2. Welcome to Interactive Chalkboard Pre-Algebra Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240

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4. Contents Lesson 6-1Ratios and Rates Lesson 6-2Using Proportions Lesson 6-3Scale Drawings and Models Lesson 6-4Fractions, Decimals, and Percents Lesson 6-5Using the Percent Proportion Lesson 6-6Finding Percents Mentally Lesson 6-7Using Percent Equations Lesson 6-8Percent of Change Lesson 6-9Probability and Predictions

5. Lesson 1 Contents Example 1Write Ratios as Fractions Example 2Write Ratios as Fractions Example 3Find Unit Rate Example 4Convert Rates

6. Example 1-1a Express the ratio 10 roses out of 12 flowers as a fraction in simplest form. Divide the numerator and denominator by the GCF, 2. The ratio of roses to flowers is 5 to 6. This means that for every 6 flowers, 5 of them are roses. Answer:

7. Example 1-1b Express the ratio 8 golden retrievers out of 12 dogs as a fraction in simplest form. Answer:

8. Example 1-2a Express the ratio 21 inches to 2 yards as a fraction in simplest form. Convert 2 yards to inches. Written in simplest form, the ratio is 7 to 24. Answer: Divide the numerator and denominator by the GCF, 3.

9. Example 1-2b Express the ratio 4 feet to 18 inches as a fraction in simplest form. Answer:

10. Example 1-3a Shopping A 12-oz bottle of cleaner costs $4.50. A 16-oz bottle of cleaner costs $6.56. Which costs less per ounce? Find and compare the unit rates of the bottles. Divide the numerator and denominator by 12 to get a denominator of 1. For the 12-oz bottle, the unit rate is $0.38 per ounce.

11. Example 1-3b Divide the numerator and denominator by 16 to get a denominator of 1. For the 16-oz bottle, the unit rate is $0.41 per ounce. Answer: The 12-oz bottle has the lower cost per ounce.

12. Example 1-3c Shopping A 6-pack of a soft drink costs $1.50. A 12-pack of a soft drink costs $2.76. Which pack costs less per can? Answer: The 12-pack costs less per can.

13. Example 1-4a Animals A snail moved 30 feet in 2 hours. How many inches per minute did the snail move? You need to convert to. There are 12 inches in 1 foot and 60 minutes in 1 hour. Write 30 feet in 2 hours as..

14. Divide the common factors and units.. Example 1-4b 6 1 1 2 3 1 Convert feet to inches and hours to minutes.. The reciprocal of is..

15. Example 1-4c Simplify. Answer:30 feet in 2 hours is equivalent to 3 inches per minute.

16. Example 1-4d Jogging Dave jogs 2 miles in 22 minutes. How many feet per second is this? Answer:This is 8 feet per second.

17. End of Lesson 1

18. Lesson 2 Contents Example 1Identify Proportions Example 2Solve Proportions Example 3Use a Proportion to Solve a Problem Example 4Convert Measurements

19. Example 2-1a Determine whether the pair of ratios forms a proportion. Write a proportion. Cross products Simplify. Answer: So,.

20. Example 2-1b Determine whether the pair of ratios forms a proportion. Write a proportion. Cross products Simplify. Answer:So,.

21. Determine whether each pair of ratios forms a proportion. a. b. Example 2-1c Answer:

22. Example 2-2a Solve the proportion. Cross products Multiply. Divide. Answer:The solution is 21.6.

23. Example 2-2b Solve the proportion. Cross products Multiply. Divide. Answer:The solution is 5.

24. Solve each proportion. a. b. Example 2-2c Answer: 4.5 Answer: 7

25. Example 2-3a Architecture An architect builds a model of a building before the actual building is built. The model is 8 inches tall and the actual building will be 22 feet tall. The model is 20 inches wide. Find the width of the actual building. ExploreYou know the actual height of the building and the corresponding height of the model. You need to find the actual width of the building that corresponds with a model width of 20 inches. PlanWrite and solve a proportion using ratios that compare the actual building to the model. Let w represent the actual width of the building.

26. Example 2-3b Solve Write a proportion. Cross products Multiply. Divide. Simplify.

27. Example 2-3c Answer: The actual width of the building is 55 feet. Examine Check the cross products. Since and, the answer is correct.

