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Chapter PretestThe pretest below contains problems that are representative of the problems you will find in the chapter.Change each percent to a decimal.1. 68%0.682. 2%0.023. 21.5%0.215Change each decimal to a percent.4. 0.3939%5. 0.38638.6%6. 3.98398%Change each percent to a fraction or mixed number in lowest terms.7. 33%8. 45%9. 8.5%_33100 _ 920 _ 17200 Change each fraction or mixed number to a percent.10. _67100 67%13. What number is 5% of 24?1.211. _4 5 80%14. What percent of 40 is 6?15%12. 2_1 4 225%15. 12 is 24% of what number?50Getting Ready for Chapter 5The problems below review material covered previously that you need to know in order to be successful in Chapter5. If you have any difficulty with the problems here, you need to go back and review before going on to Chapter 5.Perform the indicated operations.1. 136 + 5.44 141.44 2. 300 − 75 225 3. 1,793,000 − 315,568 1,477,4324. _652 × _ 1100 13 _ 5. 0.2 × 100 20 6. 4.89 × 100 489407. 0.15 ⋅ 63 9.45 8. _35.2 0.352 9. 3.62 ÷ 100 0.036210010. _34 (Round to the nearest tenth.) 117.2 11. 600 × 0.04 × 60_0.29 360 4Reduce.12. 36 _100 9 _25 13. 45 _1000 9 _Change each fraction or mixed number to a decimal.200 14. Change 321 _to an improper2fraction._652 15. 3 _8 0.375 16. 5 _12 0.416– 17. 2 1 _2 2.5Solve30218. 25 = 0.40 ⋅ n 62.5 19. 0.12n = 1,836 15,300 20. 1.075x = 3,200 (Round to thenearest hundredth.) 2976.74Chapter 5 <strong>Percent</strong>
Introduction . . .<strong>Percent</strong>s, Decimals, and FractionsThe sizes of categories in the pie chart below are given as percents. The whole piechart is represented by 100%. In general, 100% of something is the whole thing.In this section we will look at the meaning of percent. To begin, we learn tochange decimals to percents and percents to decimals.Factors Producing More Traffic TodayIncrease in trip lengths 35%Increase in population 13%Fewer occupants travelingin vehicles 17%Switch to driving from othermodes of transportation 17%Increase in trips taken 18%A The Meaning of <strong>Percent</strong><strong>Percent</strong> means “per hundred.” Writing a number as a percent is a way of comparingthe number with 100. For example, the number 42% (the % symbol is read“percent”) is the same as 42 one-hundredths. That is:42% = _42100 <strong>Percent</strong>s are really fractions (or ratios) with denominator 100.Here are some examples that show the meaning of percent.Example 150% = _50100 Example 275% = _75100 Example 325% = _25100 Example 433% = _33100 Example 56% = _6100 Example 6160% = _160100 5.1ObjectivesA Change percents to fractions.B Change percents to decimals.C Change decimals to percents.D Change percents to fractions inlowest terms.E Change fractions to percents.Practice ProblemsWrite each number as an equivalentfraction without the % symbol.1. 40%2. 80%3. 15%4. 37%5. 8%6. 150%5.1 <strong>Percent</strong>s, Decimals, and FractionsAnswers1. _40 2. 100 80 _ 3. 15_100 1004. _37 5. _ 8100 100 6. 150_ 100303
304Chapter 5 <strong>Percent</strong>BChanging <strong>Percent</strong>s to DecimalsTo change a percent to a decimal number, we simply use the meaning of percent.7. Change to a decimal.a. 25.2%b. 2.52%Example 7Change 35.2% to a decimal.Solution We drop the % symbol and write 35.2 over 100.35.2% = _35.2 Use the meaning of % to convert to100a fraction with denominator 100= 0.352 Divide 35.2 by 100We see from Example 7 that 35.2% is the same as the decimal 0.352. The resultis that the % symbol has been dropped and the decimal point has been movedtwo places to the left. Because % always means “per hundred,” we will always endup moving the decimal point two places to the left when we change percents todecimals. Because of this, we can write the following rule.RuleTo change a percent to a decimal, drop the % symbol and move the decimalpoint two places to the left, inserting zeros as placeholders if needed.Here are some examples to illustrate how to use this rule.Change each percent to a decimal.8. 40%Example 825% = 0.25Example 975% = 0.759. 80% Notice that the results in Examples 8, 9,and 10 are consistent with the results inExamples 1, 2, and 310. 15%Example 1050% = 0.50Example 116.8% = 0.06811. 5.6% Notice here that we put a 0 in front ofthe 6 so we can move the decimal pointtwo places to the left12. 4.86%Example 123.62% = 0.0362Example 130.4% = 0.00413. 0.6% This time we put two 0s in front of the4 in order to be able to move thedecimal point two places to the left14. 0.58%Example 14Cortisone cream is 0.5% hydrocortisone. Writing this numberas a decimal, we have0.5% = 0.005Answers7. a. 0.252 b. 0.0252 8. 0.409. 0.80 10. 0.15 11. 0.05612. 0.0486 13. 0.006 14. 0.0058
5.1 <strong>Percent</strong>s, Decimals, and Fractions305CChanging Decimals to <strong>Percent</strong>sNow we want to do the opposite of what we just did in Examples 7–14. We wantto change decimals to percents. We know that 42% written as a decimal is 0.42,which means that in order to change 0.42 back to a percent, we must move thedecimal point two places to the right and use the % symbol:0.42 = 42% Notice that we don’t show the new decimalpoint if it is at the end of the numberRuleTo change a decimal to a percent, we move the decimal point two places tothe right and use the % symbol.Examples 15–20 show how we use this rule.Example 150.27 = 27%Example 164.89 = 489%Write each decimal as a percent.15. 0.3516. 5.77Example 170.2 = 0.20 = 20%Notice here that we put a 0 after the2 so we can move the decimal pointtwo places to the right17. 0.4Example 180.09 = 09% = 9%Notice that we can drop the 0 at theleft without changing the value ofthe number18. 0.03Example 1925 = 25.00 = 2,500%Here, we put two 0s after the 5 sothat we can move the decimal pointtwo places to the right19. 45Example 20A softball player has a batting average of 0.650. As a percent,this number is 0.650 = 65.0%.20. 0.69Eyewire/Getty ImagesAs you can see from the examples above, percent is just a way of comparingnumbers to 100. To multiply decimals by 100, we move the decimal point twoplaces to the right. To divide by 100, we move the decimal point two places to theleft. Because of this, it is a fairly simple procedure to change percents to decimalsand decimals to percents.Answers15. 35% 16. 577% 17. 40%18. 3% 19. 4,500% 20. 69%
