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4. Contents Lesson 2-1Fractions and Decimals Lesson 2-2Comparing and Ordering Rational Numbers Lesson 2-3Multiplying Rational Numbers Lesson 2-4Dividing Rational Numbers Lesson 2-5Adding and Subtracting Like Fractions Lesson 2-6Adding and Subtracting Unlike Fractions Lesson 2-7Solving Equations with Rational Numbers Lesson 2-8Powers and Exponents Lesson 2-9Scientific Notation

5. Lesson 1 Contents Example 1Write a Fraction as a Decimal Example 2Write a Mixed Number as a Decimal Example 3Round a Repeating Decimal Example 4Write a Terminating Decimal as a Fraction Example 5Write a Repeating Decimal as a Fraction

6. Example 1-1a Write as a decimal. Add a decimal point and zeros to the dividend: Division ends when the remainder is 0. 0.1875

7. Example 1-1b You can also use a calculator. Answer: The fraction can be written as 0.1875. 3 16 0.1875 ENTER

8. Example 1-1c Answer: 0.0625 Write as a decimal.

9. Example 1-2a Write as a decimal. means To change to a decimal, divide 2 by 11. The remainder after each step is 2 or 9. 0.1818 The three dots means the one and eight keep repeating. …

10. Example 1-2b You can also use a calculator. Answer: The mixed number can be written as 3 + 0.1818… or 3.1818…. 2 11 0.1818181818 ENTER

11. Example 1-2c Answer: 5.1111… Write as a decimal.

12. Example 1-3a AGRICULTURE A Florida farmer lost the fruit on 8 of 15 orange trees because of unexpected freezing temperatures. Find the fraction of the orange trees that did not produce fruit. Express your answer as a decimal to the nearest thousandth. To find his fraction of trees that did not produce fruit, divide the number of lost trees, 8, by the total number of trees, 15. Look at the digit to the right of the thousandths place. Round down since 3 < 5. Answer: The fraction of fruit trees that did not produce fruit was 0.533. 8 15 0.5333333333 ENTER

13. Example 1-3b SCHOOL In Mrs. Townley’s eighth grade science class, 4 out of 22 students did not turn in their homework. Find the fraction of the students who did not turn in their homework. Express your answer as a decimal to the nearest thousandth. Answer: 0.182

14. Example 1-4a Write 0.32 as a fraction. Answer: The decimal 0.32 can be written as 0.32 is 32 hundredths. Simplify. Divide by the greatest common factor of 32 and 100, 4.

15. Example 1-4b Answer: Write 0.16 as a fraction.

16. Example 1-5a ALGEBRA Write as a mixed number. Multiply N by 10 because 1 digit repeats. Let Then Subtract to eliminate the repeating part, 0.777….

17. Example 1-5b Answer: The decimal can be written as Divide each side by 9. Simplify.

18. Example 1-5c Answer: ALGEBRA Write as a mixed number.

19. End of Lesson 1

20. Lesson 2 Contents Example 1Compare Rational Numbers Example 2Compare Negative Rational Numbers Example 3Order Rational Numbers

21. Example 2-1a Method 1 Write as fractions with the same denominator. For the least common denominator is 91. Replace with, or to make a true sentence.

22. Example 2-1a Method 2 Write as decimals. Write as decimals. Use a calculator. Answer: 8 13 0.61538… ENTER 3 7 0.42857… ENTER Since

23. Example 2-1b Answer: Replace with, or to make a true sentence.

24. Example 2-2a Write as a decimal.. Since Replace with, or to make a true sentence.

25. Example 2-2b Check Use a number line to check the answer. The answer is correct. Answer:

26. Example 2-2c Answer: Replace with, or to make a true sentence.

27. Example 2-3a CHEMISTRY The values for the approximate densities of various substances are shown in the table. Order the densities from least to greatest. Source: CRC Handbook of Chemistry and Physics rubber neoprene 2.32pyrex glass 1.1nylon marble fused silica crown glass brick 1.87beryllium 2.7aluminum Density (g/cm 3 ) Substance

28. Example 2-3b Answer: From the least to the greatest, the densities are So, nylon is the least dense, and aluminum is the most dense. Write each fraction as a decimal.

