Exercises: Fractions (1.4)

Domenic Spilotro, MSc

Exercises: Fractions (1.4)

Exercises: Find Equivalent Fractions

Instructions: For questions 1-4, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.

1. $\frac{3}{8}$

Solution

$\frac{6}{16}, \frac{9}{24}, \frac{12}{32}$ (Answers may vary)

2. $\frac{5}{8}$

3. $\frac{5}{9}$

Solution

$\frac{10}{18}, \frac{15}{27}, \frac{20}{36}$ (Answers may vary)

4. $\frac{1}{8}$

Exercises: Simplify Fractions

Instructions: For questions 5-14, simplify.

5. $- \frac{40}{88}$

Solution

$- \frac{5}{11}$

6. $- \frac{63}{99}$

7. $- \frac{108}{63}$

Solution

$- \frac{12}{7}$

8. $- \frac{104}{48}$

9. $\frac{120}{252}$

Solution

$\frac{10}{21}$

10. $\frac{182}{294}$

11. $- \frac{3 x}{12 y}$

Solution

$- \frac{x}{4 y}$

12. $- \frac{4 x}{32 y}$

13. $\frac{14 x^{2}}{21 y}$

Solution

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

14. $\frac{24 a}{32 b^{2}}$

Exercises: Multiply Fractions

Instructions: For questions 15-30, multiply.

15. $\frac{3}{4} \cdot \frac{9}{10}$

Solution

$\frac{27}{40}$

16. $\frac{4}{5} \cdot \frac{2}{7}$

17. $- \frac{2}{3} (- \frac{3}{8})$

Solution

$\frac{1}{4}$

18. $- \frac{3}{4} (- \frac{4}{9})$

19. $- \frac{5}{9} \cdot \frac{3}{10}$

Solution

$- \frac{1}{6}$

20. $- \frac{3}{8} \cdot \frac{4}{15}$

21. $(- \frac{14}{15}) (\frac{9}{20})$

Solution

$- \frac{21}{50}$

22. $(- \frac{9}{10}) (\frac{25}{33})$

23. $(- \frac{63}{84}) (- \frac{44}{90})$

Solution

$\frac{11}{30}$

24. $(- \frac{63}{60}) (- \frac{40}{88})$

25. $4 \cdot \frac{5}{11}$

Solution

$\frac{20}{11}$

26. $5 \cdot \frac{8}{3}$

27. $\frac{3}{7} \cdot 21 n$

Solution

$9 n$

28. $\frac{5}{6} \cdot 30 m$

29. $(- 8) (\frac{17}{4})$

Solution

$- 34$

30. $(- 1) (- \frac{6}{7})$

Exercises: Divide Fractions

Instructions: For questions 31-44, divide.

31. $\frac{3}{4} \div \frac{2}{3}$

Solution

$\frac{9}{8}$

32. $\frac{4}{5} \div \frac{3}{4}$

33. $- \frac{7}{9} \div (- \frac{7}{4})$

Solution

$1$

34. $- \frac{5}{6} \div (- \frac{5}{6})$

35. $\frac{3}{4} \div \frac{x}{11}$

Solution

$\frac{33}{4 x}$

36. $\frac{2}{5} \div \frac{y}{9}$

37. $\frac{5}{18} \div (- \frac{15}{24})$

Solution

$- \frac{4}{9}$

38. $\frac{7}{18} \div (- \frac{14}{27})$

39. $\frac{8 u}{15} \div \frac{12 v}{25}$

Solution

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

40. $\frac{12 r}{25} \div \frac{18 s}{35}$

41. $- 5 \div \frac{1}{2}$

Solution

$- 10$

42. $- 3 \div \frac{1}{4}$

43. $\frac{3}{4} \div (- 12)$

Solution

$- \frac{1}{16}$

44. $- 15 \div (- \frac{5}{3})$

Exercises: Simplify by Dividing

Instructions: For questions 45-50, simplify.

45. $\frac{- \frac{8}{21}}{\frac{12}{35}}$

Solution

$- \frac{10}{9}$

46. $\frac{- \frac{9}{16}}{\frac{33}{40}}$

47. $\frac{- \frac{4}{5}}{2}$

Solution

$- \frac{2}{5}$

48. $\frac{5}{\frac{3}{10}}$

49. $\frac{\frac{m}{3}}{\frac{n}{2}}$

Solution

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

50. $\frac{- \frac{3}{8}}{- \frac{y}{12}}$

Exercises: Simplify Expressions Written with a Fraction Bar

Instructions: For questions 51-70, simplify.