306Chapter 5 <strong>Percent</strong>Who Pays Health Care BillsD Changing <strong>Percent</strong>s to FractionsTo change a percent to a fraction, drop the % symbol and write the original numberover 100.Example 21The pie chart in the margin shows who pays health carebills. Change each percent to a fraction.Patient 19%Private insurance 36%Government 45%21. Change 82% to a fraction inlowest terms.22. Change 6.5% to a fraction inlowest terms.SolutionIn each case, we drop the percent symbol and write the number over100. Then we reduce to lowest terms if possible.19% = _19 45% = 45_100100 = 9 _20hreduce 36% = 36_100 = 9 _25 hExample 22Change 4.5% to a fraction in lowest terms.Solution We begin by writing 4.5 over 100:4.5% = 4.5 _100 reduceWe now multiply the numerator and the denominator by 10 so the numerator willbe a whole number:_4.5 = 4.5_ × 10 Multiply the numerator and100 100 × 10the denominator by 10= 45 _1,000 = _9 Reduce to lowest terms20023. Change 42_1 % to a fraction in2lowest terms.Example 23Change 32 1 _% to a fraction in lowest terms.2Solution Writing 32_1 2 % over 100 produces a complex fraction. We change 321 _2 to an improper fraction and simplify:32_1 32_12 % = _ 2 100 _652 = _100 Change 321 _to the improper fraction 65_2 2 = 65 _2 × 1 _100 Dividing by 100 is the same as multiplying by 1 _100 Instructor NoteEvery time I have the chance toshow division as multiplication bythe reciprocal, I do so.= _5⋅ 13 ⋅ 12 ⋅ 5⋅ 20 Multiplication= _13 Reduce to lowest terms40Note that we could have changed our original mixed number to a decimal firstand then changed to a fraction:Answers21. _41 22. 50 13 _ 23. 17_200 4032_1 2 % = 32.5% = 32.5 _= 32.5_ × 10= 325_100 100 × 10 1000The result is the same in both cases.= 5∙⋅ 5∙ ⋅ 13 _5∙ ⋅ 5∙ ⋅ 40= 13_40
5.1 <strong>Percent</strong>s, Decimals, and Fractions307E Changing Fractions to <strong>Percent</strong>sTo change a fraction to a percent, we can change the fraction to a decimal andthen change the decimal to a percent.Example 24Suppose the price your bookstore pays for your textbookis _710 of the price you pay for your textbook. Write 7 _as a percent.10Solution We can change _7 to a decimal by dividing 7 by 10:100.7____10) 7.0 7 0024. Change to a percent.a. _910 b. 9 _20 We then change the decimal 0.7 to a percent by moving thedecimal point two places to the right and using the % symbol:0.7 = 70%You may have noticed that we could have saved some time in Example 24by simply writing _7 as an equivalent fraction with denominator 100. That is:10_ 7 = 7_ ⋅ 10= 70_= 70%10 10 ⋅ 10 100This is a good way to convert fractions like _7 to percents. It works well for fractionswith denominators of 2, 4, 5, 10, 20, 25, and 50, because they are easy to10change to fractions with denominators of 100.Example 25Change 3 _to a percent.8Solution We write _3 as a decimal by dividing 3 by 8. We then change the8decimal to a percent by moving the decimal point two places to the right andusing the % symbol.25. Change to a percent.a. _5 8 b. 9 _8 _ 3 = 0.375 = 37.5%8_____ .3758) 3.000 2—460—564040 —026. Change to a percent.Example 26Change _5 to a percent.12Solution We begin by dividing 5 by 12:_______ .416612) 5.0000 4 8201280728072a. 7 _12 b. 13 _12 Answers24. a. 90% b. 45%25. a. 62.5% b. 112.5%
308Chapter 5 <strong>Percent</strong>NoteWhen rounding off, let’sagree to round off tothe nearest thousandthand then move the decimal point.Our answers in percent form willthen be accurate to the nearesttenth of a percent, as in Example 26.Because the 6s repeat indefinitely, we can use mixed number notation to write_ 5 _12 = 0.41 6= 41_2 3 %Or, rounding, we can write_ 5 = 41.7% To the nearest tenth of a percent1227. Change to a percent.a. 3_3 4 b. 3_7 8 Example 27Change 2 1 _Solutionto a percent.2We first change to a decimal and then to a percent:2_1 = 2.52= 250%Table 1 lists some of the most commonly used fractions and decimals and theirequivalent percents.Table 1Fraction Decimal percent_1 0.5 50%2_1 0.25 25%4_3 0.75 75%4 1 _3 0.3 331 _3 % 2 _3 0.6 662 _3 %_1 0.2 20%5_2 0.4 40%5_3 0.6 60%5_4 0.8 80%5Answers26. a. 58_1 % ≈ 58.3%3b. 108_1 % ≈ 108.3%327. a. 375% b. 387.5%Getting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.1. What is the relationship between the word percent and the number100?2. Explain in words how you would change 25% to a decimal.3. Explain in words how you would change 25% to a fraction.4. After reading this section you know that _1 , 0.5, and 50% are equivalent.2Show mathematically why this is true.
5.1 Problem Set309Problem Set 5.1A Write each percent as a fraction with denominator 100. [Examples 1–6]1. 20%2. 40%3. 60%4. 80%5. 24%6. 48%_20100 _ 40100 _ 60100 _ 80100 _ 24100 _ 48100 7. 65%8. 35%_65100 _ 35100 B Change each percent to a decimal. [Examples 7–14]9. 23%0.2310. 34%0.3411. 92%0.9212. 87%0.8713. 9%0.0914. 7%0.0715. 3.4%0.03416. 5.8%0.05817. 6.34%0.063418. 7.25%0.072519. 0.9%0.00920. 0.6%0.006C Change each decimal to a percent. [Examples 15–20]21. 0.2323%22. 0.3434%23. 0.9292%24. 0.8787%25. 0.4545%26. 0.5454%27. 0.033%28. 0.044%29. 0.660%30. 0.990%31. 0.880%32. 0.550%33. 0.2727%34. 0.6262%35. 1.23123%36. 2.34234%
310Chapter 5 <strong>Percent</strong>D Change each percent to a fraction in lowest terms. [Examples 21–23]37. 60%38. 40%39. 75%40. 25%41. 4%42. 2%_3 5 _ 2 5 _ 3 4 _ 1 4 _ 125 _ 150 43. 26.5% 44. 34.2% 45. 71.87% 46. 63.6% 47. 0.75% 48. 0.45%_53200 _ 171500 _ 7,18710,000 _ 159250 _ 3400 9_2,000 49. 6_1 4 %50. 5_1 4 %51. 33_1 3 % 52. 66_2 3 %_116 _ 21400 _ 1 3 _ 2 3 E Change each fraction or mixed number to a percent. [Examples 24–27]53. 1 _2 50%54. 1 _4 25%55. 3 _4 75%56. _2 3 57. _1 3 58. _1 5 66_ 2 3 % 33_ 1 3 % 20%59. 4 _5 80%60. _1 6 61. _7 8 16_ 2 3 % 87.5%62. 1 _8 12.5%63. 7 _50 14%64. 9 _25 36%65. 3 1 _4 66. 2 1 _8 67. 1 1 _2 68. 1 3 _4 325%212.5%150%175%69. _21 to the nearest tenth of a percent4348.8%70. _36 to the nearest tenth of a percent4973.5%