29. Example 2-3c Source: www.coasterglobe.com AMUSEMENT PARKS The ride times for five amusement park attractions are shown in the table below. Order the lengths from least to greatest. Answer: 1.8Mind Eraser Serial Thriller X–Flight 1.5Double Loop Big Dipper Ride Time (min) Coaster

30. End of Lesson 2

31. Lesson 3 Contents Example 1Multiply Fractions Example 2Multiply Negative Fractions Example 3Multiply Mixed Numbers Example 4Evaluate an Algebraic Expression Example 5Use Dimensional Analysis

32. Example 3-1a Divide 3 and 9 by their GCF, 3. Simplify. Answer: Multiply the denominators. Multiply the numerators. Find Write in simplest form. 1 3

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

34. Example 3-2a Divide –3 and 12 by their GCF, 3. The factors have different signs, so the product is negative. Multiply the denominators. Multiply the numerators. Find Write in simplest form. Answer: –1 4

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

36. Example 3-3a Divide 16 and 4 by their GCF, 4. Simplify. Multiply the denominators. Multiply the numerators. Find Write in simplest form. 4 1

37. Example 3-3b Answer: Check is about 3, and is about 2. Therefore, is about or 6. The answer is reasonable.

38. Example 3-3c Answer: Find Write in simplest form.

39. Example 3-4a Divide out common factors. Simplify. Answer: Replace p with, r with, and q with 1 1 1 ALGEBRA Evaluate prq if

40. Example 3-4b Answer: ALGEBRA Evaluate prq if

41. Example 3-5a WATER USE Low–flow showerheads use gallons of water per minute. If family members shower a total of hours per week, how much water does the family use for showers each week? Water used equals the time multiplied by the water flow rate. Words Variables Equation w = t  r w = hours  gallons per minute

42. Example 3-5b Write the equation. Divide by common factors and units. 70 1

43. Example 3-5c Check The problem asks for the amount of water used a week. When you divide the common units, the answer is expressed in gallons. So, the answer is reasonable. Answer: If the family showers hours per week at a rate of gallons per minute, they will use about 350 gallons of water.

44. Example 3-5d Answer: about 897 gallons WATER USE Some showerheads use gallons of water per minute. If family members shower a total of hours per week, how much water does the family use for showers each week?

45. End of Lesson 3

46. Lesson 4 Contents Example 1Find a Multiplicative Inverse Example 2Divide Fractions Example 3Divide by a Whole Number Example 4Divide by Negative Fractions Example 5Divide Mixed Numbers Example 6Use Dimensional Analysis

47. Example 4-1a Write the multiplicative inverse of Answer: Since the multiplicative inverse of Write as an improper fraction.

48. Example 4-1b Answer: Write the multiplicative inverse of

49. Example 4-2a Answer: Divide 5 and 10 by their GCF, 5. Simplify. Find Write in simplest form. Multiply by the multiplicative inverse of which is 1 2

50. Example 4-2b Answer: Find Write in simplest form.

51. Example 4-3a Answer: Multiply by the multiplicative inverse of 12, which is Divide 6 and 12 by their GCF, 6. Simplify. Find Write in simplest form. Write 12 as 1 2

52. Example 4-3b Answer: Find Write in simplest form.

53. Example 4-4a Answer: Divide 2 and 8 by their GCF, 2. The fractions have different signs, so the quotient is negative. Find Write in simplest form. Multiply by the multiplicative inverse of which is. 1 4

54. Example 4-4b Answer: Find Write in simplest form.

55. Example 4-5a Divide 4 and 8 by their GCF, 4. Simplify. Find Write in simplest form. The multiplicative inverse of is. 2 1

56. Example 4-5b Answer: Check Since is about 3 and is about –2, you can estimate the answer to be about which is. The answer seems reasonable because

57. Example 4-5c Answer: Find Write in simplest form.

58. Example 4-6a MOVIES One summer day, a cinema shows a hit action movie repeatedly for hours. It takes about 150 minutes to run the movie and prepare for the next showing. How many times does the cinema show the movie that day?

59. Example 4-6b Divide by common factors. Simplify. Since 150 minutes equals hours, divide by Multiply by the multiplicative inverse of which is 1 1 5 1

60. Example 4-6c Check Use dimensional analysis to examine the units. Answer: The cinema shows the movie 5 times that day. Convert hours to minutes. Divide out the units. Simplify. The result is expressed as showings. This agrees with your answer of 5 showings that day.

61. Example 4-6d Answer: 6 times ZOOS A zoo has a reptile show that runs repeatedly for hours. It takes about 105 minutes to conduct the show and prepare for the next one. How many times does the zoo run the reptile show each day?

