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4. Contents Lesson 5-1Prime Factorization Lesson 5-2Greatest Common Factor Lesson 5-3Simplifying Fractions Lesson 5-4Fractions and Decimals Lesson 5-5Fractions and Percents Lesson 5-6Percents and Decimals Lesson 5-7Least Common Multiple Lesson 5-8Comparing and Ordering Rational Numbers

5. Lesson 1 Contents Example 1Identify Numbers as Prime or Composite Example 2Identify Numbers as Prime or Composite Example 3Find the Prime Factorization Example 4Factor an Algebraic Expression

6. Example 1-1a Determine whether the number 63 is prime or composite. Answer: The number 63 has six factors: 1, 3, 7, 9, 21, and 63. So, it is composite.

7. Example 1-1b Determine whether the number 41 is prime or composite. Answer: prime

8. Example 1-2a Determine whether the number 29 is prime or composite. Answer: The number 29 has only two factors, 1 and 29, so it is prime.

9. Example 1-2b Determine whether the number 24 is prime or composite. Answer: composite

10. Example 1-3a Find the prime factorization of 100. Method 1 Use a factor tree.

11. Example 1-3a Method 2 Divide by prime numbers. Start here. Answer: The prime factorization of 100 is

12. Example 1-3b Find the prime factorization of 72. Answer:

13. Example 1-4a ALGEBRA Factor Answer:

14. Example 1-4b ALGEBRA Factor Answer:

15. End of Lesson 1

16. Lesson 2 Contents Example 1Find the GCF by Listing Factors Example 2Find the GCF Using Prime Factors Example 3Find the GCF Using Prime Factors Example 4Find the GCF of an Algebraic Expression Example 5Use the GCF to Solve a Problem

17. Example 2-1a Find the GCF of 28 and 42. First, list the factors of 28 and 42. factors of 28: 1, 2, 4, 7, 14, 28 factors of 42: 1, 2, 3, 6, 7, 14, 21, 42 Notice that 1, 2, 7, and 14 are common factors of 28 and 42. So, the GCF is 14.

18. Example 2-1a Check You can draw a Venn diagram to check your answer. Answer: 14

19. Example 2-1b Find the GCF of 18 and 45. Answer: 9

20. Example 2-2a Find the GCF of 20 and 32. Method 1 Write the prime factorization. The common prime factors are 2 and 2.

21. Example 2-2a Answer: The GCF of 20 and Method 2 Divide by prime numbers. Divide both 20 and 32 by 2. Then divide the quotients by 2. Start here.

22. Example 2-2b Find the GCF of 24 and 36. Answer: 12

23. Example 2-3a Find the GCF of 21, 42, and 63. Circle the common factors. The common prime factors are 3 and 7. Answer: The GCF is 3  7, or 21.

24. Example 2-3b Find the GCF of 24, 48, and 60. Answer: 12

25. Example 2-4a ALGEBRA Find the GCF of 12p 2 and 30p 3. Factor each expression. Answer: The GCF is 2 Circle the common factors.

26. Example 2-4b ALGEBRA Find the GCF of Answer: 7mn

27. Example 2-5a ART Searra wants to cut a 15-centimeter by 25-centimeter piece of tag board into squares for an art project. She does not want to waste any of the tag board and she wants the largest squares possible. What is the length of the side of the squares she should use? The largest length of side possible is the GCF of the dimensions of the tag board. The GCF of 15 and 25 is 5. Answer: Searra should use squares with sides measuring 5 centimeters.

28. Example 2-5b CANDY Alice is making candy baskets using chocolate hearts and lollipops. She has 32 chocolate hearts and 48 lollipops. She wants to have an equal number of chocolate hearts and lollipops in each basket. Find the greatest number of chocolate hearts and lollipops Alice can put in each basket. Answer: 16

29. End of Lesson 2

30. Lesson 3 Contents Example 1Write Fractions in Simplest Form Example 2Write Fractions in Simplest Form Example 3Use Fractions to Solve a Problem

31. Example 3-1a Write in simplest form. First, find the GCF of the numerator and denominator. factors of 12: 1, 2, 3, 4, 6, 12 factors of 45: 1, 3, 5, 9, 15, 45 The GCF of 12 and 45 is 3. Then, divide the numerator and the denominator by the GCF.