5.1 Problem Set311Applying the Concepts71. Mothers The chart shows the percentage of womenwho continue working after having a baby.72. U.S. Energy The pie chart shows where Americans gettheir energy.Working Women with Infants1997 50.6%1998 53.4%1999 53.6%2000 52.7%2001 51.0%2002 56.1%2003 53.7%Source: U.S. Department of LaborRole of Renewable Energy In the U.S.Natural Gas 23%Coal 23%Renewable Energy 7%Nuclear Energy 8%Petroleum 40%Source: Energy Information Adminstration 2006Using the chart, convert the percentage for the followingyears to a decimal.a. 1997 0.506b. 2000 0.527c. 2003 0.537Using the chart, convert the percentage to a fraction forthe following types of energy. Reduce to lowest terms.a. Natural Gas _23100 b. Nuclear Energy 2 _25 c. Petroleum 2 _5 73. Paying Bills According to Pew Research, a non-politicalorganization that provides information on the issues,attitudes and trends shaping America, most peoplestill pay their monthly bills by check.Paying Bills74. Pizza Ingredients The pie chart below shows the decimalrepresentation of each ingredient by weight that is usedto make a sausage and mushroom pizza. We see thathalf of the pizza’s weight comes from the crust. Changeeach decimal to a percent.Mushroom and Sausage PizzaCheck 54%Electronic/Online 28%Cash 15%Other 3%Crust 0.5Cheese 0.25Sausage 0.075Mushrooms 0.05Tomato Sauce 0.125a. Convert each percent to a fraction._2750 , 7 _25 , _ 320 , _ 3100 b. Convert each percent to a decimal.0.54, 0.28, 0.15, 0.03c. About how many times more likely are you to paya bill with a check than by electronic or onlinemethods?About 2 times as likely.crust 50.0%cheese 25.0%sausage 7.5%mushrooms 5.0%tomato sauce 12.5%
312Chapter 5 <strong>Percent</strong>Calculator ProblemsUse a calculator to write each fraction as a decimal, and then change the decimal to a percent. Round all answers to thenearest tenth of a percent.75. 29 _37 76. 18 _83 77. 6 _51 78. 8 _95 79. 236 _327 80. 568 _732 78.4%21.7%11.8%8.4%72.2%77.6%Getting Ready for the Next SectionMultiply.81. 0.25(74)18.582. 0.15(63)9.4583. 0.435(25)10.87584. 0.635(45)28.575Divide. Round the answers to the nearest thousandth, if necessary.85. _2142 86. _2184 87. _250.4 88. 31.9 _78 0.50.2562.50.409Solve for n.89. 42n = 210.590. 25 = 0.40n62.5Maintaining Your SkillsWrite as a decimal.91. _1 8 0.12592. _3 8 0.37593. _5 8 0.62594. _7 8 0.87595. _116 0.062596. _316 0.187597. _516 0.312598. _716 0.4375Divide.99. 1 _8 ÷ 1 _16 2100. 3 _8 ÷ 3 _16 2101. 5 _8 ÷ 5 _16 2102. 7 _8 ÷ 7 _16 2103. 0.125 ÷ 0.0625104. 0.375 ÷ 0.1875105. 0.625 ÷ 0.3125106. 0.875 ÷ 0.43752222
Introduction . . .Basic <strong>Percent</strong> ProblemsThe American Dietetic Association (ADA) recommends eating foods in which thenumber of calories from fat is less than 30% of the total number of calories. Foodsthat satisfy this requirement are considered healthy foods. Is the nutrition labelshown below from a food that the ADA would consider healthy? This is the typeof question we will be able to answer after we have worked through the examplesin this section.5.2ObjectivesA Solve the three types of percentproblems.B Solve percent problems involvingfood labels.C Solve percent problems usingproportions.Nutrition FactsServing Size 1/2 cup (65g)Servings Per Container: 8Amount Per ServingCalories 150Calories from fat 90% Daily Value*Total Fat 10g16%Saturated Fat 6g32%Cholesterol 35mg12%Sodium 30mg1%Total Carbohydrate 14g5%Dietary Fiber 0g0%Sugars 11gProtein 2gVitamin A 6% • Vitamin C 0%Calcium 6% •Iron 0%*<strong>Percent</strong> Daily Values are based on a 2,000calorie diet.Figure 1Nutrition label from vanilla ice creamThis section is concerned with three kinds of word problems that are associatedwith percents. Here is an example of each type:Type A: What number is 15% of 63?Type B: What percent of 42 is 21?Type C: 25 is 40% of what number?A Solving <strong>Percent</strong> Problems Using EquationsThe first method we use to solve all three types of problems involves translatingthe sentences into equations and then solving the equations. The following translationsare used to write the sentences as equations:Instructor NoteThe problems in this section are thebasis for all the application problemsthat follow in this chapter. I usethis method of solving percent problemsbecause it is more algebraic innature than other methods.Englishmathematicsis =of⋅ (multiply)a numbernwhat numbernwhat percentnThe word is always translates to an = sign. The word of almost always meansmultiply. The number we are looking for can be represented with a letter, such asn or x.5.2 Basic <strong>Percent</strong> Problems313
314Chapter 5 <strong>Percent</strong>Practice Problems1. a. What number is 25% of 74?b. What number is 50% of 74?Example 1What number is 15% of 63?SolutionWe translate the sentence into an equation as follows:What number is 15% of 63?gg gg gn = 0.15 ⋅ 63To do arithmetic with percents, we have to change to decimals. That is why 15%is rewritten as 0.15. Solving the equation, we haven = 0.15 ⋅ 63n = 9.4515% of 63 is 9.452. a. What percent of 84 is 21?b. What percent of 84 is 42?Example 2What percent of 42 is 21?SolutionWe translate the sentence as follows:What percent of 42 is 21?n ⋅ 42 = 21We solve for n by dividing both sides by 42._ n ⋅ 42 = _2142 42 n = _2142 n = 0.50Because the original problem asked for a percent, we change 0.50 to a percent:n = 50%21 is 50% of 42ggg g g3. a. 35 is 40% of what number?b. 70 is 40% of what number?Example 325 is 40% of what number?SolutionFollowing the procedure from the first two examples, we have25 is 40% of what number?g gg25 = 0.40 ⋅ nAgain, we changed 40% to 0.40 so we can do the arithmetic involved in the problem.Dividing both sides of the equation by 0.40, we have_ 250.40 = _ 0.40⋅ n 0.40_ 250.40 = n62.5 = ng25 is 40% of 62.5gAnswers1. a. 18.5 b. 372. a. 25% b. 50%3. a. 87.5 b. 175As you can see, all three types of percent problems are solved in a similar manner.We write is as =, of as ⋅, and what number as n. The resulting equation is thensolved to obtain the answer to the original question.
