1.3E: Exercises

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Practice Makes Perfect

Simplify Fractions

In the following exercises, simplify.

1. $−\dfrac{108}{63}$

Answer: $−\dfrac{12}{7}$

2. $−\dfrac{104}{48}$

3. $\dfrac{120}{252}$

Answer: $\dfrac{10}{21}$

4. $\dfrac{182}{294}$

5. $\dfrac{14x^2}{21y}$

Answer: $\dfrac{2x^2}{3y}$

6. $\dfrac{24a}{32b^2}$

7. $−\dfrac{210a^2}{110b^2}$

Answer: $−\dfrac{21a^2}{11b^2}$

8. $−\dfrac{30x^2}{105y^2}$

Multiply and Divide Fractions

In the following exercises, perform the indicated operation.

9. $−\dfrac{3}{4}\left(−\dfrac{4}{9}\right)$

Answer: $\dfrac{1}{3}$

10. $−\dfrac{3}{8}⋅\dfrac{4}{15}$

11. $\left(−\dfrac{14}{15}\right)\left(\dfrac{9}{20}\right)$

Answer: $−\dfrac{21}{50}$

12. $\left(−\dfrac{9}{10}\right)\left(\dfrac{25}{33}\right)$

13. $\left(−\dfrac{63}{84}\right)\left(−\dfrac{44}{90}\right)$

Answer: $\dfrac{11}{30}$

14. $\left(−\dfrac{33}{60}\right)\left(−\dfrac{40}{88}\right)$

15. $\dfrac{3}{7}⋅21n$

Answer: $9n$

16. $\dfrac{5}{6}⋅30m$

17. $\dfrac{3}{4}÷\dfrac{x}{11}$

Answer: $\dfrac{33}{4x}$

18. $\dfrac{2}{5}÷\dfrac{y}{9}$

19. $\dfrac{5}{18}÷\left(−\dfrac{15}{24}\right)$

Answer: $−\dfrac{4}{9}$

20. $\dfrac{7}{18}÷\left(−\dfrac{14}{27}\right)$

21. $\dfrac{8u}{15}÷\dfrac{12v}{25}$

Answer: $\dfrac{10u}{9v}$

22. $\dfrac{12r}{25}÷\dfrac{18s}{35}$

23. $\dfrac{3}{4}÷(−12)$

Answer: $−\dfrac{1}{16}$

24. $−15÷\left(−\dfrac{5}{3}\right)$

In the following exercises, simplify.

25. $−\dfrac{\dfrac{8}{21} }{\dfrac{12}{35}}$

Answer: $−\dfrac{10}{9}$

26. $− \dfrac{\dfrac{9}{16} }{\dfrac{33}{40}}$

27. $−\dfrac{\dfrac{4}{5}}{2}$

Answer: $−\dfrac{2}{5}$

28. $\dfrac{\dfrac{5}{3}}{10}$

29. $\dfrac{\dfrac{m}{3}}{\dfrac{n}{2}}$

Answer: $\dfrac{2m}{3n}$

30. $\dfrac{−\dfrac{3}{8}}{−\dfrac{y}{12}}$

Add and Subtract Fractions

In the following exercises, add or subtract.