28. Example 2-3d Planes A model of a jet airplane has a length of 9 inches and a wingspan of 6 inches. Find the wingspan of the actual plane if the length is 120 feet. Answer: The wingspan of the actual plane is 80 feet.

29. Example 2-4a Attractions The Circleville Pumpkin Show in Circleville, Ohio, boasts the world’s largest pumpkin pie. The pie weighs 350 pounds and is 5 feet in diameter. Find the diameter of the pie in centimeters if 1 foot = 30.48 centimeters. Let x represent the diameter in centimeters. customary measurement metric measurement customary measurement metric measurement Cross products Simplify. Answer: The diameter of the pie is 152.4 centimeters.

30. Example 2-4b School The gymnasium in a new school building measures 55 feet in length. Find the length of the gymnasium in centimeters if 1 ft = 30.48 centimeters. Answer: The length of the gymnasium is 1676.4 centimeters.

31. End of Lesson 2

32. Lesson 3 Contents Example 1Find Actual Measurements Example 2Determine the Scale Example 3Construct a Scale Drawing

33. Example 3-1a Map A map has a scale of 1 inch = 8 miles. Two towns are 3.25 inches apart on the map. What is the actual distance between the two towns? Let x represent the actual distance between the two towns. Write and solve a proportion. map distance actual distance map distance actual distance Find the cross products. Simplify. Answer:The actual distance between the two towns is 26 miles.

34. Example 3-1b Map A map has a scale of 1 inch = 8 miles. Two towns are 3.25 inches apart on the map. What is the scale factor? To find the scale factor, write the ratio of 1 inch to 8 miles in simplest form. Convert 8 miles to inches. Answer:The scale factor is. That is, each measurement on the map is the actual measurement.

35. Example 3-1c Scale Drawing A scale drawing of a new house has a scale of 1 inch = 4 feet. The height of the living room ceiling is 2.75 inches on the scale drawing. a. What is the actual height of the ceiling? b.What is the scale factor? Answer:11 feet Answer:

36. Example 3-2a Model Car A model car is 4 inches long. The actual car is 12 feet long. What is the scale of the model? Write the ratio of the length of the model to the actual length of the car. Then solve a proportion in which the length of the model is 1 inch and the actual length is x feet. model length actual length model length actual length Find the cross products. Simplify.

37. Example 3-2b Divide each side by 4. Simplify. Answer:The scale is 1 inch = 3 feet.

38. Example 3-2c Log Cabin A model log cabin is 12 inches high. The actual log cabin is 42 feet high. What is the scale of the model? Answer:The scale is 1 inch = 3.5 feet.

39. Example 3-3a Patio Design Sheila is designing a patio that is 16 feet long and 14 feet wide. Make a scale drawing of the patio. Use a scale of 0.5 inch = 4 feet. Step 1Find the measure of the patio’s length on the drawing. Let x represent the length. drawing length actual length drawing length actual length Find the cross products. Simplify.

40. Example 3-3b Divide each side by 4. On the drawing, the length is 2 inches. Step 2Find the measure of the patio’s width on the drawing. Let w represent the width. drawing width actual width drawing width actual width Find the cross products. Simplify.

41. Example 3-3c Divide each side by 4. Simplify. On the drawing, the width is 1.75 or inches.

42. Example 3-3d Step 3Make the scale drawing. Use -inch grid paper. Since 2 inches = 8 squares and inches = 7 squares, draw a rectangle that is 8 squares by 7 squares.

43. Example 3-3e Answer:

44. Example 3-3f Gardening Bob is designing a garden that is 18 feet long and 14 feet wide. Make a scale drawing of the garden. Use a scale of 0.5 inch = 4 feet. Answer: The length is inches = 9 squares and the width is inches = 7 squares.

45. Example 3-3g Answer:

46. End of Lesson 3

47. Lesson 4 Contents Example 1Percents as Fractions Example 2Fractions as Percents Example 3Percents as Decimals Example 4Decimals as Percents Example 5Fractions as Percents Example 6Compare Numbers

48. Example 4-1a Express 60% as a fraction in simplest form. Answer:

49. Example 4-1b Express 104% as a fraction in simplest form. Answer:

50. Example 4-1c Express 0.3% as a fraction in simplest form. Answer: Multiply by to eliminate the decimal in the numerator.