51. $\frac{22 + 3}{10}$

Solution

$\frac{5}{2}$

52. $\frac{19 - 4}{6}$

53. $\frac{48}{24 - 15}$

Solution

$\frac{16}{3}$

54. $\frac{46}{4 + 4}$

55. $\frac{- 6 + 6}{8 + 4}$

Solution

$0$

56. $\frac{- 6 + 3}{17 - 8}$

57. $\frac{4 \cdot 3}{6 \cdot 6}$

Solution

$\frac{1}{3}$

58. $\frac{6 \cdot 6}{9 \cdot 2}$

59. $\frac{4^{2} - 1}{25}$

Solution

$\frac{3}{5}$

60. $\frac{7^{2} + 1}{60}$

61. $\frac{8 \cdot 3 + 2 \cdot 9}{14 + 3}$

Solution

$2 \frac{8}{17}$

62. $\frac{9 \cdot 6 - 4 \cdot 7}{22 + 3}$

63. $\frac{5 \cdot 6 - 3 \cdot 4}{4 \cdot 5 - 2 \cdot 3}$

Solution

$\frac{3}{5}$

64. $\frac{8 \cdot 9 - 7 \cdot 6}{5 \cdot 6 - 9 \cdot 2}$

65. $\frac{5^{2} - 3^{2}}{3 - 5}$

Solution

$- 8$

66. $\frac{6^{2} - 4^{2}}{4 - 6}$

67. $\frac{7 \cdot 4 - 2 (8 - 5)}{9 \cdot 3 - 3 \cdot 5}$

Solution

$\frac{11}{6}$

68. $\frac{9 \cdot 7 - 3 (12 - 8)}{8 \cdot 7 - 6 \cdot 6}$

69. $\frac{9 (8 - 2) - 3 (15 - 7)}{6 (7 - 1) - 3 (17 - 9)}$

Solution

$\frac{5}{2}$

70. $\frac{8 (9 - 2) - 4 (14 - 9)}{7 (8 - 3) - 3 (16 - 9)}$

Exercises: Translate Phrases to Expressions with Fractions

Instructions: For questions 71-74, translate each English phrase into an algebraic expression.

71. the quotient of $r$ and the sum of $s$ and $10$

Solution

$\frac{r}{s + 10}$

72. the quotient of $A$ and the difference of $3$ and $B$

73. the quotient of the difference of $x$ and $y$ , and $- 3$

Solution

$\frac{x - y}{- 3}$

74. the quotient of the sum of $m$ and $n$ , and $4 q$

Exercises: Everyday Math

Instructions: For questions 75-78, answer the given everyday math word problems.

75. Baking. A recipe for chocolate chip cookies calls for $\frac{3}{4}$ cup brown sugar. Imelda wants to double the recipe.

a. How much brown sugar will Imelda need? Show your calculation.
b. Measuring cups usually come in sets of $\frac{1}{4}, \frac{1}{3}, \frac{1}{2}, and 1$ cup. Draw a diagram to show two different ways that Imelda could measure the brown sugar needed to double the cookie recipe.

Solution

a. $1 \frac{1}{2}$ cups
b. Answers will vary

76. Baking. Nina is making $4$ pans of fudge to serve after a music recital. For each pan, she needs $\frac{2}{3}$ cup of condensed milk.

a. How much condensed milk will Nina need? Show your calculation.
b. Measuring cups usually come in sets of $\frac{1}{4}, \frac{1}{3}, \frac{1}{2}, and 1$ cup. Draw a diagram to show two different ways that Nina could measure the condensed milk needed for $4$ pans of fudge.

77. Portions. Don purchased a bulk package of candy that weighs $5$ pounds. He wants to sell the candy in little bags that hold $\frac{1}{4}$ pound. How many little bags of candy can he fill from the bulk package?

Solution

$20$ bags

78. Portions. Kristen has $\frac{3}{4}$ yards of ribbon that she wants to cut into $6$ equal parts to make hair ribbons for her daughter’s $6$ dolls. How long will each doll’s hair ribbon be?