31. $\dfrac{7}{12}+\dfrac{5}{8}$

Answer: $\dfrac{29}{24}$

32. $\dfrac{5}{12}+\dfrac{3}{8}$

33. $\dfrac{7}{12}−\dfrac{9}{16}$

Answer: $\dfrac{1}{48}$

34. $\dfrac{7}{16}−\dfrac{5}{12}$

35. $−\dfrac{13}{30}+\dfrac{25}{42}$

Answer: $\dfrac{17}{105}$

36. $−\dfrac{23}{30}+\dfrac{5}{48}$

37. $−\dfrac{39}{56}−\dfrac{22}{35}$

Answer: $−\dfrac{53}{40}$

38. $−\dfrac{33}{49}−\dfrac{18}{35}$

39. $−\dfrac{2}{3}−\left(−\dfrac{3}{4}\right)$

Answer: $\dfrac{1}{12}$

40. $−\dfrac{3}{4}−\left(−\dfrac{4}{5}\right)$

41. $\dfrac{x}{3}+\dfrac{1}{4}$

Answer: $\dfrac{4x+3}{12}$

42. $\dfrac{x}{5}−\dfrac{1}{4}$

43. ⓐ $\dfrac{2}{3}+\dfrac{1}{6}$

ⓑ $\dfrac{2}{3}÷\dfrac{1}{6}$

Answer: ⓐ $\dfrac{5}{6}$ ⓑ $4$

44. ⓐ $−\dfrac{2}{5}−\dfrac{1}{8}$

ⓑ $−\dfrac{2}{5}·\dfrac{1}{8}$

45. ⓐ $\dfrac{5n}{6}÷\dfrac{8}{15}$

ⓑ $\dfrac{5n}{6}−\dfrac{8}{15}$

Answer: ⓐ $\dfrac{25n}{16}$ ⓑ $\dfrac{25n−16}{30}$

46. ⓐ $\dfrac{3a}{8}÷\dfrac{7}{12}$

ⓑ $\dfrac{3a}{8}−\dfrac{7}{12}$

47. ⓐ $−\dfrac{4x}{9}−\dfrac{5}{6}$

ⓑ $−\dfrac{4k}{9}⋅\dfrac{5}{6}$

Answer: ⓐ $\dfrac{−8x−15}{18}$ ⓑ $−\dfrac{10k}{27}$

48. ⓐ $−\dfrac{3y}{8}−\dfrac{4}{3}$

ⓑ $−\dfrac{3y}{8}⋅\dfrac{4}{3}$

49. ⓐ $−\dfrac{5a}{3}+\left(−\dfrac{10}{6}\right)$

ⓑ $−\dfrac{5a}{3}÷\left(−\dfrac{10}{6}\right)$

Answer: ⓐ $\dfrac{−5(a+1)}{3}$ ⓑ $a$

50. ⓐ $\dfrac{2b}{5}+\dfrac{8}{15}$

ⓑ $\dfrac{2b}{5}÷\dfrac{8}{15}$

Use the Order of Operations to Simplify Fractions

In the following exercises, simplify.

51. $\dfrac{5⋅6−3⋅4}{4⋅5−2⋅3}$

Answer: $\dfrac{9}{7}$

52. $\dfrac{8⋅9−7⋅6}{5⋅6−9⋅2}$

53. $\dfrac{5^2−3^2}{3−5}$

Answer: $−8$

54. $\dfrac{6^2−4^2}{4−6}$

55. $\dfrac{7⋅4−2(8−5)}{9⋅3−3⋅5}$

Answer: $\dfrac{11}{6}$

56. $\dfrac{9⋅7−3(12−8)}{8⋅7−6⋅6}$

57. $\dfrac{9(8−2)−3(15−7)}{6(7−1)−3(17−9)}$

Answer: $\dfrac{5}{2}$

58. $\dfrac{8(9−2)−4(14−9)}{7(8−3)−3(16−9)}$

59. $\dfrac{2^3+4^2}{\left(\dfrac{2}{3}\right)^2}$

Answer: $54$

60. $\dfrac{3^3−3^2}{\left(\dfrac{3}{4}\right)^2}$

61. $\dfrac{\left(\dfrac{3}{5}\right)^2}{\left(\dfrac{3}{7}\right)^2}$

Answer: $\dfrac{49}{25}$

62. $\dfrac{\left(\dfrac{3}{4}\right)^2}{\left(\dfrac{5}{8}\right)^2}$

63. $\dfrac{2}{\dfrac{1}{3}+\dfrac{1}{5}}$

Answer: $\dfrac{15}{4}$

64. $\dfrac{5}{\dfrac{1}{4}+\dfrac{1}{3}}$

65. $\dfrac{\dfrac{7}{8}−\dfrac{2}{3}}{\dfrac{1}{2}+\dfrac{3}{8}}$

Answer: $\dfrac{5}{21}$

66. $\dfrac{\dfrac{3}{4}−\dfrac{3}{5}}{\dfrac{1}{4}+\dfrac{2}{5}}$

Mixed Practice

In the following exercises, simplify.