51. Example 4-1d The fraction bar indicates division. Express as a fraction in simplest form. Answer: 9 4

52. Example 4-1e Express each percent as a fraction in simplest form. a. b. c. d. Answer:

53. Example 4-2a Express as a percent. Answer:

54. Example 4-2b Answer: Express as a percent.

55. Example 4-2c Express each fraction as a percent. a. b. Answer: 68% Answer: 140%

56. Example 4-3a Express 60% as a decimal. Divide by 100 and remove the %. Answer:

57. Example 4-3b Express 2% as a decimal. Divide by 100 and remove the %. Answer:

58. Example 4-3c Express 658% as a decimal. Divide by 100 and remove the %. Answer:

59. Example 4-3d Express 0.4% as a decimal. Divide by 100 and remove the %. Answer:

60. Example 4-3e Express each percent as a decimal. a. 84% b. 7% c. 302% d. 0.9% Answer: 0.84 Answer: 0.07 Answer: 3.02 Answer: 0.009

61. Example 4-4a Express 0.4 as a percent. Answer: Multiply by 100 and add the %.

62. Example 4-4b Express 0.05 as a percent. Answer: Multiply by 100 and add the %.

63. Example 4-4c Express 0.0008 as a percent. Answer: Multiply by 100 and add the %.

64. Example 4-4d Express 7.3 as a percent. Answer: Multiply by 100 and add the %.

65. Example 4-4e Express each decimal as a percent. a. 0.84 b. 0.01 c. 0.004 d. 2.39 Answer: 84% Answer: 1% Answer: 0.4% Answer: 239%

66. Example 4-5a Express as a percent. Round to the nearest tenth percent, if necessary. Answer:

67. Expressas a percent. Round to the nearest tenth percent, if necessary. Example 4-5b Answer:

68. Express as a percent. Round to the nearest tenth percent, if necessary. Example 4-5c Answer:

69. Example 4-5d Express as a percent. Round to the nearest tenth percent, if necessary. Answer:

70. Express each fraction as a percent. Round to the nearest tenth percent, if necessary. a. b. c. d. Example 4-5e Answer: 37.5% Answer: 41.7% Answer: 1.3% Answer: 123.5%

71. Example 4-6a Bakery A baker says that 25% of his customers buy only bread and of his customers buy only cookies. Which group is larger? Write as a percent. Then compare. Answer: Since 40% is greater than 25%, the group that buys only cookies is larger.

72. Example 4-6b School The school principal states that of the students are involved in instrumental music while 42% are involved in vocal music. Which group is larger? Answer: The group involved in vocal music is larger.

73. End of Lesson 4

74. Lesson 5 Contents Example 1Find the Percent Example 2Find the Percent Example 3Apply the Percent Proportion Example 4Find the Part Example 5Apply the Percent Proportion Example 6Find the Base

75. Example 5-1a Twenty is what percent of 25 ? Twenty is being compared to 25. So, 20 is the part and 25 is the base. Let p represent the percent. Replace a with 20 and b with 25. Find the cross products. Simplify.

76. Example 5-1b Divide each side by 25. Simplify. Answer: 20 is 80% of 25.

77. Example 5-1c Twelve is what percent of 40 ? Answer: 12 is 30% of 40.

78. Example 5-2a What percent of 8 is 12 ? Twelve is being compared to 8. So, 12 is the part and 8 is the base. Let p represent the percent. Replace a with 12 and b with 8. Simplify. Find the cross products.

79. Example 5-2b Answer: 150% of 8 is 12. Divide each side by 8. Simplify.

80. Answer: 175% of 20 is 35. Example 5-2c What percent of 20 is 35 ?

81. Example 5-3a Tennis From the years 1988 through 1996 Steffi Graf won the U.S. Open tennis championships five times and Wimbledon seven times. What percent of both tournaments combined during those years was Steffi Graf the women’s champion? Compare the number of Steffi Graf’s wins, 12, to the total number of tournaments played, 18. Let a represent the part, 12, and let b represent the base, 18, in the percent proportion. Let p represent the percent. Replace a with 12 and b with 18.