32. Example 3-1a CheckMultiply the numerator and denominator of the answer by the GCF. The result should be the original fraction. Answer: So, written in simplest form is

33. Example 3-1b Write in simplest form. Answer:

34. Example 3-2a Write in simplest form. Answer: written in simplest form is

35. Example 3-2b Write in simplest form. Answer:

36. Example 3-3a MUSIC Two notes form a perfect fifth if the simplified fraction of the frequencies of the notes equals If note Hertz and note Hertz, do they form a perfect fifth? 111 111 The slashes mean that part of the numerator and part of the denominator are both divided by the same number. For example,

37. Example 3-3a Answer: The fraction of the frequency of the notes D and G is So, the two notes do form a perfect fifth.

38. Example 3-3b MARBLES In a bag of 96 marbles, 18 of the marbles are black. Write the fraction of black marbles in simplest form. Answer:

39. End of Lesson 3

40. Lesson 4 Contents Example 1Write Fractions as Decimals Example 2Write Fractions as Decimals Example 3Write Fractions as Repeating Decimals Example 4Write Fractions as Repeating Decimals Example 5Write Decimals as Fractions

41. Example 4-1a Write as a decimal. The fraction indicates Method 1 Use paper and pencil. Division ends when the remainder is 0.

42. Example 4-1a Method 2 Use a calculator. 180.125 ENTER Answer:

43. Example 4-1b Write as a decimal. Answer: 0.4

44. Example 4-2a Write as a decimal. Method 1 Use paper and pencil. The mixed number Write as a sum. Add.

45. Example 4-2a Method 2 Use a calculator. Answer: 357.67 ENTER

46. Example 4-2b Write as a decimal. Answer: 3.625

47. Example 4-3a Write as a decimal. Method 1 Use paper and pencil.

48. Example 4-3a Method 2 Use a calculator. Answer: 1110.090909 ENTER

49. Example 4-3b Write as a decimal. Answer:

50. Example 4-4a Write as a decimal. Method 1 Use paper and pencil. Write as a sum. Write the fraction as a decimal. Add. Method 2 Use a calculator. Answer: 496.4444…6 ENTER

51. Example 4-4b Answer: Write as a decimal.

52. Example 4-5a Write 0.72 as a fraction in simplest form. The 2 is in the hundredths place. Simplify. Answer:

53. Example 4-5b Write 0.85 as a fraction in simplest form. Answer:

54. End of Lesson 4

55. Lesson 5 Contents Example 1Write Ratios as Percents Example 2Write Ratios as Percents Example 3Write a Fraction as a Percent Example 4Write a Percent as a Fraction Example 5Use Percent to Solve a Problem

56. Example 5-1a Write the ratio as a percent. Diana scored 63 goals out of 100 attempts. You can represent 63 out of 100 with a model. Answer:

57. Example 5-1b Write the ratio as a percent. Alicia sold 34 of the 100 cookies at the bake sale. Answer:

58. Example 5-2a Write the ratio as a percent. 31.9 out of 100 people bought crunchy peanut butter. Answer:

59. Example 5-2b Write the ratio as a percent. 73.4 out of 100 people preferred the chicken instead of the roast beef. Answer:

60. Example 5-3a Write as a percent..… multiply the numerator and denominator by 4.

61. Example 5-3a Answer: So,

62. Example 5-3b Write as a percent. Answer:

63. Example 5-4a Write 22% as a fraction in simplest form. Definition of percent Simplify. Answer:

64. Example 5-4b Write 84% as a fraction in simplest form. Answer:

65. Example 5-5a RECREATION The graphic shows the most popular outdoor activities according to parents with children ages 4 – 14. What fraction of parents prefer hiking as a favorite outdoor activity? In the bar graph, 15 parents chose hiking. Find the total number of responses.

66. Example 5-5a Answer: So, of the parents chose hiking.

67. Example 5-5b SPORTS Each member of the football team is supposed to get playing time in each game. By the end of the third quarter of a game, 17 of the 25 players had already been in the game. What percent of the players is this? Answer:

68. End of Lesson 5

69. Lesson 6 Contents Example 1Write Percents as Decimals Example 2Write Percents as Decimals Example 3Write Percents as Decimals Example 4Write Percents as Decimals Example 5Write Decimals as Percents Example 6Write Decimals as Percents Example 7Write Decimals as Percents Example 8Write Decimals as Percents

70. Example 6-1a POPULATION According to the Administration on Aging, about 28% of the population of the United States is 19 years of age or younger. Write 28% as a decimal. Write the fraction as a decimal. Write the percent as a fraction. Answer:

71. Example 6-1b AMUSEMENT PARK A popular amusement park reports that 17% of its visitors will return at least three times during the year. Write 17% as a decimal. Answer: 0.17

72. Example 6-2a Write 47.8% as a decimal. Multiply by 10 to remove the decimal in the numerator. Simplify. Write the fraction as a decimal. Write the percent as a fraction. Answer:

73. Example 6-2b Write 83.2% as a decimal. Answer: 0.832

74. Example 6-3a Write 95.3% as a decimal. Divide by 100. Remove the %. Answer:

75. Example 6-3b Write 38% as a decimal. Answer: 0.38

76. Example 6-4a Write as a decimal. Divide by 100. Answer: 0.082 Remove the %.

77. Example 6-4b Answer: 0.2775 Write as a decimal.

78. Example 6-5a POPULATION In 1790, about 0.05 of the population of the United States lived in an urban setting. Write 0.05 as a percent. Answer: Definition of decimal Definition of percent

79. Example 6-5b POPULATION In 2000, the population of Illinois had increased by 0.086 from 1990. Write 0.086 as a percent. Answer: 8.6%

80. Example 6-6a Write 0.121 as a percent. Definition of decimal Divide both numerator and denominator by 10. Definition of percent Answer:

81. Example 6-6b Write 0.364 as a percent. Answer: 36.4%

82. Example 6-7a Write 0.33 as a percent. Multiply by 100. Add the %. Answer:

83. Example 6-7b Write 0.52 as a percent. Answer: 52%

84. Example 6-8a Write 0.419 as a percent. Multiply by 100. Add the %. Answer:

85. Example 6-8b Write 0.869 as a percent. Answer: 86.9%

86. End of Lesson 6

87. Lesson 7 Contents Example 1Find the LCM by Listing Multiples Example 2Find the LCM Using Prime Factors Example 3Find the LCM by Using Prime Factors

88. Example 7-1a Find the LCM of 4 and 6. First, list the multiples of 4 and 6. multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36... multiples of 6: 6, 12, 18, 24, 30, 36,... Notice that 12, 24,..., are common multiples. Answer: The LCM of 4 and 6 is 12.

89. Example 7-1b Find the LCM of 8 and 12. Answer: 24

90. Example 7-2a Find the LCM of 4 and 15. Write the prime factorization. The prime factors of 4 and 15 are 2, 3, and 5. Multiply the greatest power of 2, 3, and 5. Answer: The LCM of 4 and 15 is 60.

91. Example 7-2b Find the LCM of 6 and 14. Answer: 42

92. Example 7-3a Find the LCM of 18, 24, and 48. Answer: The LCM of 18, 24, and 48 is 144. LCM:

93. Example 7-3b Find the LCM of 12, 20, and 45. Answer: 180

94. End of Lesson 7

95. Lesson 8 Contents Example 1Compare Fractions Example 2Compare Ratios Example 3Order Ratios Example 4Identify Rational Numbers

96. Example 8-1a GRADES Enrique and his younger brother both had a math test last Friday. Enrique scored 48 points out of 60 and his brother scored 55 points out of 75. Who got the better score, Enrique or his brother? Method 1Rename using the LCD. Then compare numerators. Enrique: Brother: The LCD of 60 and 75 is 300. Since

97. Example 8-1a Method 2Write each fraction as a decimal. Then compare decimals. Enrique: Brother: Answer: Enrique got the better score. Since,

98. Example 8-1b HOCKEY During the hockey season, Kyle scored 14 goals out of 24 shots taken and his teammate David scored 18 goals out of 30 shots taken. Who had the higher scoring percentage? Answer: David

99. Example 8-2a DOGS According to the Pet Food Manufacturer’s Association, 11 out of 25 people own large dogs and 13 out of 50 people own medium dogs. Do more people own large or medium dogs? Write and as decimals and compare. Answer: A greater fraction of people own large dogs than own medium dogs. Since

100. Example 8-2b SEASONS A survey showed that 21 out of 50 people stated that summer is their favorite season and 13 out of 25 people stated that fall is their favorite season. Do more people prefer summer or fall? Answer: fall

101. Example 8-3a Write and 72% as decimals and then compare all three decimals. Since 0.6 < 0.7 < 0.72, 0.6 < <72%.

102. Example 8-3a CheckYou can change 0.6 and 72% to fractions, then compare all three fractions using the LCD. Answer:

103. Example 8-3b Answer:

104. Example 8-4a MULTIPLE-CHOICE TEST ITEM Find the number that is rational. A 0.12345…B 0.123123C  D 0.102030… Read the Test Item To find the number that is rational, identify three numbers that are not rational. Solve the Test Item 0.12345..., , and 0.102030... are all irrational numbers. So, 0.123123..., a repeating decimal, is the rational number. Answer: B

105. Example 8-4b Answer: D MULTIPLE-CHOICE TEST ITEM Find the number that is not rational. A 0.121212… B 0.75 C D 0.61626364…

106. End of Lesson 8

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