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4. Contents Lesson 5-1Writing Fractions as Decimals Lesson 5-2Rational Numbers Lesson 5-3Multiplying Rational Numbers Lesson 5-4Dividing Rational Numbers Lesson 5-5Adding and Subtracting Like Fractions Lesson 5-6Least Common Multiple Lesson 5-7Adding and Subtracting Unlike Fractions Lesson 5-8Measures of Central Tendency Lesson 5-9Solving Equations with Rational Numbers Lesson 5-10Arithmetic and Geometric Sequences

5. Lesson 1 Contents Example 1Write a Fraction as a Terminating Decimal Example 2Write a Mixed Number as a Decimal Example 3Write Fractions as Repeating Decimals Example 4Compare Fractions and Decimals Example 5Compare Fractions to Solve a Problem

6. Example 1-1a Write as a decimal. Method 1 Use paper and pencil. Answer: 0.0625

7. Example 1-1a Method 2 Use a calculator. Answer: 0.0625 ENTER 1160.0625 Write as a decimal.

8. Example 1-1b Answer: 0.375 Write as a decimal.

9. Example 1-2a Write as a decimal. Write as the sum of an integer and a fraction. Add. Answer: 1.25

10. Example 1-2b Write as a decimal. Answer: 2.6

11. Example 1-3a Write as a decimal. The digits 12 repeat. Answer:

12. Example 1-3a Write as a decimal. The digits 18 repeat. Answer:

13. Example 1-3b Answer: a. Write as a decimal. b. Write as a decimal. Answer:

14. Example 1-4a Answer: On a number line, 0.7 is to the right of 0.65, so. Write the sentence.Write as a decimal. In the tenths place,. Replace  with, or = to make a true sentence.

15. Example 1-4b Answer: < Replace  with, or = to make a true sentence.

16. Example 1-5a Grades Jeremy got a score of on his first quiz and on his second quiz. Which grade was the higher score? Write the fractions as decimals and then compare the decimals. Answer: The scores were the same, 0.80. Quiz #1: Quiz #2:

17. Example 1-5b Answer: the second recipe Baking One recipe for cookies requires of a cup of butter and a second recipe for cookies requires of a cup of butter. Which recipe uses less butter?

18. End of Lesson 1

19. Lesson 2 Contents Example 1Write Mixed Numbers and Integers as Fractions Example 2Write Terminating Decimals as Fractions Example 3Write Repeating Decimals as Fractions Example 4Classify Numbers

20. Example 2-1a Write as a fraction. Write as an improper fraction. Answer:

21. Example 2-1a Write 10 as a fraction. Answer:

22. Example 2-1b Answer: a. Write as a fraction. b. Write –6 as a fraction. Answer:

23. Example 2-2a Write 0.26 as a fraction or mixed number in simplest form. 0.26 is 26 hundredths. Answer: Simplify. The GCF of 26 and 100 is 2.

24. Example 2-2a Write 2.875 as a fraction or mixed number in simplest form. 2.875 is 2 and 875 thousandths. Answer: Simplify. The GCF of 875 and 1000 is 125.

25. Example 2-2b Write each decimal as a fraction or mixed number in simplest form. a. 0.84 b. 3.625 Answer:

26. Example 2-3a Write as a fraction in simplest form. Let N represent the number. Multiply each side by 100 because two digits repeat. Subtract N from 100N to eliminate the repeating part, 0.3939...

27. Example 2-3a Divide each side by 99. Simplify. Answer: ENTER 13330.3939393939 Check

28. Example 2-3b Write as a fraction in simplest form. Answer:

29. Example 2-4a Identify all sets to which the number 15 belongs. Answer: 15 is a whole number, an integer, a natural number, and a rational number.

30. Example 2-4a Identify all sets to which the number belongs. Answer: is a rational number.

31. Example 2-4a Identify all sets to which the number 0.30303030... belongs. Answer: 0.30303030... is a nonterminating, repeating decimal. So, it is a rational number.

32. Example 2-4b Answer: integer, rational Answer: rational Identify all sets to which each number belongs. a. –7 b. c. 0.24242424...