5.2 Basic <strong>Percent</strong> Problems315Example 4What number is 43.5% of 25?g g gn = 0.435 ⋅ 25n = 10.910.9 is 43.5% of 25ggRounded to the nearest tenth4. What number is 63.5% of 45?(Round to the nearest tenth.)Example 5What percent of 78 is 31.9?g g gggn ⋅ 78 = 31.9_ n ⋅ 78 = _31.9 78 78 n = 31.9 _78 n = 0.409 Rounded to the nearest thousandthn = 40.9%5. What percent of 85 is 11.9?40.9% of 78 is 31.9Example 634 is 29% of what number?g g34 = 0.29 ⋅ n_340.29 = _ 0.29⋅ n 0.29_340.29 = nggg117.2 = n Rounded to the nearest tenth6. 62 is 39% of what number?(Round to the nearest tenth.)34 is 29% of 117.2BFood LabelsExample 7As we mentioned in the introduction to this section, theAmerican Dietetic Association recommends eating foods in which the number ofcalories from fat is less than 30% of the total number of calories. According to thenutrition label below, what percent of the total number of calories is fat calories?Nutrition FactsServing Size 1/2 cup (65g)Servings Per Container: 8Amount Per ServingCalories 150Calories from fat 90% Daily Value*Total Fat 10g16%Saturated Fat 6g32%Cholesterol 35mg12%Sodium 30mg1%Total Carbohydrate 14g5%Dietary Fiber 0g0%Sugars 11gProtein 2gVitamin A 6% • Vitamin C 0%Calcium 6% •Iron 0%*<strong>Percent</strong> Daily Values are based on a 2,000calorie diet.7. The nutrition label below isfrom a package of vanilla frozenyogurt. What percentof the total number of calories isfat calories? Round youranswer to the nearest tenth ofa percent.Nutrition FactsServing Size 1/2 cup (98g)Servings Per Container: 4Amount Per ServingCalories 160Calories from fat 25% Daily Value*Total Fat 2.5g4%Saturated Fat 1.5g7%Cholesterol 45mg15%Sodium 55mg2%Total Carbohydrate 26g9%Dietary Fiber 0g0%Sugars 19gProtein 8gVitamin A 0% • Vitamin C 0%Calcium 25% •Iron 0%*<strong>Percent</strong> Daily Values are based on a 2,000calorie diet.Answers4. 28.6 5. 14% 6. 159.0Figure 2Nutrition label from vanilla ice cream
316Chapter 5 <strong>Percent</strong>SolutionTo solve this problem, we must write the question in the form of oneof the three basic percent problems shown in Examples 1–6. Because there are 90calories from fat and a total of 150 calories, we can write the question this way:90 is what percent of 150?Now that we have written the question in the form of one of the basic percentproblems, we simply translate it into an equation. Then we solve the equation.90 is what percent of 150?g gg90 = n ⋅ 150_90150 = nn = 0.60 = 60%m8888888m8888888The number of calories from fat in this package of ice cream is 60% of the totalnumber of calories. Thus the ADA would not consider this to be a healthy food.CSolving <strong>Percent</strong> Problems Using ProportionsWe can look at percent problems in terms of proportions also. For example, weknow that 24% is the same as _24100 , which reduces to 6 _. That is25_24100 = 6 _25 h{24 is to 100 as 6 is to 25hWe can illustrate this visually with boxes of proportional lengths:h{24610025In general, we say_ <strong>Percent</strong>100 = Amount _Basehh{<strong>Percent</strong> is to 100 as Amount is to Baseh{Answers7. 15.6% of the calories are fromfat. (So far as fat content isconcerned, the frozen yogurt isa healthier choice than the icecream.)
5.2 Basic <strong>Percent</strong> Problems317Example 8What number is 15% of 63?Solution This is the same problem we worked in Example 1. We let n be thenumber in question. We reason that n will be smaller than 63 because it is only15% of 63. The base is 63 and the amount is n. We compare n to 63 as we compare15 to 100. Our proportion sets up as follows:8. Rework Practice Problem 1using proportions.15 is to 100 as n is to 63{h{hh_15100 = n _63 15n10063Solving the proportion, we have15 ⋅ 63 = 100n Extremes/means property945 = 100n Simplify the left side9.45 = n Divide each side by 100This gives us the same result we obtained in Example 1.Example 9What percent of 42 is 21?Solution This is the same problem we worked in Example 2. We let n be thepercent in question. The amount is 21 and the base is 42. We compare n to 100 aswe compare 21 to 42. Here is our reasoning and proportion:9. Rework Practice Problem 2using proportions.n is to 100 as 21 is to 42{{h hh_n100 = 21 _42 n2110042Solving the proportion, we have42n = 21 ⋅ 100 Extremes/means property42n = 2,100 Simplify the right siden = 50 Divide each side by 42Since n is a percent, our answer is 50%, giving us the same result we obtained inExample 2.Answers8. a. 18.5 b. 379. a. 25% b. 50%
318Chapter 5 <strong>Percent</strong>10. Rework Practice Problem 3using proportions.Example 1025 is 40% of what number?Solution This is the same problem we worked in Example 3. We let n be thenumber in question. The base is n and the amount is 25. We compare 25 to n aswe compare 40 to 100. Our proportion sets up as follows:40 is to 100 as 25 is to n{40h h25{h_40100 = 25 _n 100 nNoteWhen you work theproblems in the problemset, use whichevermethod you like, unless yourinstructor indicates that you areto use one method instead of theother.Solving the proportion, we have40 ⋅ n = 25 ⋅ 100 Extremes/means property40 ⋅ n = 2,500 Simplify the right siden = 62.5 Divide each side by 40So, 25 is 40% of 62.5, which is the same result we obtained in Example 3.Getting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.1. When we translate a sentence such as “What number is 15% of 63?” intosymbols, what does each of the following translate to?a. is b. of c. what number2. Look at Example 1 in your text and answer the question below.The number 9.45 is what percent of 63?3. Show that the answer to the question below is the same as the answerto the question in Example 2 of your text.The number 21 is what percent of 42?4. If 21 is 50% of 42, then 21 is what percent of 84?Answer10. a. 87.5 b. 175
5.2 Problem Set319Problem Set 5.2A C Solve each of the following problems. [Examples 1–6]1. What number is 25% of 32?82. What number is 10% of 80?83. What number is 20% of 120?244. What number is 15% of 75?11.255. What number is 54% of 38?20.526. What number is 72% of 200?1447. What number is 11% of 67?7.378. What number is 2% of 49?0.989. What percent of 24 is 12?50%10. What percent of 80 is 20?25%11. What percent of 50 is 5?10%12. What percent of 20 is 4?20%13. What percent of 36 is 9?25%14. What percent of 70 is 14?20%15. What percent of 8 is 6?75%16. What percent of 16 is 9?56.25%17. 32 is 50% of what number?6418. 16 is 20% of what number?8019. 10 is 20% of what number?5020. 11 is 25% of what number?4421. 37 is 4% of what number?92522. 90 is 80% of what number?112.523. 8 is 2% of what number?40024. 6 is 3% of what number?200
320Chapter 5 <strong>Percent</strong>A C The following problems can be solved by the same method you used in Problems 1–24. [Examples 1–6]25. What is 6.4% of 87?5.56826. What is 10% of 102?10.227. 25% of what number is 30?12028. 10% of what number is 22?22029. 28% of 49 is what number?13.7230. 97% of 28 is what number?27.1631. 27 is 120% of what number?22.532. 24 is 150% of what number?1633. 65 is what percent of 130?50%34. 26 is what percent of 78?33_ 1 3 %35. What is 0.4% of 235,671?942.68436. What is 0.8% of 721,423?5,771.38437. 4.89% of 2,000 is what number?97.838. 3.75% of 4,000 is what number?15039. Write a basic percent problem, the solution to whichcan be found by solving the equation n = 0.25(350).What number is 25% of 350?40. Write a basic percent problem, the solution to which canbe found by solving the equation n = 0.35(250).What number is 35% of 250?41. Write a basic percent problem, the solution to whichcan be found by solving the equation n ⋅ 24 = 16.What percent of 24 is 16?42. Write a basic percent problem, the solution to which canbe found by solving the equation n ⋅ 16 = 24.What percent of 16 is 24?43. Write a basic percent problem, the solution to whichcan be found by solving the equation 46 = 0.75 ⋅ n.46 is 75% of what number?44. Write a basic percent problem, the solution to which canbe found by solving the equation 75 = 0.46 ⋅ n.75 is 46% of what number?