Exercises: Writing Exercises

Instructions: For questions 79-82, answer the given writing exercises.

79. Rafael wanted to order half a medium pizza at a restaurant. The waiter told him that a medium pizza could be cut into $6$ or $8$ slices. Would he prefer $3$ out of $6$ slices or $4$ out of $8$ slices? Rafael replied that since he wasn’t very hungry, he would prefer $3$ out of $6$ slices. Explain what is wrong with Rafael’s reasoning.

Solution

Answers may vary

80. Give an example from everyday life that demonstrates how $\frac{1}{2} \cdot \frac{2}{3}$ is $\frac{1}{3}$ .

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

Solution

Answers may vary

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

Exercises: Add Fractions with a Common Denominator

Instructions: For questions 83-92, add.

83. $\frac{6}{13} + \frac{5}{13}$

Solution

$\frac{11}{13}$

84. $\frac{4}{15} + \frac{7}{15}$

85. $\frac{x}{4} + \frac{3}{4}$

Solution

$\frac{x + 3}{4}$

86. $\frac{8}{q} + \frac{6}{q}$

87. $- \frac{3}{16} + (- \frac{7}{16})$

Solution

$- \frac{5}{8}$

88. $- \frac{5}{16} + (- \frac{9}{16})$

89. $- \frac{8}{17} + \frac{15}{17}$

Solution

$\frac{7}{17}$

90. $- \frac{9}{19} + \frac{17}{19}$

91. $\frac{6}{13} + (- \frac{10}{13}) + (- \frac{12}{13})$

Solution

$- \frac{16}{13}$

92. $\frac{5}{12} + (- \frac{7}{12}) + (- \frac{11}{12})$

Exercises: Subtract Fractions with a Common Denominator

Instructions: For questions 93-106, subtract.

93. $\frac{11}{15} - \frac{7}{15}$

Solution

$\frac{4}{15}$

94. $\frac{9}{13} - \frac{4}{13}$

95. $\frac{11}{12} - \frac{5}{12}$

Solution

$\frac{1}{2}$

96. $\frac{7}{12} - \frac{5}{12}$

97. $\frac{19}{21} - \frac{4}{21}$

Solution

$\frac{5}{7}$

98. $\frac{17}{21} - \frac{8}{21}$

99. $\frac{5 y}{8} - \frac{7}{8}$

Solution

$\frac{5 y - 7}{8}$

100. $\frac{11 z}{13} - \frac{8}{13}$

101. $- \frac{23}{u} - \frac{15}{u}$

Solution

$- \frac{38}{u}$

102. $- \frac{29}{v} - \frac{26}{v}$

103. $- \frac{3}{5} - (- \frac{4}{5})$

Solution

$\frac{1}{5}$

104. $- \frac{3}{7} - (- \frac{5}{7})$

105. $- \frac{7}{9} - (- \frac{5}{9})$

Solution

$- \frac{2}{9}$

106. $- \frac{8}{11} - (- \frac{5}{11})$

Exercises: Mixed Practice

Instructions: For questions 107-114, simplify.

107. $- \frac{5}{18} \cdot \frac{9}{10}$

Solution

$- \frac{1}{4}$

108. $- \frac{3}{14} \cdot \frac{7}{12}$

109. $\frac{n}{5} - \frac{4}{5}$

Solution

$\frac{n - 4}{5}$

110. $\frac{6}{11} - \frac{s}{11}$

111. $- \frac{7}{24} + \frac{2}{24}$

Solution

$- \frac{5}{24}$

112. $- \frac{5}{18} + \frac{1}{18}$

113. $\frac{8}{15} \div \frac{12}{5}$

Solution

$\frac{2}{9}$

114. $\frac{7}{12} \div \frac{9}{28}$

Exercises: Add or Subtract Fractions with Different Denominators

Instructions: For questions 115-138, add or subtract.

