}{16}-\frac{1}{3}$

115. $\frac{2-3{\left(6-3^2\right)}^2}{4\cdot 5-5^2}$

116. $\frac{{\left(2\cdot 8-6^2\right)}^2-{10}^2}{7^3-\left(2{\left(-5\right)}^3-7\right)}$

117. $8-5\left|3\cdot 4-{\left(-2\right)}^4\right|$

118. $\left|14-\left|-3-5^2\right|\right|$

Find the distance between the given numbers on a number line.

119. −14 and 22

120. −42 and −2

121. $\frac{7}{8}$ and $-\frac{1}{5}$

122. $-5\frac{1}{2}$ and $-1\frac{1}{4}$

Sample Exam

1. List three integers greater than −10.

2. Determine the unknown(s): $\left|?\right|=13$.

3. Fill in the blank with <, =, or >: $-\left|-100\right|\_\_\_9^2$.

4. Convert to a fraction: $33\frac{1}{3}\%$.

5. Convert to a percent: $2\frac{3}{4}$.

6. Reduce: $\frac{75}{225}$.

Calculate the following.

7. a. ${(-7)}^2$; b. $-{(-7)}^2$; c. $-7^2$

8. a. ${(-3)}^3$; b. $-{(-3)}^3$; c. $-3^3$

9. a. $\left|10\right|$; b. $-\left|-10\right|$; c. $-\left|10\right|$

Simplify.

10. $-\left(-\left(-1\right)\right)$

11. $\frac{2}{3}+\frac{1}{5}-\frac{3}{10}$

12. $10-\left(-12\right)+\left(-8\right)-20$

13. $-8\left(4\right)\left(-3\right)÷2$

14. $\frac{1}{2}\cdot \left(-\frac{4}{5}\right)÷\frac{14}{15}$

15. $\frac{3}{5}\cdot \frac{1}{2}-\frac{2}{3}$

16. $4\cdot 5-20÷5\cdot 2$

17. $10-7\left(3-8\right)-5^2$

18. $3+2\left|-2^2-\left(-1\right)\right|+{\left(-2\right)}^2$

19. $\frac{1}{3}\left[5^2-\left(7-\left|-2\right|\right)+15\cdot 2÷3\right]$

20. $\sqrt{\frac{1}{16}}$

21. $3\sqrt{72}$

22. Subtract 2 from the sum of 8 and −10.

23. Subtract 10 from the product of 8 and −10.

24. A student earns 9, 8, 10, 7, and 8 points on the first 5 chemistry quizzes. What is her quiz average?

25. An $8\frac{3}{4}$ foot plank is cut into 5 pieces of equal length. What is the length of each piece?

Review Exercises Answers

1:

3:

5: >

7: <

9: >

11: ±2

13: Ø, No Solution

15: $-8\le 0$

17: $-9>-10$

19: 5

21: −16

23: 9

25: −6

27: 22 units

29: 15

31: −10

33: 18,500 feet

35: 4

37: 36

39: Undefined

41: −54

43: 26 miles per gallon

45: $\frac{3}{5}$

47: $2\frac{7}{8}$

49: $-\frac{6}{35}$

51: $-\frac{7}{8}$

53: $\frac{4}{5}$

55: 1

57: $-\frac{1}{6}$

59: $\frac{4}{3}$

61: $-\frac{17}{18}$

63: 126 miles

65: $5\frac{8}{25}$

67: 8.222

69: 21.6999

71: 0.072

73: 1.47

75: 5.5%

77: 90%

79: $1.75

81: $139.41

83: 64

85: −16

87: 4/81

89: 27

91: $\frac{8}{125}$

93: −16

95: 7

97: 10

99: $15\sqrt{2}$

101: $\frac{2}{3}$

103: 9 square centimeters

105: $3\sqrt{2}$ centimeters

107: −59

109: −22

111: 31

113: 3

115: 5

117: −12

119: 36 units

121: $\frac{43}{40}$ units

Sample Exam Answers

1: {−5, 0, 5} (answers may vary)

3: <

5: 275%

7:a. 49; b. −49; c. −49

9:a. 10; b. −10; c. −10

11: $\frac{17}{30}$

13: 48

15: $-\frac{11}{30}$

17: 20

19: 10

21: $18\sqrt{2}$

23: −90

25: $1\frac{3}{4}$ feet