2.6 Chapter Review

Izabela Mazur

CHAPTER 2 Operations with Rational Numbers and Introduction to Real Numbers

2.6 Chapter Review

Review Exercises

Find Equivalent Fractions

In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.

1. $\dfrac{1}{4}$	2. $\dfrac{1}{3}$
3. $\dfrac{5}{6}$	4. $\dfrac{2}{7}$

Simplify Fractions

In the following exercises, simplify.

5. $\dfrac{7}{21}$	6. $\dfrac{8}{24}$
7. $\dfrac{15}{20}$	8. $\dfrac{12}{18}$
9. $-\dfrac{168}{192}$	10. $-\dfrac{140}{224}$
11. $\dfrac{11x}{11y}$	12. $\dfrac{15a}{15b}$

Multiply Fractions

In the following exercises, multiply.

13. $\dfrac{2}{5}\cdot \dfrac{1}{3}$	14. $\dfrac{1}{2}\cdot \dfrac{3}{8}$
15. $\dfrac{7}{12}(-\dfrac{8}{21})$	16. $\dfrac{5}{12}(-\dfrac{8}{15})$
17. $-28p(-\dfrac{1}{4})$	18. $-51q(-\dfrac{1}{3})$
19. $\dfrac{14}{5}(-15)$	20. $-1(-\dfrac{3}{8})$

Divide Fractions

In the following exercises, divide.

21. $\dfrac{1}{2}$ ÷ $\dfrac{1}{4}$	22. $\dfrac{1}{2}$ ÷ $\dfrac{1}{8}$
23. $-\dfrac{4}{5}$ ÷ $\dfrac{4}{7}$	24. $-\dfrac{3}{4}$ ÷ $\dfrac{3}{5}$
25. $\dfrac{5}{8}$ ÷ $\dfrac{a}{10}$	26. $\dfrac{5}{6}$ ÷ $\dfrac{c}{15}$
27. $\dfrac{7p}{12}$ ÷ $\dfrac{21p}{8}$	28. $\dfrac{5q}{12}$ ÷ $\dfrac{15q}{8}$
29. $\dfrac{2}{5}$ ÷ $\left(-10\right)$	30. $-18$ ÷ $-\left(\dfrac{9}{2}\right)$

In the following exercises, simplify.

31. $\dfrac{\dfrac{2}{3}}{\dfrac{8}{9}}$	32. $\dfrac{\dfrac{4}{5}}{\dfrac{8}{15}}$
33. $\dfrac{-\dfrac{9}{10}}{3}$	34. $\dfrac{2}{\dfrac{5}{8}}$
35. $\dfrac{\dfrac{r}{5}}{\dfrac{s}{3}}$	36. $\dfrac{-\dfrac{x}{6}}{-\dfrac{8}{9}}$

Simplify Expressions Written with a Fraction Bar

In the following exercises, simplify.

37. $\dfrac{4+11}{8}$	38. $\dfrac{9+3}{7}$
39. $\dfrac{30}{7-12}$	40. $\dfrac{15}{4-9}$
41. $\dfrac{22-14}{19-13}$	42. $\dfrac{15+9}{18+12}$
43. $\dfrac{5\cdot 8}{-10}$	44. $\dfrac{3\cdot 4}{-24}$
45. $\dfrac{15\cdot 5-{5}^{2}}{2\cdot 10}$	46. $\dfrac{12 \cdot 9-{3}^{2}}{3\cdot 18}$
47. $\dfrac{2+4\left(3\right)}{-3-{2}^{2}}$	48. $\dfrac{7+3\left(5\right)}{-2-{3}^{2}}$

Translate Phrases to Expressions with Fractions

In the following exercises, translate each English phrase into an algebraic expression.

49. the quotient of c and the sum of d and 9.

50. the quotient of the difference of h and k, and $-5$ .

Add and Subtract Fractions with a Common Denominator

In the following exercises, add.

51. $\dfrac{4}{9}+\dfrac{1}{9}$	52. $\dfrac{2}{9}+\dfrac{5}{9}$
53. $\dfrac{y}{3}+\dfrac{2}{3}$	54. $\dfrac{7}{p}+\dfrac{9}{p}$
55. $-\dfrac{1}{8}+\left(-\dfrac{3}{8}\right)$	56. $-\dfrac{1}{8}+\left(-\dfrac{5}{8}\right)$

In the following exercises, subtract.

57. $\dfrac{4}{5}-\dfrac{1}{5}$	58. $\dfrac{4}{5}-\dfrac{3}{5}$
59. $\dfrac{y}{17}-\dfrac{9}{17}$	60. $\dfrac{x}{19}-\dfrac{8}{19}$
61. $-\dfrac{8}{d}-\dfrac{3}{d}$	62. $-\dfrac{7}{c}-\dfrac{7}{c}$

Add or Subtract Fractions with Different Denominators

In the following exercises, add or subtract.