5.2 Problem Set321B Applying the Concepts [Example 7]Nutrition For each nutrition label in Problems 45–48, find what percent of the total number of calories comes from fat calories.Then indicate whether the label is from a food considered healthy by the American Dietetic Association. Round to thenearest tenth of a percent if necessary.45. SpaghettiNutrition FactsServing Size 2 oz. (56g per 1/8 of pkg) dryServings Per Container: 8Amount Per ServingCalories 210Calories from fat 10% Daily Value*Total Fat 1g2%Saturated Fat 0g0%Polyunsaturated Fat 0.5gMonounsaturated Fat 0gCholesterol 0mg0%Sodium 0mg0%Total Carbohydrate 42g14%Dietary Fiber 2g7%Sugars 3gProtein 7gVitamin A 0% • Vitamin C 0%Calcium 0% •Iron 10%Thiamin 30% • Riboflavin 10%Niacin 15% •*<strong>Percent</strong> Daily Values are based on a 2,000calorie diet46. Canned Italian tomatoesNutrition FactsServing Size 1/2 cup (121g)Servings Per Container: about 3 1/2Amount Per ServingCalories 25Total Fat 0gSaturated Fat 0gCholesterol 0mgSodium 300mgPotassium 145mgTotal Carbohydrate 4gDietary Fiber 1gSugars 4g0% calories from fat; healthyCalories from fat 0% Daily Value*0%0%0%12%4%2%4%Protein 1gVitamin A 20% • Vitamin C 15%Calcium 4% •Iron 15%*<strong>Percent</strong> Daily Values are based on a 2,000calorie diet.4.8% calories from fat; healthy47. Shredded Romano cheeseNutrition FactsServing Size 2 tsp (5g)Servings Per Container: 34Amount Per ServingCalories 20Total Fat 1.5gSaturated Fat 1gCholesterol 5mgSodium 70mgTotal Carbohydrate 0gFiber 0gSugars 0gCalories from fat 10% Daily Value*2%5%2%3%0%0%Protein 2gVitamin A 0% • Vitamin C 0%Calcium 4% •Iron 0%*<strong>Percent</strong> Daily Values are based on a 2,000calorie diet.48. Tortilla chipsNutrition FactsServing Size 1 oz (28g/About 12 chips)Servings Per Container: about 2Amount Per ServingCalories 140Calories from fat 60% Daily Value*Total Fat 7g1%Saturated Fat 1g6%Cholesterol 0mg0%Sodium 170mg7%Total Carbohydrate 18g6%Dietary Fiber 1g4%Sugars less than 1gProtein 2gVitamin A 0% • Vitamin C 0%Calcium 4% •Iron 2%*<strong>Percent</strong> Daily Values are based on a 2,000calorie diet.50% calories from fat; not healthy42.9% calories from fat; not healthy
Introduction . . .General Applications of <strong>Percent</strong>As you know from watching television and reading the newspaper, we encounterpercents in many situations in everyday life. A recent newspaper article discussingthe effects of a cholesterol-lowering drug stated that the drug in question“lowered levels of LDL cholesterol by an average of 35%.” As we progress throughthis chapter, we will become more and more familiar with percent. As a result, wewill be better equipped to understand statements like the one above concerningcholesterol.In this section we continue our study of percent by doing more of the translationsthat were introduced in Section 5.2. The better you are at working the problemsin Section 5.2, the easier it will be for you to get started on the problems inthis section.AApplications Involving <strong>Percent</strong>Example 1On a 120-question test, a student answered 96 correctly.What percent of the problems did the student work correctly?Solution We have 96 correct answers out of a possible 120. The problem canbe restated as96 is what percent of 120?m8m8m8m888888888m88888888896 = n ⋅ 120_96 = n_ ⋅ 120 Divide both sides by 120120 120n = _96 Switch the left and right120sides of the equationn = 0.80 Divide 96 by 120= 80% Rewrite as a percent5.3ObjectivesA Solve application problemsinvolving percent.Practice Problems1. On a 150-question test, a studentanswered 114 correctly.What percent of the problemsdid the student work correctly?Instructor NoteAs you can see, the key to thismethod of solving application problemsis to translate the problem intoone of the three basic percent questionsfrom the previous section.When we write a test score as a percent, we are comparing the original score toan equivalent score on a 100-question test. That is, 96 correct out of 120 is thesame as 80 correct out of 100.Example 2How much HCl (hydrochloric acid) is in a 60-milliliter bottlethat is marked 80% HCl?Solution If the bottle is marked 80% HCl, that means 80% of the solution isHCl and the rest is water. Because the bottle contains 60 milliliters, we can restatethe question as:2. How much HCl is in a40-milliliter bottle that ismarked 75% HCl?What is 80% of 60?gn = 0.80 ⋅ 60g ggn = 48gHCL 80%60 mlThere are 48 milliliters of HCl in 60 milliliters of 80% HCl solution.5.3 General Applications of <strong>Percent</strong>Answers1. 76% 2. 30 milliliters323
324Chapter 5 <strong>Percent</strong>3. If 42% of the students in a certaincollege are female andthere are 3,360 female students,what is the total number of studentsin the college?Example 3If 48% of the students in a certain college are female andthere are 2,400 female students, what is the total number of students in thecollege?Solution We restate the problem as:There are 5,000 students.2,400 is 48% of what number?g g g g2,400 = 0.48 ⋅ ng_2,4000.48 = _ 0.48⋅ n Divide both sides by 0.48 0.48n = _2,400 Switch the left and right0.48sides of the equationn = 5,0004. Suppose in Example 4 that 35%of the students receive a gradeof A. How many of the 300 studentsis that?Example 4If 25% of the students in elementary algebra coursesreceive a grade of A, and there are 300 students enrolled in elementary algebrathis year, how many students will receive A’s?Solution After reading the question a few times, we find that it is the same asthis question:What number is 25% of 300?g g ggn = 0.25 ⋅ 300n = 75Thus, 75 students will receive A’s in elementary algebra.gAlmost all application problems involving percents can be restated as one ofthe three basic percent problems we listed in Section 5.2. It takes some practicebefore the restating of application problems becomes automatic. You may have toreview Section 5.2 and Examples 1–4 above several times before you can translateword problems into mathematical expressions yourself.Getting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.1. On the test mentioned in Example 1, how many questions would thestudent have answered correctly if she answered 40% of the questionscorrectly?2. If the bottle in Example 2 contained 30 milliliters instead of 60, whatwould the answer be?3. In Example 3, how many of the students were male?4. How many of the students mentioned in Example 4 received a gradelower than A?Answers3. 8,000 students 4. 105 students