82. Example 5-3b Find the cross products. Simplify. Divide each side by 18. Simplify. Answer:Steffi Graf won about 66.7% of the tournaments.

83. Example 5-3c Bake Sale At the school bake sale, 23 chocolate chip cookies, 18 oatmeal raisin cookies, and 7 peanut butter cookies were sold. If the sale started with a total of 90 cookies, what percent of the cookies were sold? Answer:About 53.3% of the cookies were sold.

84. Example 5-4a What number is 8.8% of 20 ? The percent is 8.8, and the base is 20. Let a represent the part. Answer: 8.8% of 20 is 1.76. Replace b with 20 and p with 8.8. Find the cross products. Simplify. Mentally divide each side by 100.

85. Example 5-4b What number is 42.5% of 90 ? Answer: 42.5% of 90 is 38.25.

86. Example 5-5a Recycling At Kington Middle School, 600 students were asked if their families recycle cans and/or newspapers. 32% said they recycle both; 25% said they recycle cans; 19% said they recycle newspapers; 24% said they do not recycle. How many families recycle both cans and newspapers? The total number of families is 600. So, 600 is the base. The percent is 32%. To find 32% of 600, let b represent the base, 600, and let p represent the percent, 32%, in the percent proportion. Let a represent the part.

87. Example 5-5b Replace b with 600 and p with 32. Find the cross products. Simplify. Mentally divide each side by 100. Answer:192 families recycle both cans and newspapers.

88. Example 5-5c Market Research A survey of 500 teenagers found that 37% prefer a soft drink with dinner; 48% prefer water; and 15% prefer milk. How many of the teenagers prefer milk with dinner? Answer:75 teenagers prefer milk with dinner.

89. Example 5-6a Seventy is 28% of what number? The percent is 28% and the part is 70. Let b represent the base. Replace a with 70 and p with 28. Find the cross products. Simplify.

90. Example 5-6b Divide each side by 28. Simplify. Answer: 70 is 28% of 250.

91. Example 5-6c Ninety is 24% of what number? Answer: 90 is 24% of 375.

92. End of Lesson 5

93. Lesson 6 Contents Example 1Find Percent of a Number Mentally Example 2Estimate Percents Example 3Use Estimation to Solve a Problem

94. Example 6-1a Find 50% of 46 mentally. Think:. Think:of 46 is 23. Answer: 50% of 46 is 23.

95. Think:of 88 is 22. Think:. Example 6-1b Find 25% of 88 mentally. Answer: 25% of 88 is 22.

96. Think:. Example 6-1c Find 70% of 110 mentally. Answer: 70% of 110 is 77. Think:of 110 is 11. So, of 110 is 77.

97. Example 6-1d Find the percent of each number mentally. a. 50% of 82 b. 25% of 36 c. 80% of 60 Answer: 41 Answer: 9 Answer: 48

98. Example 6-2a Estimate 22% of 494. 22% is about 20% or. 494 is about 500. of 500 is 100. Answer: 22% of 494 is about 100.

99. Example 6-2b Estimate 63% of 788. 63% is about 60% or. 788 is about 800. of 800 is 480. Answer: 63% of 788 is about 480.

100. Example 6-2c 1% of 1200 is 12. Estimateof 1219. 1219 is almost 1200. of 1219 is about or 3. Answer:

101. Example 6-2d Estimate 155% of 38. 155% means about 150 for every 100 or about 15 for every 10. 38 has about 4 tens. Answer: 155% of 38 is about 60.

102. a.Estimate 38% of 400. b.Estimate 72% of 318. c.Estimate of 2482. d.Estimate 183% of 93. Example 6-2e Answer: 160 Answer: 210 Answer: 5 Answer: 162

103. Example 6-3a Money A restaurant bill totals $21.35. You want to leave a 15% tip. What is a reasonable amount for the tip? $21.35 is about $21. So, 15% is about or $3.15. Answer:A reasonable amount for the tip would be $3. 10% of $21 is $2.10. Move the decimal point 1 place to the left. 5% of $21 is $1.05. 5% is one half of 10%.

104. Example 6-3b Money A restaurant bill totals $59.05. You want to leave a 15% tip. What is a reasonable amount for the tip? Answer:A reasonable amount for the tip would be $9.