62. End of Lesson 4

63. Lesson 5 Contents Example 1Add Like Fractions Example 2Subtract Like Fractions Example 3Add Mixed Numbers Example 4Subtract Mixed Numbers

64. Example 5-1a Answer: Simplify. Add the numerators. The denominators are the same. Find Write in simplest form.

65. Example 5-1b Answer: Find Write in simplest form.

66. Example 5-2a Answer: Subtract the numerators. The denominators are the same. Find Write in simplest form. Rename

67. Example 5-2b Answer: Find Write in simplest form.

68. Example 5-3a Answer: Add the whole numbers and fractions separately. Add the numerators. Simplify. Find Write in simplest form.

69. Example 5-3b Answer: Find Write in simplest form.

70. Example 5-4a HEIGHTS In the United States, the average height of a 9–year–old girl is inches. The average height of a 16–year–old girl is inches. How much does an average girl grow from age 9 to age 16?

71. Example 5-4b Estimate 64 – 54 = 10 Write the mixed numbers as improper fractions. Subtract the numerators. The denominators are the same. Answer: The average girl grows inches from age 9 to age 16. Rename

72. Example 5-4c Answer: inches HEIGHTS Ainsley was inches tall when she was 4 years old. When she was 10 years old, she was inches tall. How much did she grow between the ages of 4 and 10?

73. End of Lesson 5

74. Lesson 6 Contents Example 1Subtract Unlike Fractions Example 2Add Mixed Numbers Example 3Estimate the Sum of Mixed Numbers Example 4Evaluate Expressions

75. Example 6-1a The LCD is 7  5 or 35. Rename each fraction using the LCD. Find Write in simplest form. Subtract by adding its inverse,

76. Example 6-1b Answer: Add the numerators. Simplify.

77. Example 6-1c Answer: Find Write in simplest form.

78. Example 6-2a Write the mixed numbers as fractions. The LCD is or 24. Find Write in simplest form. Rename each fraction using the LCD.

79. Example 6-2b Answer: Add the numerators. Simplify.

80. Example 6-2c Answer: Find Write in simplest form.

81. Example 6-3a MULTIPLE- CHOICE TEST ITEM During January, Nikki watched television hours the first week, hours the second week, hours the third week, and hours the fourth week. How much time did she watch television during the four-week period? A hours B hours C hours D hours

82. Example 6-3b Solve the Test Item It would take some time to change each of the fractions to ones with a common denominator. However, notice that all four of the numbers are about 2. Since 2 x 4 = 8, the answer will be about 8. Notice that only one of the choices is close to 8. Answer: B Read the Test Item You need to find the sum of four mixed numbers.

83. Example 6-3c Answer: C MULTIPLE- CHOICE TEST ITEM During July, Carsten watched television hours the first week, hours the second week, and hours the third week. How much time did she watch television during the 3-week period? A hours B hours C hours D hours

84. Example 6-4a Answer: Rename each fraction using the LCD, 45. Subtract the numerators. Simplify. Replace p with and q with ALGEBRA Find the value of p – q if and

85. Example 6-4b Answer: ALGEBRA Find the value of

86. End of Lesson 6

87. Lesson 7 Contents Example 1Solve by Using Addition or Subtraction Example 2Solve by Using Addition or Subtraction Example 3Solve by Using Multiplication or Division Example 4Solve by Using Multiplication or Division Example 5Write an Equation to Solve a Problem

88. Example 7-1a Answer: 0.78 Write the equation. Subtract 2.84 from each side. Simplify. Solve. Check your solution. Write the equation. Replace g with 0.78. Simplify. Check

89. Example 7-1b Answer: 3.08 Solve. Check your solution.

90. Example 7-2a Solve Write the equation. Simplify. Add to each side.

91. Example 7-2b Answer: Rename each fraction using the LCD, 15. Simplify.

92. Example 7-2c Solve Answer:

93. Example 7-3a Write the equation. Simplify. Multiply each side by Solve Check your solution.

94. Example 7-3b Check Write the equation. Replace c with –33. Simplify. Answer: –33

95. Example 7-3c Answer: –45 Solve Check your solution.

96. Example 7-4a Solve Write the equation. Divide each side by 9.7. Simplify. Check the solution. Answer: –7

97. Example 7-4b Solve 3.4t = –27.2. Answer: –8

98. Example 7-5a Rate equals distance divided by time. Words Variables Equation PHYSICS You can determine the rate an object is traveling by dividing the distance it travels by the time it takes to cover the distance If an object travels at a rate of 14.3 meters per second for 17 seconds, how far does it travel? r= 14.3=

99. Example 7-5b Answer: The object travels 243.1 meters. Write the equation. Multiply each side by 17. Simplify.

100. Example 7-5c Answer: 379.6 miles PHYSICS You can determine the rate an object is traveling by dividing the distance it travels by the time it takes to cover the distance If an object travels at a rate of 73 miles per hour for 5.2 hours, how far does it travel?