33. End of Lesson 2

34. Lesson 3 Contents Example 1Multiply Fractions Example 2Simplify Before Multiplying Example 3Multiply Negative Fractions Example 4Multiply Mixed Numbers Example 5Multiply Algebraic Fractions Example 6Use Dimensional Analysis

35. Example 3-1a Find. Write the product in simplest form. Multiply the numerators. Multiply the denominators. Answer: Simplify. The GCF of 10 and 40 is 10.

36. Example 3-1b Find. Answer:

37. Example 3-2a Find. Write the product in simplest form. Multiply the numerators and multiply the denominators. Answer: Simplify. Divide 8 and 6 by their GCF, 2.

38. Example 3-2b Find. Write the product in simplest form. Answer:

39. Example 3-3a Find. Write the product in simplest form. Multiply the numerators and multiply the denominators. Answer: Simplify. Divide 2 and 4 by their GCF, 2.

40. Example 3-3b Find. Answer:

41. Example 3-4a Find. Write the product in simplest form.. Rename and rename Divide by the GCF, 3. Multiply. Simplify. Answer:

42. Example 3-4b Find. Write the product in simplest form. Answer:

43. Example 3-5a Find. Write the product in simplest form. The GCF of and q is q. Answer: Simplify.

44. Example 3-5b Find. Write the product in simplest form. Answer:

45. Example 3-6a Track The track at Cole’s school is mile around. If Cole runs one lap in two minutes, how far (in miles) does he run in 30 minutes? WordsDistance equals the rate multiplied by the time. VariablesLet d = distance, r = rate, and t = time. Formula d = rt

46. Example 3-6a Multiply. Simplify. Answer: Cole runs miles in 30 minutes. Write the formula. mile per 2 minutes 30 minutes Divide by the common factors and units. 15 1

47. Example 3-6a Check The problem asks for the distance. When you divide the common units, the answer is expressed in miles. So, the answer is reasonable.

48. Example 3-6b Walking Bob walksmile in 12 minutes. How far does he walk in 30 minutes? Answer:

49. End of Lesson 3

50. Lesson 4 Contents Example 1Find Multiplicative Inverses Example 2Divide by a Fraction Example 3Divide by a Whole Number Example 4Divide by a Mixed Number Example 5Divide by an Algebraic Fraction Example 6Use Dimensional Analysis

51. Example 4-1a Find the multiplicative inverse of. The product is 1. Answer: The multiplicative inverse or reciprocal of.

52. Example 4-1a Find the multiplicative inverse of. Write as an improper fraction. Answer: The reciprocal of. The product is 1.

53. Find the multiplicative inverse of each number. a. b. Example 4-1b Answer:

54. Example 4-2a Find. Write the quotient in simplest form. Multiply by the multiplicative inverse of,. Divide 5 and 10 by their GCF, 5. Answer: Simplify.

55. Example 4-2b Find. Write the quotient in simplest form. Answer:

56. Example 4-3a Find. Write the quotient in simplest form. Write 3 as. Multiply by the multiplicative inverse of,. Answer: Multiply the numerators and multiply the denominators.

57. Example 4-3b Find. Write the quotient in simplest form. Answer:

58. Example 4-4a Find. Write the quotient in simplest form. Rename the mixed numbers as improper fractions. Multiply by the multiplicative inverse of,. Divide out common factors. Answer: Simplify.

59. Example 4-4b Find. Write the quotient in simplest form. Answer:

60. Example 4-5a Find. Write the quotient in simplest form. Multiply by the multiplicative inverse of,. Divide out common factors. Answer: Simplify.

61. Example 4-5b Find. Write the quotient in simplest form. Answer:

62. Example 4-6a Travel How many gallons of gas are needed to travel miles if a car gets miles per gallon? To find how many gallons, divide.

63. Example 4-6a Write as improper fractions. Multiply by the reciprocal of. Simplify. Divide out common factors. 105 1 217

64. Example 4-6a Answer: gallons of gas are needed. Check Use dimensional analysis to examine the units. Simplify. Divide out the units. The result is expressed as gallons. This agrees with your answer of gallons of gas.

65. Example 4-6b Sewing Emily has yards of fabric. She wants to make pillows which each require yards of fabric to complete. How many pillows can Emily make? Answer: or 8 pillows

66. End of Lesson 4

67. Lesson 5 Contents Example 1Add Fractions Example 2Add Mixed Numbers Example 3Subtract Fractions Example 4Subtract Mixed Numbers Example 5Add Algebraic Fractions

68. Example 5-1a Find. Write the sum in simplest form. Estimate The denominators are the same. Add the numerators. Answer: Simplify and rename as a mixed number.

