# 1.4E: Exercises

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

## Practice Makes Perfect

Simplify Fractions

In the following exercises, simplify.

1. $$−\dfrac{108}{63}$$

$$−\dfrac{12}{7}$$

2. $$−\dfrac{104}{48}$$

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

$$\dfrac{10}{21}$$

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

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

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

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

7. $$−\dfrac{210a^2}{110b^2}$$

$$−\dfrac{21a^2}{11b^2}$$

8. $$−\dfrac{30x^2}{105y^2}$$

Multiply and Divide Fractions

In the following exercises, perform the indicated operation.

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

$$\dfrac{1}{3}$$

10. $$−\dfrac{3}{8}⋅\dfrac{4}{15}$$

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

$$−\dfrac{21}{50}$$

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

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

$$\dfrac{11}{30}$$

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

15. $$\dfrac{3}{7}⋅21n$$

$$9n$$

16. $$\dfrac{5}{6}⋅30m$$

17. $$\dfrac{3}{4}÷\dfrac{x}{11}$$

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

18. $$\dfrac{2}{5}÷\dfrac{y}{9}$$

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

$$−\dfrac{4}{9}$$

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

21. $$\dfrac{8u}{15}÷\dfrac{12v}{25}$$

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

22. $$\dfrac{12r}{25}÷\dfrac{18s}{35}$$

23. $$\dfrac{3}{4}÷(−12)$$

$$−\dfrac{1}{16}$$

24. $$−15÷\left(−\dfrac{5}{3}\right)$$

In the following exercises, simplify.

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

$$−\dfrac{10}{9}$$

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

27. $$−\dfrac{\dfrac{4}{5}}{2}$$

$$−\dfrac{2}{5}$$

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

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

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

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

In the following exercises, add or subtract.

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

$$\dfrac{29}{24}$$

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

33. $$\dfrac{7}{12}−\dfrac{9}{16}$$

$$\dfrac{1}{48}$$

34. $$\dfrac{7}{16}−\dfrac{5}{12}$$

35. $$−\dfrac{13}{30}+\dfrac{25}{42}$$

$$\dfrac{17}{105}$$

36. $$−\dfrac{23}{30}+\dfrac{5}{48}$$

37. $$−\dfrac{39}{56}−\dfrac{22}{35}$$

$$−\dfrac{53}{40}$$

38. $$−\dfrac{33}{49}−\dfrac{18}{35}$$

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

$$\dfrac{1}{12}$$

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

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

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

42. $$\dfrac{x}{5}−\dfrac{1}{4}$$

43. ⓐ $$\dfrac{2}{3}+\dfrac{1}{6}$$

ⓑ $$\dfrac{2}{3}÷\dfrac{1}{6}$$

ⓐ $$\dfrac{5}{6}$$ ⓑ $$4$$

44. ⓐ $$−\dfrac{2}{5}−\dfrac{1}{8}$$

ⓑ $$−\dfrac{2}{5}·\dfrac{1}{8}$$

45. ⓐ $$\dfrac{5n}{6}÷\dfrac{8}{15}$$

ⓑ $$\dfrac{5n}{6}−\dfrac{8}{15}$$

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

46. ⓐ $$\dfrac{3a}{8}÷\dfrac{7}{12}$$

ⓑ $$\dfrac{3a}{8}−\dfrac{7}{12}$$

47. ⓐ $$−\dfrac{4x}{9}−\dfrac{5}{6}$$

ⓑ $$−\dfrac{4k}{9}⋅\dfrac{5}{6}$$

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

48. ⓐ $$−\dfrac{3y}{8}−\dfrac{4}{3}$$

ⓑ $$−\dfrac{3y}{8}⋅\dfrac{4}{3}$$

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

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

ⓐ $$\dfrac{−5(a+1)}{3}$$ ⓑ $$a$$

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. $$\dfrac{5⋅6−3⋅4}{4⋅5−2⋅3}$$

$$\dfrac{9}{7}$$

52. $$\dfrac{8⋅9−7⋅6}{5⋅6−9⋅2}$$

53. $$\dfrac{5^2−3^2}{3−5}$$

$$−8$$

54. $$\dfrac{6^2−4^2}{4−6}$$

55. $$\dfrac{7⋅4−2(8−5)}{9⋅3−3⋅5}$$

$$\dfrac{11}{6}$$

56. $$\dfrac{9⋅7−3(12−8)}{8⋅7−6⋅6}$$

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

$$\dfrac{5}{2}$$

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

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

$$54$$

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

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

$$\dfrac{49}{25}$$

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

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

$$\dfrac{15}{4}$$

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

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

$$\dfrac{5}{21}$$

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

Mixed Practice

In the following exercises, simplify.

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

$$\dfrac{5}{4}$$

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

69. $$−\dfrac{3}{8}+\dfrac{5}{12}$$

$$\dfrac{1}{24}$$

70. $$−\dfrac{1}{8}+\dfrac{7}{12}$$

71. $$−\dfrac{7}{15}−\dfrac{y}{4}$$

$$\dfrac{−28−15y}{60}$$

72. $$−\dfrac{3}{8}−\dfrac{x}{11}$$

73. $$\dfrac{11}{12a}⋅\dfrac{9a}{16}$$

$$\dfrac{33}{64}$$

74. $$\dfrac{10y}{13}⋅\dfrac{8}{15y}$$

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

$$\dfrac{7}{9}$$

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

77. $$1−\dfrac{3}{5}÷\dfrac{1}{10}$$

$$−5$$

78. $$1−\dfrac{5}{6}÷\dfrac{1}{12}$$

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

$$\dfrac{23}{24}$$

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

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

$$\dfrac{11}{5}$$

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

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

$$1$$

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

​​​​​​​

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

$$\dfrac{13}{3}$$

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 ⓐ $$w=\dfrac{1}{2}$$ ⓑ $$w=−\dfrac{1}{2}$$

ⓐ $$\dfrac{1}{5}$$ ⓑ $$\dfrac{6}{5}$$

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

​​​​​​​​​​​​​​

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

$$−\dfrac{1}{9}$$

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

​​​​​​​​​​​​​​

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

$$−\dfrac{5}{11}$$

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.

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

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

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.

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

This page titled 1.4E: Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.