101. End of Lesson 7

102. Lesson 8 Contents Example 1Write an Expression Using Powers Example 2Evaluate Powers Example 3Evaluate Powers Example 4Evaluate Powers

103. Example 8-1a Answer: p 4 q 3 Commutative Property Associative Property Definition of exponents Write using exponents.

104. Example 8-1b Answer: x 3 y 4 Write using exponents.

105. Example 8-2a Answer: 59,049 Definition of exponents Simplify. Check using a calculator. Evaluate. 9 5 59049 ENTER

106. Example 8-2b Answer: 7,776 Evaluate.

107. Example 8-3a Definition of exponents Simplify. Answer: Evaluate

108. Example 8-3b Answer: Evaluate

109. Example 8-4a Answer: 2,048 Replace x with 4 and y with 2. Definition of exponents Simplify. ALGEBRA Evaluate if and

110. Example 8-4b Answer: 2,304 ALGEBRA Evaluate if andALGEBRA Evaluate

111. End of Lesson 8

112. Lesson 9 Contents Example 1Express Numbers in Standard Form Example 2Express Numbers in Standard Form Example 3Write Numbers in Scientific Notation Example 4Write Numbers in Scientific Notation Example 5Compare Numbers in Scientific Notation

113. Example 9-1a Notice that the decimal place moves 5 places to the right. Answer: 962,000 Write in standard form. or 100,000 100,000

114. Example 9-1b Answer: 53,200 Write in standard form.

115. Example 9-2a Answer: 0.00000285 Write in standard form. Notice that the decimal point moves 6 places to the left.

116. Example 9-2b Answer: 0.000381 Write in standard form.

117. Example 9-3a Write 931,500,000 in scientific notation. Answer: The decimal point moves 8 places. The exponent is positive.

118. Example 9-3b Write 35,600,000 in scientific notation. Answer:

119. Example 9-4a Write 0.00443 in scientific notation. Answer: The decimal point moves 3 places. The exponent is negative.

120. Example 9-4b Write 0.000653 in scientific notation. Answer:

121. Example 9-5a PLANETS The following table lists the average radius at the equator for each of the planets in our solar system. Order the planets according to radius from largest to smallest. PlanetRadius (km) Earth6.38 x 10 3 Jupiter7.14 x 10 4 Mars3.40 x 10 3 Mercury2.44 x 10 3 Neptune2.43 x 10 4 Pluto1.5 x 10 3 Saturn6.0 x 10 4 Uranus2.54 x 10 4 Venus6.05 x 10 3 Source: CRC Handbook of Chemistry and Physics

122. Example 9-5b First order the numbers according to their exponents. Then order the numbers with the same exponents by comparing the factors. Step 1 7.14 x 10 4 2.43 x 10 4 6.0 x 10 4 2.54 x 10 4 Jupiter, Neptune, Saturn, Uranus 6.38 x 10 3 3.40 x 10 3 2.44 x 10 3 1.5 x 10 3 6.05 x 10 3 > Earth, Mars, Mercury, Pluto, Venus

123. Example 9-5c Step 2 7.14 x 10 4 > 6.0 x 10 4 > 2.54 x 10 4 > 2.43 x 10 4 6.38 x 10 3 > 6.05 x 10 3 > 3.40 x 10 3 > 2.44 x 10 3 > 1.5 x 10 3 JupiterSaturnUranus Neptune EarthVenusMarsMercuryPluto Answer: The order from largest to smallest is Jupiter, Saturn, Uranus, Neptune, Earth, Venus, Mars, Mercury, and Pluto.

124. Example 9-5d PLANETS The following table lists the mass for each of the planets in our solar system. Order the planets according to mass from largest to smallest. PlanetMass (in tons) Mercury3.64 x 10 20 Venus5.37 x 10 21 Earth6.58 x 10 21 Mars7.08 x 10 20 Jupiter2.09 x 10 24 Saturn6.25 x 10 23 Uranus9.57 x 10 22 Neptune1.13 x 10 23 Pluto1.38 x 10 19 Source: nssdc.gsfc.nasa.gov

125. Example 9-5e Answer: Jupiter, Saturn, Neptune, Uranus, Earth, Venus, Mars, Mercury, and Pluto

126. End of Lesson 9

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