69. Example 5-1b Find. Write the sum in simplest form. Answer:

70. Example 5-2a Find. Write the sum in simplest form. Add the whole numbers and fractions separately. Add the numerators. Simplify. Answer:

71. Example 5-2b Find. Write the sum in simplest form. Answer:

72. Example 5-3a Find. Write the difference in simplest form. Estimate The denominators are the same. Subtract the numerators. Answer: Simplify.

73. Example 5-3b Find. Write the difference in simplest form. Answer:

74. Example 5-4a Evaluate. Estimate Replace r with and q with. Write the mixed numbers as improper fractions. Subtract the numerators. Answer: Simplify.

75. Example 5-4b Evaluate. Answer:

76. Example 5-5a Find. Write the sum in simplest form. The denominators are the same. Add the numerators. Add the numerators. Answer: Simplify.

77. Example 5-5b Find. Write the sum in simplest form. Answer:

78. End of Lesson 5

79. Lesson 6 Contents Example 1Find the LCM Example 2The LCM of Monomials Example 3Find the LCD Example 4Find the LCD of Algebraic Fractions Example 5Compare Fractions

80. Example 6-1a Find the LCM of 168 and 180. NumberPrime Factorization Exponential Form 168 180 The prime factors of both numbers are 2, 3, 5, and 7. Multiply the greatest powers of 2, 3, 5, and 7 appearing in either factorization. Answer: The LCM of 168 and 180 is 2520.

81. Example 6-1b Find the LCM of 144 and 96. Answer: 288

82. Example 6-2a Find the LCM of. Answer: The LCM of. Multiply the greatest power of each prime factor.

83. Find the LCM of. Example 6-2b Answer:

84. Example 6-3a Find the LCD of. Answer: The LCD of. Multiply. Write the prime factorization of 8 and 20. Highlight the greatest power of each prime factor.

85. Example 6-3b Find the LCD of. Answer: 36

86. Example 6-4a Find the LCD of. Answer: The LCD of.

87. Example 6-4b Answer: Find the LCD of

88. Example 6-5a Replacewith, or = to make a true statement. The LCD of the fractions is. Rewrite the fractions using the LCD and then compare the numerators. Multiply the fraction by to make the denominator 105.

89. Example 6-5a Answer: Since, then. is to the left of on the number line.

90. Example 6-5b Replacewith, or = to make a true statement. Answer: <

91. End of Lesson 6

92. Lesson 7 Contents Example 1Add Unlike Fractions Example 2Add Fractions Example 3Add Mixed Numbers Example 4Subtract Fractions Example 5Subtract Mixed Numbers Example 6Use Fractions to Solve a Problem

93. Example 7-1a Use as the common denominator. Find. Rename each fraction with the common denominator. Add the numerators.Answer:

94. Example 7-1b Find. Answer:

95. Example 7-2a Find. Estimate Rename each fraction with the LCD. Add the numerators. Simplify. Answer: The LCD is 30.

96. Example 7-2b Find. Answer:

97. Example 7-3a Find. Write in simplest form. Write the mixed numbers as improper fractions. Simplify. Rename fractions using the LCD, 24.

98. Example 7-3a Simplify. Answer: Add the numerators.

99. Example 7-3b Find. Answer:

100. Example 7-4a Find. The LCD is 16. Rename using the LCD. Answer: Subtract the numerators.

101. Example 7-4b Find. Answer:

102. Example 7-5a Find. Write the mixed numbers as improper fractions. Rename the fractions using the LCD. Simplify. Answer: Subtract.

103. Example 7-5b Find. Answer:

104. Example 7-6a ExploreYou know the distances Juyong jogged each day. PlanAdd the daily distances together to find the total distance. Estimate your answer. Jogging Juyong jogged three days this week. She jogged miles, miles, and miles. How far did she jog altogether?

105. Example 7-6a Solve Rename the fractions with LCD, 20. Add the like fractions. Examine Since is close to, the answer is reasonable. Answer: Juyong jogged miles.