5.3 Problem Set325Problem Set 5.3A Solve each of the following problems by first restating it as one of the three basic percent problems of Section 5.2. Ineach case, be sure to show the equation. [Examples 1–4]1. Test Scores On a 120-question test a student answered84 correctly. What percent of the problems did the studentwork correctly?70%2. Test Scores An engineering student answered 81 questionscorrectly on a 90-question trigonometry test. Whatpercent of the questions did she answer correctly? Whatpercent were answered incorrectly?90% correct; 10% incorrect3. Basketball A basketball player made 63 out of 75 freethrows. What percent is this?84%4. Family Budget A family spends $450 every month onfood. If the family’s income each month is $1,800, whatpercent of the family’s income is spent on food?25%5. Chemistry How much HCl (hydrochloric acid) is in a60-milliliter bottle that is marked 75% HCl?45mL6. Chemistry How much acetic acid is in a 5-liter containerof acetic acid and water that is marked 80% acetic acid?How much is water?4 liters acetic acid; 1 liter water7. Farming A farmer owns 28 acres of land. Of the 28acres, only 65% can be farmed. How many acres areavailable for farming? How many are not available forfarming?18.2 acres for farming; 9.8 acres are not available for farming8. Number of Students Of the 420 students enrolled in abasic math class, only 30% are first-year students. Howmany are first-year students? How many are not?126 are first-year students; 294 are not9. Determining a Tip Servers and wait staff are often paidminimum wage and depend on tips for much of theirincome. It is common for tips to be 15% to 20% of thebill. After dinner at a local restaurant the total bill is$56.00. Since your service was above average youdecide to give a 20% tip. Determine the amount of thetip you leave for your server.$11.2010. Determining a Tip Suppose you decide to leave a 15% tipfor services after your dinner out in the preceding problem.How much of a tip did you leave your server? Howmuch smaller was the tip?$8.40; $2.80 less11. Voting In the 2004 Presidential election, George Bushreceived 53.25% of the total electoral votes and JohnKerry received 46.75% of the total electoral votes. Ifthere were 537 total votes cast by the Electoral Collegehow many electoral votes did each candidate receive?Round to the nearest vote.Bush 286; Kerry 25112. Census Data According to the U.S. Census Bureau,national population estimates grouped by age andgender for July, 2006, approximately 7.4% of the147,512,152 males in our population are between theages of 15 and 19 years old. How many males are in thisage group? Round to the nearest person.10,915,899
326Chapter 5 <strong>Percent</strong>13. Bachelors According to the U.S. Census Bureau data forthe number of marriages in 2004 approximately 31.2%of the 109,830,000 males age 15 years or older havenever been married. How many males age 15 years orolder have never been married?34,266,96014. Bachelorettes According to the U.S. Census Bureau datafor the number of marriages in 2004, approximately25.8% of the 117,677,000 females age 15 years or olderhave never been married. How many females age 15years or older have never been married?30,360,66615. Number of Students If 48% of the students in a certaincollege are female and there are 1,440 female students,what is the total number of students in thecollege?3,000 students16. Mixture Problem A solution of alcohol and water is 80%alcohol. The solution is found to contain 32 millilitersof alcohol. How many milliliters total (both alcohol andwater) are in the solution?40 mL17. Number of Graduates Suppose 60% of the graduatingclass in a certain high school goes on to college. If 240students from this graduating class are going on tocollege, how many students are there in the graduatingclass?400 students18. Defective Parts In a shipment of airplane parts, 3% areknown to be defective. If 15 parts are found to bedefective, how many parts are in the shipment?500 parts19. Number of Students There are 3,200 students at ourschool. If 52% of them are female, how many femalestudents are there at our school?1,664 female students20. Number of Students In a certain school, 75% of the studentsin first-year chemistry have had algebra. If thereare 300 students in first-year chemistry, how many ofthem have had algebra?225 students21. Population In a city of 32,000 people, there are 10,000people under 25 years of age. What percent of thepopulation is under 25 years of age?31.25%22. Number of Students If 45 people enrolled in a psychologycourse but only 35 completed it, what percent of the studentscompleted the course? (Round to the nearest tenthof a percent.)77.8%
5.3 Problem Set327Calculator ProblemsThe following problems are similar to Problems 1–22. They should be set up the same way. Then the actual calculationsshould be done on a calculator.23. Number of People Of 7,892 people attending an outdoorconcert in Los Angeles, 3,972 are over 18 years of age.What percent is this? (Round to the nearest wholenumberpercent.)50%24. Manufacturing A car manufacturer estimates that 25%of the new cars sold in one city have defective enginemounts. If 2,136 new cars are sold in that city, how manywill have defective engine mounts?534 cars25. Population The map shows the most populated citiesin the United States. If the population of New YorkCity is about 42% of the state’s population, what is theapproximate population of the state?About 19.2 million26. Prom The graph shows how much girls plan to spendon the prom. If 5,086 girls were surveyed, how manyare planning on spending less than $200 on the prom?Round to the nearest whole number.1,475 girlsWhere Is Everyone?The Cost of Looking GoodLos Angeles, CA3.80San Diego, CA 1.26Phoeniz, AZ 1.37Dallas, TX 1.21Houston, TX 2.01Chicago, IL2.89Philadelphia, PA 1.49New York City, NY* Polpulation in MillionsSource: U.S. Census Bureau8.08Less than $200$200 - $400$400 - $60019%More than $600 11%Takin’ out a loan 7%Source: www.thepromsite.com 5,086 total votes29%34%Multiply.Getting Ready for the Next Section27. 0.06(550)3328. 0.06(625)37.529. 0.03(289,500)8,68530. 0.03(115,900)3,477Divide. Write your answers as decimals.31. 5.44 ÷ 0.0413632. 4.35 ÷ 0.0314533. 19.80 ÷ 3960.0534. 11.82 ÷ 1970.0635. 1,836 _0.12 15,30036. 115 _0.1 1,15037. 90 _600 0.1538. 105 _750 0.14
328Chapter 5 <strong>Percent</strong>Maintaining Your SkillsThe problems below review multiplication with fractions and mixed numbers.Multiply.39. _1 2 ⋅ 2 _5 40. _3 4 ⋅ 1 _3 41. _3 4 ⋅ 5 _9 42. _5 6_1 5 _ 1 4 _ 512 _ 1013 ⋅ 12_13 43. 2 ⋅ _3 8 44. 3 ⋅ _512 45. 1_1 4 ⋅ 8 _15 46. 2_1 3 ⋅ 9 _10 _3 4 _ 5 4 _ 2 3 2_ 110 Extending the Concepts: Batting AveragesBatting averages in baseball are given as decimal numbers, rounded to the nearest thousandth.For example, at the end of June 2008, Milton Bradley had the highest batting average in theAmerican League. At that time, he had 76 hits in 235 times at bat. His batting average was .323,which is found by dividing the number of hits by the number of times he was at bat and thenrounding to the nearest thousandth.number of hitsBatting average = ___= 76_= 0.323number of times at bat 235Because we can write any decimal number as a percent, we can convert batting averages topercents and use our knowledge of percent to solve problems. Looking at Milton Bradley’s battingaverage as a percent, we can say that he will get a hit 32.3% of the times he is at bat.Each of the following problems can be solved by converting batting averages to percents andtranslating the problem into one of our three basic percent problems. (All numbers are fromthe end of June 2008.)47. Chipper Jones had the highest batting average in theNational League with 100 hits in 254 times at bat.What percent of the time Chipper Jones is at bat canwe expect him to get a hit?39.4%, to the nearest tenth of a percent48. Sammy Sosa had 104 hits in 412 times at bat. What percentof the time can we expect Sosa to get a hit?25.2%, to the nearest tenth of a percent49. Barry Bonds was batting .276. If he had been at bat340 times, how many hits did he have? (Rememberhis batting average has been rounded to the nearestthousandth.)94 hits50. Joe Mauer was batting .321. If he had been at bat265 times, how many hits did he have? (Remember,his batting average has been rounded to the nearestthousandth.)85 hits51. How many hits must Milton Bradley have in his next50 times at bat to maintain a batting average of atleast .323?At least 16 hitsProblems 51 and 52 are easiest to solve if students think in terms ofcumulative batting average instead of the batting average for the next50 times at bat.52. How many hits must Chipper Jones have in his next 50times at bat to maintain a batting average of at least.394?At least 20 hits