115. $\frac{1}{2} + \frac{1}{7}$

Solution

$\frac{9}{14}$

116. $\frac{1}{3} + \frac{1}{8}$

117. $\frac{1}{3} - (- \frac{1}{9})$

Solution

$\frac{4}{9}$

118. $\frac{1}{4} - (- \frac{1}{8})$

119. $\frac{7}{12} + \frac{5}{8}$

Solution

$\frac{29}{24}$

120. $\frac{5}{12} + \frac{3}{8}$

121. $\frac{7}{12} - \frac{9}{16}$

Solution

$\frac{1}{48}$

122. $\frac{7}{16} - \frac{5}{12}$

123. $\frac{2}{3} - \frac{3}{8}$

Solution

$\frac{7}{24}$

124. $\frac{5}{6} - \frac{3}{4}$

125. $- \frac{11}{30} + \frac{27}{40}$

Solution

$\frac{37}{120}$

126. $- \frac{9}{20} + \frac{17}{30}$

127. $- \frac{13}{30} + \frac{25}{42}$

Solution

$\frac{17}{105}$

128. $- \frac{23}{30} + \frac{5}{48}$

129. $- \frac{39}{56} - \frac{22}{35}$

Solution

$- \frac{53}{40}$

130. $- \frac{33}{49} - \frac{18}{35}$

131. $- \frac{2}{3} - (- \frac{3}{4})$

Solution

$\frac{1}{12}$

132. $- \frac{3}{4} - (- \frac{4}{5})$

133. $1 + \frac{7}{8}$

Solution

$\frac{15}{8}$

134. $1 - \frac{3}{10}$

135. $\frac{x}{3} + \frac{1}{4}$

Solution

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

136. $\frac{y}{2} + \frac{2}{3}$

137. $\frac{y}{4} - \frac{3}{5}$

Solution

$\frac{4 y - 12}{20}$

138. $\frac{x}{5} - \frac{1}{4}$

Exercises: Mixed Practice

Instructions: For questions 139-152, simplify.

139.

a. $\frac{2}{3} + \frac{1}{6}$
b. $\frac{2}{3} \div \frac{1}{6}$

Solution

a. $\frac{5}{6}$
b. $4$

140.

a. $- \frac{2}{5} - \frac{1}{8}$
b. $- \frac{2}{5} \cdot \frac{1}{8}$

141.

a. $\frac{5 n}{6} \div \frac{8}{15}$
b. $\frac{5 n}{6} - \frac{8}{15}$

Solution

a. $\frac{25 n}{16}$
b. $\frac{25 n - 16}{30}$

142.

a. $\frac{3 a}{8} \div \frac{7}{12}$
b. $\frac{3 a}{8} - \frac{7}{12}$

143. $- \frac{3}{8} \div (- \frac{3}{10})$

Solution

$\frac{5}{4}$

144. $- \frac{5}{12} \div (- \frac{5}{9})$

145. $- \frac{3}{8} + \frac{5}{12}$

Solution

$\frac{1}{24}$

146. $- \frac{1}{8} + \frac{7}{12}$

147. $\frac{5}{6} - \frac{1}{9}$

Solution

$\frac{13}{18}$

148. $\frac{5}{9} - \frac{1}{6}$

149. $- \frac{7}{15} - \frac{y}{4}$

Solution

$\frac{- 28 - 15 y}{60}$

150. $- \frac{3}{8} - \frac{x}{11}$

151. $\frac{11}{12 a} \cdot \frac{9 a}{16}$

Solution

$\frac{33}{64}$

152. $\frac{10 y}{13} \cdot \frac{8}{15 y}$

Exercises: Use the Order of Operations to Simplify Complex Fractions

Instructions: For questions 153-174, simplify.