106. Example 7-6b Gardening Howard’s tomato plants grew inches during the first week after sprouting, inches during the second week, and inches the third week. Find the total growth during the first three weeks after sprouting. Answer:inches

107. End of Lesson 7

108. Lesson 8 Contents Example 1Find the Mean, Median, and Mode Example 2Use a Line Plot Example 3Find Extreme Values that Affect the Mean Example 4Use, Mean, Median, and Mode to Analyze Data Example 5Work Backward

109. Example 8-1a Movies The revenue of the 10 highest grossing movies as of June 2000 are given in the table. Find the mean, median, and mode of the revenues. Answer: The mean revenue is $379.8 million. Top 10 Movie Revenues (millions of $) 601330 461313 431309 400306 357290

110. Example 8-1a To find the median, order the numbers from least to greatest. Answer: The median revenue is $343.5 million. 290, 306, 309, 313, 330, 357, 400, 431, 461, 601 Answer: There is no mode because each number in the set occurs once. There is an even number of items. Find the mean of the two middle numbers.

111. Example 8-1b Test Scores The test scores for a class of nine students are 85, 93, 78, 99, 62, 83, 90, 75, 85. Find the mean, median, and mode of the test scores. Answer: mean, 83.3 ; median, 85 ; mode, 85

112. Example 8-2a Olympics The line plot below shows the number of gold medals earned by each country that participated in the 1998 Winter Olympic games in Nagano, Japan. Find the mean, median, and mode for the gold medals won. Answer: The mean is 2.875.

113. Example 8-2a There are 24 numbers. The median number is the average of the 12 th and 13 th numbers. Answer: The median is 2. The number 0 occurs most frequently in the set of data. Answer: The mode is 0.

114. Example 8-2b Families A survey of school-age children shows the family sizes displayed in the line plot below. Find the mean, median, and mode. Answer: mean, 4.3 ; median, 5 ; mode, 5

115. Example 8-3a Quiz scores The quiz scores for a math class are 8, 7, 6, 10, 8, 8, 9, 8, 7, 9, 8, 0, and 10. Identify an extreme value and describe how it affects the mean. The data value 0 appears to be an extreme value. Calculate the mean with and without the extreme value to find how it affects the mean. mean with extreme valuemean without extreme value Answer: The extreme value 0 decreases the mean by 8.2 – 7.5 or about 0.7.

116. Example 8-3b Birth Weight The birth weights of ten newborn babies are given in pounds: 7.3, 8.4, 9.1, 7.9, 8.8, 6.5, 7.9, 4.1, 8.0, 7.5. Identify an extreme value and describe how it affects the mean. Answer: 4.1 ; it decreases the mean by about 0.4.

117. Bob’s Books The Reading Place 12901400 1450 14001550 1600 36502000 Example 8-4a The table shows the monthly salaries of the employees at two bookstores. Find the mean, median, and mode for each set of data. Based on the averages, which bookstore pays its employees better? mean: Bob’s Books median: 1290, 1400, 1400, 1600, 3650 median mode: $1400

118. Bob’s Books The Reading Place 12901400 1450 14001550 1600 36502000 Example 8-4a mean: The Reading Place mode: none median: 1400, 1450, 1550, 1600, 2000 median Answer: The $3650 salary at Bob’s Books is an extreme value that increases the mean salary. The employees at The Reading Place are generally better paid as shown by the median.

119. MenWomen 51 26 34 82 121 48 Example 8-4b The number of hours spent exercising each week by men and women are given in the table. Find the mean, median, and mode for each set of data. Based on the averages, which gender exercises more? Answer: Men: mean, 5.7 ; median, 4.5 ; mode, none Women: mean, 3.7 ; median, 3 ; mode, 1 Men seem to exercise more.

120. Example 8-5a Grid–In Test Item Jenny’s bowling average is 146. Today she bowled 138, 140, and 145. What does she need to score on her fourth game to maintain her average? Read the Test Item Find the sum of the first three games. Then write an equation to find the score needed on the fourth game.

121. Example 8-5a Solve the Test Item Step 1Find the sum of the first three games x. mean of the first three scores Multiply each side by 3. Simplify. sum of first three scores

122. Example 8-5a Step 2 Find the fourth score, x. mean Write an equation. Multiply each side by 4 and simplify. Subtract 423 from each side and simplify. Substitution.