105. End of Lesson 6

106. Lesson 7 Contents Example 1Find the Part Example 2Find the Percent Example 3Find the Base Example 4Find Discount Example 5Apply Simple Interest Formula

107. Example 7-1a Find 38% of 22. Estimate: 40% of 20 is 8. You know that that the base is 22 and the percent is 38%. Let n represent the part. Write 38% as the decimal 0.38. Simplify. Answer: 38% of 22 is 8.36.

108. Example 7-1b Find 64% of 48. Answer: 64% of 48 is 30.72.

109. Example 7-2a 19 is what percent of 25 ? You know that the base is 25 and the part is 19. Let n represent the percent. Divide each side by 25. Estimate:, which is 80%.

110. Example 7-2b Answer: 19 is 76% of 25. Simplify. The answer makes sense compared to the estimate.

111. Example 7-2c 8 is what percent of 25 ? Answer: 8 is 32% of 25.

112. Example 7-3a 84 is 16% of what number? Estimate: 80 is 16% of 500. You know that the part is 84 and the percent is 16%. Let n represent the base. Divide each side by 0.16. Write 16% as the decimal 0.16.

113. Example 7-3b Answer: 84 is 16% of 525. Simplify. The answer is reasonable since it is close to the estimate.

114. Example 7-3c 315 is 42% of what number? Answer: 315 is 42% of 750.

115. Example 7-4a Jewelry The regular price of a ring is $495. It is on sale at a 20% discount. What is the sale price of the ring? Method 1 First, use the percent equation to find 20% of 495. Estimate: Let d represent the discount. The base is 495 and the percent is 20%. Simplify.

116. Example 7-4b Then, find the sale price. Subtract the discount from the original price. Method 2 Let s represent the sale price. A discount of 20% means the ring will cost or 80% of the original price. Use the percent equation to find 80% of 495.

117. Example 7-4c The base is 495 and the percent is 80%. Answer:The sale price of the ring will be $396.

118. Example 7-4d Retail The regular price of a stereo system is $1295. The system is on sale at a 15% discount. Find the sale price of the stereo system. Answer:The sale price of the stereo system is $1100.75.

119. Example 7-5a Banking Suppose you invest $2000 at an annual interest rate of 4.5%. How long will it take for it to earn $495? Write the simple interest formula. Replace I with 495, p with 2000, and r with 0.045. Simplify.

120. Example 7-5b Divide each side by 90. Simplify. Answer:It will take 5.5 years to earn $495.

121. Example 7-5c Banking Suppose you invest $3500 at an annual interest rate of 6.25%. How long will it take for it to earn $875? Answer:It will take 4 years to earn $875.

122. End of Lesson 7

123. Lesson 8 Contents Example 1Find Percent of Change Example 2Find Percent of Increase Example 3Find Percent of Increase Example 4Find Percent of Decrease

124. Example 8-1a Find the percent of change from 325 to 390. Step 1Subtract to find the amount of change. Step 2Write a ratio that compares the amount of change to the original amount. Express the ratio as a percent. Substitution.

125. Example 8-1b Write the decimal as a percent. Answer:The percent of change from 325 to 390 is 20%.

126. Example 8-1c Find the percent of change from 84 to 105. Answer:The percent of change from 84 to 105 is 25%.

127. Example 8-2a Tuition In 1965, when John entered college, the tuition per year was $7500. In 2000, when his daughter went to the same school, the tuition was $25,500. Find the percent of change. Step 1Subtract to find the amount of change. Step 2Write a ratio that compares the amount of change to the original tuition. Express the ratio as a percent.

128. Example 8-2b Answer:The percent of change is 240%. In this case, the percent of change is a percent of increase. Substitution. Write the decimal as a percent.

129. Example 8-2c Textbooks In 1990, the price of a textbook was $38. In 2000, the price of the same textbook was $81. Find the percent of change. Answer:The percent of change was about 113%. In this case, the percent of change is a percent of increase.

130. Example 8-3a Multiple-Choice Test Item Refer to the table shown. Which county had the least percent of increase in population from 1990 to 2000? County19902000 Breckenridge16,31218,648 Bracken77668279 Calloway30,73534,177 Fulton82717752 A BreckinridgeB Bracken C CallowayD Fulton

131. Example 8-3a Read the Test Item Percent of increase tells how much the population has increased in relation to 1990. Solve the Test Item Use a ratio to find each percent of increase. Then compare the percents.