153. $\frac{2^{3} + 4^{2}}{{(\frac{2}{3})}^{2}}$

Solution

$54$

154. $\frac{3^{3} - 3^{2}}{{(\frac{3}{4})}^{2}}$

155. $\frac{{(\frac{3}{5})}^{2}}{{(\frac{3}{7})}^{2}}$

Solution

$\frac{49}{25}$

156. $\frac{{(\frac{3}{4})}^{2}}{{(\frac{5}{8})}^{2}}$

157. $\frac{2}{\frac{1}{3} + \frac{1}{5}}$

Solution

$\frac{15}{4}$

158. $\frac{5}{\frac{1}{4} + \frac{1}{3}}$

159. $\frac{\frac{7}{8} - \frac{2}{3}}{\frac{1}{2} + \frac{3}{8}}$

Solution

$\frac{5}{21}$

160. $\frac{\frac{3}{4} - \frac{3}{5}}{\frac{1}{4} + \frac{2}{5}}$

161. $\frac{1}{2} + \frac{2}{3} \cdot \frac{5}{12}$

Solution

$\frac{7}{9}$

162. $\frac{1}{3} + \frac{2}{5} \cdot \frac{3}{4}$

163. $1 - \frac{3}{5} \div \frac{1}{10}$

Solution

$- 5$

164. $1 - \frac{5}{6} \div \frac{1}{12}$

165. $\frac{2}{3} + \frac{1}{6} + \frac{3}{4}$

Solution

$\frac{19}{12}$

166. $\frac{2}{3} + \frac{1}{4} + \frac{3}{5}$

167. $\frac{3}{8} - \frac{1}{6} + \frac{3}{4}$

Solution

$\frac{23}{24}$

168. $\frac{2}{5} + \frac{5}{8} - \frac{3}{4}$

169. $12 (\frac{9}{20} - \frac{4}{15})$

Solution

$\frac{11}{5}$

170. $8 (\frac{15}{16} - \frac{5}{6})$

171. $\frac{\frac{5}{8} + \frac{1}{6}}{\frac{19}{24}}$

Solution

$1$

172. $\frac{\frac{1}{6} + \frac{3}{10}}{\frac{14}{30}}$

173. $(\frac{5}{9} + \frac{1}{6}) \div (\frac{2}{3} - \frac{1}{2})$

Solution

$\frac{13}{3}$

174. $(\frac{3}{4} + \frac{1}{6}) \div (\frac{5}{8} - \frac{1}{3})$

Exercises: Evaluate Variable Expressions with Fractions

Instructions: For questions 175-184, evaluate.

175. $x + (- \frac{5}{6})$ when

a. $x = \frac{1}{3}$
b. $x = - \frac{1}{6}$

Solution

a. $- \frac{1}{2}$
b. $- 1$

176. $x + (- \frac{11}{12})$ when

a. $x = \frac{11}{12}$
b. $x = \frac{3}{4}$

177. $x - \frac{2}{5}$ when

a. $x = \frac{3}{5}$
b. $x = - \frac{3}{5}$

Solution

a. $\frac{1}{5}$
b. $- 1$

178. $x - \frac{1}{3}$ when

a. $x = \frac{2}{3}$
b. $x = - \frac{2}{3}$

179. $\frac{7}{10} - w$ when

a. $w = \frac{1}{2}$
b. $w = - \frac{1}{2}$

Solution

a. $\frac{1}{5}$
b. $\frac{6}{5}$

180. $\frac{5}{12} - w$ when

a. $w = \frac{1}{4}$
b. $w = - \frac{1}{4}$

181. $2 x^{2} y^{3}$ when $x = - \frac{2}{3}$ and $y = - \frac{1}{2}$

Solution

$- \frac{1}{9}$

182. $8 u^{2} v^{3}$ when $u = - \frac{3}{4}$ and $v = - \frac{1}{2}$

183. $\frac{a + b}{a - b}$ when $a = - 3, b = 8$

Solution

$- \frac{5}{11}$

184. $\frac{r - s}{r + s}$ when $r = 10, s = - 5$

Exercises: Everyday Math

Instructions: For questions 185-186, answer the given everyday math word problems.

185. Decorating. Laronda is making covers for the throw pillows on her sofa. For each pillow cover, she needs $\frac{1}{2}$ yard of print fabric and $\frac{3}{8}$ yard of solid fabric. What is the total amount of fabric Laronda needs for each pillow cover?

Solution

$\frac{7}{8}$ yard

186. Baking. Vanessa is baking chocolate chip cookies and oatmeal cookies. She needs $\frac{1}{2}$ cup of sugar for the chocolate chip cookies and $\frac{1}{4}$ of sugar for the oatmeal cookies. How much sugar does she need altogether?

Exercises: Writing Exercises

Instructions: For questions 187-188, answer the given writing exercises.

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

Solution

Answers may vary

188. How do you find the LCD of $2$ fractions?

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Fanshawe Pre-Health Sciences Mathematics 1 Copyright © 2022 by Domenic Spilotro, MSc is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License, except where otherwise noted.

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