63. $\dfrac{1}{3}+\dfrac{1}{5}$	64. $\dfrac{1}{4}+\dfrac{1}{5}$
65. $\dfrac{1}{5}-\left(-\dfrac{1}{10}\right)$	66. $\dfrac{1}{2}-\left(-\dfrac{1}{6}\right)$
67. $\dfrac{2}{3}+\dfrac{3}{4}$	68. $\dfrac{3}{4}+\dfrac{2}{5}$
69. $\dfrac{11}{12}-\dfrac{3}{8}$	70. $\dfrac{5}{8}-\dfrac{7}{12}$
71. $-\dfrac{9}{16}-\left(-\dfrac{4}{5}\right)$	72. $-\dfrac{7}{20}-\left(-\dfrac{5}{8}\right)$
73. $1+\dfrac{5}{6}$	74. $1-\dfrac{5}{9}$

Use the Order of Operations to Simplify Complex Fractions

In the following exercises, simplify.

75. $\dfrac{{\left(\dfrac{1}{5}\right)}^{2}}{2+{3}^{2}}$	76. $\dfrac{{\left(\dfrac{1}{3}\right)}^{2}}{5+{2}^{2}}$
77. $\dfrac{\dfrac{2}{3}+\dfrac{1}{2}}{\dfrac{3}{4}-\dfrac{2}{3}}$	78. $\dfrac{\dfrac{3}{4}+\dfrac{1}{2}}{\dfrac{5}{6}-\dfrac{2}{3}}$

Evaluate Variable Expressions with Fractions

In the following exercises, evaluate.

79. $x+\dfrac{1}{2}$ when a) $x=-\dfrac{1}{8}$ b) $x=-\dfrac{1}{2}$	80. $x+\dfrac{2}{3}$ when a) $x=-\dfrac{1}{6}$ b) $x=-\dfrac{5}{3}$
81. $4{p}^{2}q$ when $p=-\dfrac{1}{2}$ and $q=\dfrac{5}{9}$	82. $5{m}^{2}n$ when $m=-\dfrac{2}{5}$ and $n=\dfrac{1}{3}$
83. $\dfrac{u+v}{w}$ when $u=-4,v=-8,w=2$	84. $\dfrac{m+n}{p}$ when $m=-6,n=-2,p=4$

Name and Write Decimals

In the following exercises, write as a decimal.

85. Eight and three hundredths	86. Nine and seven hundredths
87. One thousandth	88. Nine thousandths

In the following exercises, name each decimal.

89. 7.8	90. 5.01
91. 0.005	92. 0.381

Round Decimals

In the following exercises, round each number to the nearest a) hundredth b) tenth c) whole number.

93. 5.7932	94. 3.6284
95. 12.4768	96. 25.8449

Add and Subtract Decimals

In the following exercises, add or subtract.

97. $18.37+9.36$	98. $256.37-85.49$
99. $15.35-20.88$	100. $37.5+12.23$
101. $-4.2+\left(-9.3\right)$	102. $-8.6+\left(-8.6\right)$
103. $100-64.2$	104. $100-65.83$
105. $2.51+40$	106. $9.38+60$

Multiply and Divide Decimals

In the following exercises, multiply.

107. $\left(0.3\right)\left(0.4\right)$	108. $\left(0.6\right)\left(0.7\right)$
109. $\left(8.52\right)\left(3.14\right)$	110. $\left(5.32\right)\left(4.86\right)$
111. $\left(0.09\right)\left(24.78\right)$	112. $\left(0.04\right)\left(36.89\right)$

In the following exercises, divide.

113. 0.15 ÷ 5	114. 0.27 ÷ 3
115. $8.49 ÷ 12	116. $16.99 ÷ 9
117. 12 ÷ 0.08	118. 5 ÷ 0.04

Convert Decimals and Fractions

In the following exercises, write each decimal as a fraction.

119. 0.08	120. 0.17
121. 0.425	122. 0.184
123. 1.75	124. 0.035

In the following exercises, convert each fraction to a decimal.

125. $\dfrac{2}{5}$	126. $\dfrac{4}{5}$
127. $-\dfrac{3}{8}$	128. $-\dfrac{5}{8}$
129. $\dfrac{5}{9}$	130. $\dfrac{2}{9}$
131. $\dfrac{1}{2}+6.5$	132. $\dfrac{1}{4}+10.75$

Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

In the following exercises, write as the ratio of two integers.

133. a) 9 b) 8.47

134. a) $-15$ b) 3.591

In the following exercises, list the a) rational numbers, b) irrational numbers.