132. Example 8-3b Breckinridge Bracken

133. Example 8-3c Calloway Fulton Answer:Bracken County had the least percent of increase in population from 1990 to 2000. The answer is B. Eliminate this choice because the population decreased.

134. Example 8-3d Multiple-Choice Test Item The table shows test scores on the first two math tests of the semester for four students. Which student had the greatest percent of increase from test 1 to test 2? A Holly B Ben C Sally D Max Student NameTest 1 ScoreTest 2 Score Holly 8491 Ben 9389 Sally 6579 Max 7398

135. Example 8-3e Answer:Max had the greatest percent of increase from test 1 to test 2. The answer is D.

136. Example 8-4a Clothing A $110 sweater is on sale for $88. What is the percent of change? Step 2Compare the amount of change to the original price. Step 1Subtract to find the amount of change.

137. Example 8-4b Substitution. Write the decimal as a percent. Answer:The percent of change is –20%. In this case, the percent of change is a percent of decrease.

138. Example 8-4c Shoes A $145 pair of tennis shoes is on sale for $105. What is the percent of change? Answer:The percent of change is about –27.6%.

139. End of Lesson 8

140. Lesson 9 Contents Example 1Find Probability Example 2Find Probability Example 3Find Experimental Probability Example 4Make a Prediction

141. Example 9-1a Suppose a number cube is rolled. What is the probability of rolling a 4 or a 5? There are 2 numbers on the number cube that are a 4 or a 5: 4 and 5. There are 6 possible outcomes: 1, 2, 3, 4, 5, and 6. Answer:The probability of rolling a 4 or a 5 is

142. Example 9-1b Suppose a number cube is rolled. What is the probability of rolling a number that is divisible by 3? Answer:The probability of rolling a number that is divisible by 3 is

143. Example 9-2a Suppose that two number cubes are rolled. Find the probability of rolling two identical numbers. Make a table showing the sample space when rolling two number cubes. 123456 1(1, 1)(1, 2)(1, 3)(1, 4)(1, 5)(1, 6) 2(2, 1)(2, 2)(2, 3)(2, 4)(2, 5)(2, 6) 3(3, 1)(3, 2)(3, 3)(3, 4)(3, 5)(3, 6) 4(4, 1)(4, 2)(4, 3)(4, 4)(4, 5)(4, 6) 5(5, 1)(5, 2)(5, 3)(5, 4)(5, 5)(5, 6) 6(6, 1)(6, 2)(6, 3)(6, 4)(6, 5)(6, 6)

144. Example 9-2b There are 6 outcomes in which the two numbers are identical. Answer: This means there is a chance of rolling two identical numbers.

145. Example 9-2c Suppose that two number cubes are rolled. Find the probability of rolling two numbers whose sum is 8. Answer:There is a chance of rolling two numbers whose sum is 8.

146. Example 9-3a Use the table below to determine the experimental probability of landing on heads for this experiment. Answer:The experimental probability of landing on heads in this case is

147. Example 9-3b Basketball Brian is shooting baskets with a basketball. He makes 13 shots and misses 9 shots. Determine the experimental probability of Brian making a shot. Answer:The experimental probability of making a shot in this case is

148. Example 9-4a Sports Miss Newman surveyed her class to see which sport they preferred watching. 44% preferred football, 28% basketball, 20% soccer, and 8% tennis. Out of 560 students in the entire school, how many would you expect to say they prefer watching basketball? The total number of students is 560. So, 560 is the base. The percent is 28%. To find 28% of 560, let b represent the base, 560, and let p represent the percent, 28%, in the percent proportion. Let a represent the part.

149. Example 9-4b part base percent Simplify. Mentally divide each side by 100. Answer:You can expect about 157 students to say they prefer watching basketball.

150. Example 9-4c Colors The students in an art class were surveyed about their favorite color. 32% preferred blue, 29% preferred red, 23% preferred yellow, and 16% preferred green. Out of 450 students in the entire school, how many would you expect to say they prefer red? Answer:You can expect about 131 students to say they prefer red.

151. End of Lesson 9

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