123. Example 8-5a Answer: Jenny needs to score at least 161 to maintain her average of 146.

124. Example 8-5b Emily scored 73, 82, and 85 on her first three math tests. What score does Emily need on the fourth test to give her an average of 82 for the four tests? Answer: 88

125. End of Lesson 8

126. Lesson 9 Contents Example 1Solve by Using Addition Example 2Solve by Using Subtraction Example 3Solve by Using Division Example 4Solve by Using Multiplication

127. Example 9-1a Solve. Check your solution. Write the equation. Add to each side. Simplify. Rename the fractions using the LCD and add.

128. Example 9-1a Answer: Simplify. Check Write the original equation. Simplify. Replace y with.

129. Example 9-1b Solve. Check your solution. Answer:

130. Example 9-2a Solve. Check your solution. Write the equation. Subtract 8.6 from each side. Simplify. Check: Write the original equation. Simplify. Answer: Replace m with 2.6.

131. Example 9-2b Solve. Check your solution. Answer: 6.1

132. Divide each side by 9. Example 9-3a Solve. Check your solution. Write the equation. Simplify. Answer: Check: Write the original equation. Replace a with 0.4.

133. Example 9-3b Solve. Check your solution. Answer: 0.8

134. Example 9-4a Solve. Check your solution. Write the equation. Multiply each side by 6. Simplify. Answer:

135. Example 9-4a Check: Write the original equation. Simplify. Replace x with 48.

136. Example 9-4a Solve. Check your solution. Write the equation. Simplify. Check the solution. Answer: Multiply each side by.

137. Example 9-4a Check: Write the original equation. Simplify. Replace t with –10.

138. Example 9-4b a. Solve. Check your solution. b. Solve. Check your solution. Answer: 45 Answer: 16

139. End of Lesson 9

140. Lesson 10 Contents Example 1Identify an Arithmetic Sequence Example 2Identify an Arithmetic Sequence Example 3Identify Geometric Sequences

141. Example 10-1a State whether the sequence –5, –1, 3, 7, 11,... is arithmetic. If it is, state the common difference and write the next three terms. –5, –1, 3, 7, 11 Notice that, and so on. Answer: The terms have a common difference of, so the sequence is arithmetic. Continue the pattern to find the next three terms.

142. Example 10-1a 11, Answer: The next three terms of the sequence are 15, 19, and 23. 15, 19, 23

143. Example 10-1b State whether the sequence 12, 7, 2, –3, –8,... is arithmetic. If it is, state the common difference and write the next three terms. Answer: arithmetic; –5 ; –13, –18, –23

144. Example 10-2a State whether the sequence 0, 2, 6, 12, 20,... is arithmetic. If it is, state the common difference and write the next three terms. 0, 2,2, 6,6, 12, 20 The terms do not have a common difference. Answer: The sequence is not arithmetic. However, if the pattern continues, the next three differences will be 10, 12, and 14.

145. Example 10-2a 20, 30, 42, 56 The next three terms are 30, 42, and 56.

146. Example 10-2b State whether the sequence 20, 17, 11, 2, –10,... is arithmetic. If it is, state the common difference and write the next three terms. Answer: not arithmetic

147. Example 10-3a State whether the sequence 2, 4, 4, 8, 8, 16, 16... is geometric. If it is, state the common ratio and write the next three terms. 2, 4,4, 4,4, 8,8, 8,8, 16, 16 Answer: There is no common ratio. The sequence is not geometric.

148. Example 10-3a 27, –9, 3,3, –1, State whether the sequence 27, –9, 3, –1,,... is geometric. If it is, state the common ratio and write the next three terms. Answer: The common ratio is, so the sequence is geometric. Continue the pattern to find the next three terms.

149. Example 10-3a,,, The next three terms are,, and.

150. a. State whether the sequence 2, 8, 32, 128, 512,... is geometric. If it is, state the common ratio and write the next three terms. b. State whether the sequence 100, 20, 4,,... is geometric. If it is, state the common ratio and write the next three terms. Example 10-3b Answer: geometric; 4 ; 2048, 8192, 32,768 Answer: geometric; ;,,

151. End of Lesson 10

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