330Chapter 5 <strong>Percent</strong>3. Suppose the purchase price oftwo speakers is $197 and thesales tax is $11.82. What is thesales tax rate?Example 3Suppose the purchase price of a stereo system is $396 andthe sales tax is $19.80. What is the sales tax rate?Solution We restate the problem as:$19.80 is what percent of $396?g g19.80 = n ⋅ 396To solve this equation, we divide both sides by 396:The sales tax rate is 5%.gm888888888m88888888_ 19.80 = n_ ⋅ 396 Divide both sides by 396396 396n = _19.80 Switch the left and right396sides of the equationn = 0.05 Dividen = 5% 0.05 = 5%Instructor NoteAs was the case with sales tax andsales tax rate, students have a tendencyto confuse commission andcommission rate.B CommissionMany salespeople work on a commission basis. That is, their earnings are a percentageof the amount they sell. The commission rate is a percent, and the actualcommission they receive is a dollar amount.4. A real estate agent gets 3% ofthe price of each house shesells. If she sells a house for$115,000, how much moneydoes she earn?Example 4A real estate agent gets 3% of the price of each house shesells. If she sells a house for $289,500, how much money does she earn?Solution The commission is 3% of the price of the house, which is $289,500.We restate the problem as:What is 3% of $289,500?gn = 0.03 ⋅ 289,500n = 8,685The commission is $8,685.gg g g5. An appliance salesperson’scommission rate is 10%. If thecommission on one of the ovensis $115, what is the purchaseprice of the oven?Example 5Suppose a car salesperson’s commission rate is 12%. If thecommission on one of the cars is $1,836, what is the purchase price of the car?Solution12% of the sales price is $1,836. The problem can be restated as:12% of what number is $1,836?888n 8888n88nm888m880.12 ⋅ n = 1,836_0.12⋅ n= _1,836 Divide both sides by 0.12 0.12 0.12n = 15,300The car sells for $15,300.Answers3. 6% 4. $3,450 5. $1,150
5.4 Sales Tax and Commission331Example 6If the commission on a $600 dining room set is $90, whatis the commission rate?Solution The commission rate is a percentage of the selling price. What wewant to know is:6. If the commission on a $750sofa is $105, what is the commissionrate?$90 is what percent of $600?888888n888888n8888nm88890 = n ⋅ 600m888_90 = n_ ⋅ 600 Divide both sides by 600600 600n = _90 Switch the left and right600sides of the equationn = 0.15n = 15%DivideChange to a percentThe commission rate is 15%.Getting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.1. Explain the difference between the sales tax and the sales tax rate.2. Rework Example 1 using a sales tax rate of 7% instead of 6%.3. Suppose the bicycle in Example 2 was purchased in California, wherethe sales tax rate in 2008 was 7.25%. How much more would the bicyclehave cost?4. Suppose the car salesperson in Example 5 receives a commission of$3,672. Assuming the same commission rate of 12%, how much does thiscar sell for?Answer6. 14%
5.4 Problem Set333Problem Set 5.4A These problems should be solved by the method shown in this section. In each case show the equation needed tosolve the problem. Write neatly, and show your work. [Examples 1–3]1. Sales Tax Suppose the sales tax rate in Mississippi is 7%of the purchase price. If a new food processor sells for$750, how much is the sales tax?$52.502. Sales Tax If the sales tax rate is 5% of the purchase price,how much sales tax is paid on a television that sells for$980?$493. Sales Tax and Purchase Price Suppose the sales tax ratein Michigan is 6%. How much is the sales tax on a $45concert ticket? What is the total price?$2.70; $47.704. Sales Tax and Purchase Price Suppose the sales tax rate inHawaii is 4%. How much tax is charged on a new car ifthe purchase price is $16,400? What is the total price?$656; $17,0565. Total Price The sales tax rate is 4%. If the sales tax ona 10-speed bicycle is $6, what is the purchase price?What is the total price?$150; $1566. Total Price The sales tax on a new microwave oven is$30. If the sales tax rate is 5%, what is the purchaseprice? What is the total price?$600; $6307. Tax Rate Suppose thepurchase price of a diningroom set is $450. If thesales tax is $22.50, what isthe sales tax rate?5%8. Tax Rate If the purchase price ofa bottle of California wine is$24 and the sales tax is $1.50,what is the sales tax rate?6 _ 1 %, or 6.25%49. Energy The chart shows the cost to install either solarpanels or a wind turbine. A farmer is installing theequipment to generate energy from the wind. If helives in a state that has a 6% sales tax rate, how muchdid the farmer pay in sales tax on the total equipmentcost?$420.90Solar Versus Wind Energy CostsEquipment Cost:Modules $6200Fixed Rack $1570Charge Controller $971Cable $440TOTAL $9181Equipment Cost:Turbine $3300Tower $3000Cable $715TOTAL $701510. Prom The graph shows how much guys plan to spendon prom. The sum of the tax on all the expenses a guyhad for prom was $15.75. If he lived in a state that has asales tax rate of 7.5%, what spending bracket would hehave been in?He spent $210, so he would be in the $200 to $300 bracket.Handsome At What Cost?0 - $10019%$100 - $20027%$200 - $30020%$300 - $400Takin’ out a loan17%17%Source: www.thepromsite.com 636 total votesSource: a Limited 2006