135. $0.84,0.79132\text{…},1.\stackrel{\text{-}}{3}$

136. $2.3\stackrel{\text{-}}{8},0.572,4.93814\text{…}$

In the following exercises, list the a) whole numbers, b) integers, c) rational numbers, d) irrational numbers, e) real numbers for each set of numbers.

137. $-4,0,\dfrac{5}{6} ,17 ,5.2537\text{…}$

138. $-2, 0.\stackrel{\text{-}}{36},\dfrac{13}{3},6.9152\text{…},10\dfrac{1}{2}$

Locate Fractions on the Number Line

In the following exercises, locate the numbers on a number line.

139. $\dfrac{2}{3},\dfrac{5}{4},\dfrac{12}{5}$	140. $\dfrac{1}{3},\dfrac{7}{4},\dfrac{13}{5}$
141. $2\dfrac{1}{3},-2\dfrac{1}{3}$	142. $1\dfrac{3}{5},-1\dfrac{3}{5}$

In the following exercises, order each of the following pairs of numbers, using < or >.

143. $-1$ ___ $-\dfrac{1}{8}$	144. $-3\dfrac{1}{4}$ ___ $-4$
145. $-\dfrac{7}{9}$ ___ $-\dfrac{4}{9}$	146. $-2$ ___ $-\dfrac{19}{8}$

Locate Decimals on the Number Line

In the following exercises, locate on the number line.

147. 0.3	148. -0.2
149. -2.5	150. 2.7

In the following exercises, order each of the following pairs of numbers, using < or >.

151. 0.9 ___ 0.6	152. 0.7 ___ 0.8
153. -0.6 ___ -0.59	154. -0.27 ___ -0.3

Use the Commutative and Associative Properties

In the following exercises, use the Associative Property to simplify.

155. $-12\left(4m\right)$	156. $30\left(\dfrac{5}{6}q\right)$
157. $\left(a+16\right)+31$	158. $\left(c+0.2\right)+0.7$

In the following exercises, simplify.

159. $6y+37+\left(-6y\right)$	160. $\dfrac{1}{4}+\dfrac{11}{15}+\left(-\dfrac{1}{4}\right)$
161. $\dfrac{14}{11}\cdot \dfrac{35}{9}\cdot \dfrac{14}{11}$	162. $-18\cdot 15 \cdot \dfrac{2}{9}$
163. $\left(\dfrac{7}{12}+\dfrac{4}{5}\right)+\dfrac{1}{5}$	164. $\left(3.98d+0.75d\right)+1.25d$
165. $11x+8y+16x+15y$	166. $52m+\left(-20n\right)+\left(-18m\right)+\left(-5n\right)$

Use the Identity and Inverse Properties of Addition and Multiplication

In the following exercises, find the additive inverse of each number.

167.

a) $\dfrac{1}{3}$
b) $5.1$
c) $-14$
d) $-\dfrac{8}{5}$

168.
a) $-\dfrac{7}{8}$
b) $-0.03$
c) $17$
d) $\dfrac{12}{5}$

In the following exercises, find the multiplicative inverse of each number.

169. a) 10 b) $-\dfrac{4}{9}$ c) 0.6

170. a) $-\dfrac{9}{2}$ b) $-7$ c) 2.1

Use the Properties of Zero

In the following exercises, simplify.

171. $83\cdot 0$	172. $\dfrac{0}{9}$
173. $\dfrac{5}{0}$	174. $0$ \div $\dfrac{2}{3}$

In the following exercises, simplify.

175. $43+39+\left(-43\right)$	176. $\left(n+6.75\right)+0.25$
177. $\dfrac{5}{13}\cdot 57 \cdot \dfrac{13}{5}$	178. $\dfrac{1}{6}\cdot17\cdot12$
179. $\dfrac{2}{3}\cdot 28 \cdot \dfrac{3}{7}$	180. $9\left(6x-11\right)+15$

Simplify Expressions Using the Distributive Property

In the following exercises, simplify using the Distributive Property.

181. $7\left(x+9\right)$	182. $9\left(u-4\right)$
183. $-3\left(6m-1\right)$	184. $-8\left(-7a-12\right)$
185. $\dfrac{1}{3}\left(15n-6\right)$	186. $\left(y+10\right)\cdot p$
187. $\left(a-4\right)-\left(6a+9\right)$	188. $4\left(x+3\right)-8\left(x-7\right)$