67. $−\dfrac{3}{8}÷\left(−\dfrac{3}{10}\right)$

Answer: $\dfrac{5}{4}$

68. $−\dfrac{3}{12}÷\left(−\dfrac{5}{9}\right)$

69. $−\dfrac{3}{8}+\dfrac{5}{12}$

Answer: $\dfrac{1}{24}$

70. $−\dfrac{1}{8}+\dfrac{7}{12}$

71. $−\dfrac{7}{15}−\dfrac{y}{4}$

Answer: $\dfrac{−28−15y}{60}$

72. $−\dfrac{3}{8}−\dfrac{x}{11}$

73. $\dfrac{11}{12a}⋅\dfrac{9a}{16}$

Answer: $\dfrac{33}{64}$

74. $\dfrac{10y}{13}⋅\dfrac{8}{15y}$

75. $\dfrac{1}{2}+\dfrac{2}{3}⋅\dfrac{5}{12}$

Answer: $\dfrac{7}{9}$

76. $\dfrac{1}{3}+\dfrac{2}{5}⋅\dfrac{3}{4}$

77. $1−\dfrac{3}{5}÷\dfrac{1}{10}$

Answer: $−5$

78. $1−\dfrac{5}{6}÷\dfrac{1}{12}$

79. $\dfrac{3}{8}−\dfrac{1}{6}+\dfrac{3}{4}$

Answer: $\dfrac{23}{24}$

80. $\dfrac{2}{5}+\dfrac{5}{8}−\dfrac{3}{4}$

81. $12\left(\dfrac{9}{20}−\dfrac{4}{15}\right)$

Answer: $\dfrac{11}{5}$

82. $8\left(\dfrac{15}{16}−\dfrac{5}{6}\right)$

83. $\dfrac{\dfrac{5}{8}+\dfrac{1}{6}}{\dfrac{19}{24}}$

Answer: $1$

84. $\dfrac{\dfrac{1}{6}+\dfrac{3}{10}}{\dfrac{14}{30}}$

85. $\left(\dfrac{5}{9}+\dfrac{1}{6}\right)÷\left(\dfrac{2}{3}−\dfrac{1}{2}\right)$

Answer: $\dfrac{13}{3}$

86. $\left(\dfrac{3}{4}+\dfrac{1}{6}\right)÷\left(\dfrac{5}{8}−\dfrac{1}{3}\right)$

Evaluate Variable Expressions with Fractions

In the following exercises, evaluate.

87. $\dfrac{7}{10}−w$ when ⓐ $w=\dfrac{1}{2}$ ⓑ $w=−\dfrac{1}{2}$

Answer: ⓐ $\dfrac{1}{5}$ ⓑ $\dfrac{6}{5}$

88. $512−w$ when ⓐ $w=\dfrac{1}{4}$ ⓑ $w=−\dfrac{1}{4}$

89. $2x^2y^3$ when $x=−\dfrac{2}{3}$ and $y=−\dfrac{1}{2}$

Answer: $−\dfrac{1}{9}$

90. $8u^2v^3$ when $u=−\dfrac{3}{4}$ and $v=−\dfrac{1}{2}$

91. $\dfrac{a+b}{a−b}$ when $a=−3$ and $b=8$

Answer: $−\dfrac{5}{11}$

92. $\dfrac{r−s}{r+s}$ when $r=10$ and $s=−5$

Writing Exercises

93. Why do you need a common denominator to add or subtract fractions? Explain.

Answer: Answers will vary.

94. How do you find the LCD of 2 fractions?

95. Explain how you find the reciprocal of a fraction.

Answer: Answers will vary.

96. Explain how you find the reciprocal of a negative number.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

$This table has 4 columns, 5 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don’t get it. The first column has the following statements: simplify fractions, multiply and divide fractions, add and subtract fractions, use the order of operations to simplify fractions, evaluate variable expressions with fractions. The remaining columns are blank.$

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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Practice Makes Perfect

Writing Exercises

Self Check