Review Exercise Answers

1. $\dfrac{2}{8},\dfrac{3}{12},\dfrac{4}{16}$ answers may vary	3. $\dfrac{10}{12},\dfrac{15}{18},\dfrac{20}{24}$ answers may vary	5. $\dfrac{1}{3}$
7. $\dfrac{3}{4}$	9. $-\dfrac{7}{8}$	11. $\dfrac{x}{y}$
13. $\dfrac{2}{15}$	15. $-\dfrac{2}{9}$	17. $7p$
19. $-42$	21. 2	23. $-\dfrac{7}{5}$
25. $\dfrac{25}{4a}$	27. $\dfrac{2}{9}$	29. $-\dfrac{1}{25}$
31. $\dfrac{3}{4}$	33. $-\dfrac{3}{10}$	35. $\dfrac{3r}{5s}$
37. $\dfrac{15}{8}$	39. $-6$	41. $\dfrac{4}{3}$
43. $-4$	45. $\dfrac{5}{21}$	47. $-2$
49. $\dfrac{c}{d+9}$	51. $\dfrac{5}{9}$	53. $\dfrac{y+2}{3}$
55. $-\dfrac{1}{2}$	57. $\dfrac{3}{5}$	59. $\dfrac{y-9}{17}$
61. $-\dfrac{11}{d}$	63. $\dfrac{8}{15}$	65. $\dfrac{3}{10}$
67. $\dfrac{17}{12}$	69. $\dfrac{13}{24}$	71. $\dfrac{19}{80}$
73. $\dfrac{11}{6}$	75. $\dfrac{1}{275}$	77. 14
79. a) $\dfrac{3}{8}$ b) 0	81. $\dfrac{5}{9}$	83. $-6$
85. 8.03	87. 0.001	89. seven and eight tenths
91. five thousandths	93. a) 5.79 b) 5.8 c) 6	95. a) 12.48 b) 12.5 c) 12
97. 27.73	99. −5.53	101. −13.5
103. 35.8	105. 42.51	107. 0.12
109. 26.7528	111. 2.2302	113. 0.03
115. $0.71	117. 150	119. $\dfrac{2}{25}$
121. $\dfrac{17}{40}$	123. $\dfrac{7}{4}$	125. 0.4
127. $-0.375$	129. $0.\stackrel{\text{-}}{5}$	131. 7
133. a) $\dfrac{9}{1}$ b) $\dfrac{847}{100}$	135. a) $0.84,1.\stackrel{\text{-}}{3}$ b) $0.79132\text{…}$ ,	137. a) 0, 17 b) -4,0,17 c) $-4,0,\dfrac{5}{6},17$ d) $5.2537\text{…}$ e) $-4,0,17,\dfrac{5}{6}, 5.2537\text{…}$
139.	141.	143. <
145. >	147.	149.

151. >	153. >	155. $-48m$
157. $a+47$	159. 37	161. $\dfrac{35}{9}$
163. $1\dfrac{7}{12}$	165. $27x+23y$	167. a) $-\dfrac{1}{3}$ b) $-5.1$ c) 14 d) $\dfrac{8}{5}$
169. a) $\dfrac{1}{10}$ b) $-\dfrac{9}{4}$ c) $\dfrac{5}{3}$	171. 0	173. undefined
175. 39	177. 57	179. 8
181. $7x+63$	183. $-18m+3$	185. $5n-2$
187. $-5a-13$

Practice Test

1. Convert 1.85 to a fraction and simplify.

2. Locate $\dfrac{2}{3},-1.5$ , and $\dfrac{9}{4}$ on a number line.

In the following exercises, simplify each expression.

3. $4+10\left(3+9\right)-{5}^{2}$	4. $-85+42$
5. $-19-25$	6. ${\left(-2\right)}^{4}$
7. $-5\left(-9\right)$ ÷ $15$	8. $\dfrac{3}{8}\cdot \dfrac{11}{12}$
9. $\dfrac{4}{5}$ ÷ $\dfrac{9}{20}$	10. $\dfrac{12+3\cdot 5}{15-6}$
11. $\dfrac{m}{7}+\dfrac{10}{7}$	12. $\dfrac{7}{12}-\dfrac{3}{8}$
13. $-5.8+\left(-4.7\right)$	14. $100-64.25$
15. $\left(0.07\right)\left(31.95\right)$	16. 9 ÷ 0.05
17. $-14\left(\dfrac{5}{7}p\right)$	18. $\left(u+8\right)-9$
19. $6x+\left(-4y\right)+9x+8y$	20. $\dfrac{0}{23}$
21. $\dfrac{75}{0}$	22. $-2\left(13q-5\right)$

Practice Test Answers

1. $\dfrac{37}{20}$		3. 99
4. $-43$	5. $-44$	6. $16$
7. $3$	8. $\dfrac{11}{32}$	9. $\dfrac{16}{9}$
10 $3$	11. $\dfrac{m+10}{7}$	12. $\dfrac{5}{24}$
13. $-10.5$	14. 35.75	15. 2.2365
16. $180$	17. $-10p$	18. $u -1$
19. $15x+4y$	20. 0	21. undefined
22. $-26q + 10$