334Chapter 5 <strong>Percent</strong>B [Examples 4–6]11. Commission A real estate agent has a commission rateof 3%. If a piece of property sells for $94,000, what isher commission?$2,82012. Commission A tire salesperson has a 12% commissionrate. If he sells a set of radial tires for $400, what is hiscommission?$4813. Commission and Purchase Price Suppose a salespersongets a commission rate of 12% on the lawnmowersshe sells. If the commission on one of the mowers is$24, what is the purchase price of the lawnmower?$20014. Commission and Purchase Price If an appliance salespersongets 9% commission on all the appliances she sells, whatis the price of a refrigerator if her commission is $67.50?$75015. Commission Rate If the commission on an $800 washeris $112, what is the commission rate?14%16. Commission Rate A realtor makes a commission of$11,400 on a $190,000 house he sells. What is his commissionrate?6%17. Phone Bill You recently received your monthly phonebill for service in your local area. The total of the billwas $53.35. You pay $14.36 in surcharges and federaland local taxes. What percent of your phone bill ismade up of surcharges and taxes? Round your answerto the nearest tenth of a percent.26.9%18. Wireless Phone Bill You recently received your Verizonwireless phone bill for the month. The total monthly billis $70.52. Included in that total is $13.27 in surchargesand taxes. What percent of your wireless bill goestowards surcharges and taxes? Round your answer tothe nearest tenth of a percent.18.8%19. Gasoline Tax New York state has one of the highestgasoline taxes in the country. If gas is currently sellingat $4.27 for a gallon of regular gas and the tax rate is14.7%, how much of the price of a gallon of gas goestowards taxes?62.8 cents or $0.62820. Cigarette Tax In an effort to encourage people to quitsmoking, many states place a high tax on a pack ofcigarettes. Nine states place a tax of $2.00 or more ona pack of cigarettes, with New Jersey being the highestat $2.575 per pack. If this is 39% of the cost of a pack ofcigarettes in New Jersey, how much does a single packcost?$6.6021. Salary <strong>Plus</strong> Commission A computer salesperson earnsa salary of $425 a week and a 6% commission on allsales over $4000 each week. Suppose she was ableto sell $6,250 in computer parts and accessories oneweek. What was her salary for the week?$56022. Salary <strong>Plus</strong> Bonus The manager for a computer store ispaid a weekly salary of $650 plus a bonus amounting to1.5% of the net earnings of the store each week. Find hertotal salary for the week when earnings for the store are$26,875.56. Round your answer to the nearest cent.$1053.13
5.4 Problem Set335Calculator ProblemsThe following problems are similar to Problems 1–22. Set them up in the same way, but use a calculator for thecalculations.23. Sales Tax The sales tax rate on a certain item is 5.5%. Ifthe purchase price is $216.95, how much is the salestax? (Round to the nearest cent.)$11.9324. Purchase Price If the sales tax rate is 4.75% and the salestax is $18.95, what is the purchase price? What is thetotal price? (Both answers should be rounded to thenearest cent.)$398.95; $417.9025. Tax Rate The purchase price for a new suit is $229.50.If the sales tax is $10.33, what is the sales tax rate?(Round to the nearest tenth of a percent.)4.5%26. Commission If the commission rate for a mobile homesalesperson is 11%, what is the commission on the saleof a $15,794 mobile home?$1,737.3427. Selling Price Suppose the commission rate on the saleof used cars is 13%. If the commission on one of thecars is $519.35, what did the car sell for?$3,99528. Commission Rate If the commission on the sale of $79.40worth of clothes is $14.29, what is the commission rate?(Round to the nearest percent.)18%Getting Ready for the Next SectionMultiply.29. 0.05(22,000)1,10030. 0.176(1,793,000)315,56831. 0.25(300)7532. 0.12(450)54Divide. Write your answers as decimals.33. 4 ÷ 250.1634. 7 ÷ 350.2Subtract35. 25 − 21436. 1,793,000 − 315,5681,477,43237. 450 − 5439638. 300 − 75225Add.39. 396 + 19.8415.840. 22,000 + 1,10023,100
336Chapter 5 <strong>Percent</strong>Maintaining Your SkillsThe problems below review some basic concepts of division with fractions and mixed numbers.Divide.41. 1 _3 ÷ 2 _3 42. _2_1 2 23 ÷ 1 _3 43. 2 ÷ _3 4 44. 3 ÷ _1 2 2_ 2 3 645. 3 _8 ÷ 1 _4 1_1 2 _ 5 6 4_ 1 2 _ 1 2 46. _5 9 ÷ 2 _3 47. 2_1 4 ÷ 1 _2 48. 1_14 ÷ 21 _2
<strong>Percent</strong> Increase or Decrease and DiscountThe table and bar chart below show some statistics compiled by insurance companiesregarding stopping distances for automobiles traveling at 20 miles perhour on ice.Stopping <strong>Percent</strong>Distance Decrease5.5ObjectivesA Find the percent increase.B Find the percent decrease.C Solve application problemsinvolving the rate of discount.Regular tires 150 ft 0Snow tires 151 ft −1%Studdedsnow tiresReinforcedtire chains120 ft 20%75 ft 50%Source: Copyrighted table courtesy of The CasualtyAdjuster’s GuideStopping distance (feet)16014012010080604020150 151120750RegulartiresSnowtiresStuddedsnowtiresReinforcedtirechainsMany times it is more effective to state increases or decreases as percents, ratherthan the actual number, because with percent we are comparing everything to100.A<strong>Percent</strong> IncreaseExample 1If a person earns $22,000 a year and gets a 5% increase insalary, what is the new salary?Solution We can find the dollar amount of the salary increase by finding 5% of$22,000:Practice Problems1. A person earning $18,000 a yeargets a 7% increase in salary.What is the new salary?0.05 × 22,000 = 1,100The increase in salary is $1,100. The new salary is the old salary plus the raise:$22,000 Old salary+ 1,100 Raise (5% of $22,000)$23,100 New salary5.5 <strong>Percent</strong> Increase or Decrease and DiscountAnswer1. $19,260337
5.5 <strong>Percent</strong> Increase or Decrease and Discount339The discount is $75. The sale price is the original price less the discount:$300 Original price− 75 Less the discount (25% of $300)$225 Sale priceExample 5A man buys a washing machine on sale. The machine usuallysells for $450, but it is on sale at 12% off. If the sales tax rate is 5%, how muchis the total bill for the washer?Solution First we have to find the sale priceof the washing machine, and we begin by findingthe discount:What is 12% of $450?gn = 0.12 ⋅ 450n = 54g g ggSALE12% OFFCome in today for a30 day test trial!WASHING MACHINE5. A woman buys a new coat onsale. The coat usually sells for$45, but it is on sale at 15% off.If the sales tax rate is 5%, howmuch is the total bill for thecoat?The washing machine is marked down $54. The sale price is$450 Original price− 54 Discount (12% of $450)$396 Sale priceBecause the sales tax rate is 5%, we find the sales tax as follows:What is 5% of 396?g g g gn = 0.05 ⋅ 396n = 19.80gThe sales tax is $19.80. The total price the man pays for the washing machine is$396.00 Sale price+ 19.80 Sales tax$415.80 Total priceGetting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.1. Suppose the person mentioned in Example 1 was earning $32,000 peryear and received the same percent increase in salary. How much morewould the raise have been?2. Suppose the shoes mentioned in Example 3 were on sale for $20, insteadof $21. Calculate the new percent decrease in price.3. Suppose a store owner pays $225 for a suit, and then marks it up $75, to$300. Find the percent increase in price.4. Compare your answer to Problem 3 above with the problem given inExample 4 of your text. Do you think it is generally true that a 25% discountis equivalent to a 33_1 % markup?3Answers4. $82.50; $467.50 5. $40.16
5.5 Problem Set341Problem Set 5.5A B Solve each of these problems using the method developed in this section. [Examples 1–3]1. Salary Increase If a person earns $23,000 a year andgets a 7% increase in salary, what is the new salary?$24,6102. Salary Increase A computer programmer’s yearly incomeof $57,000 is increased by 8%. What is the dollar amountof the increase, and what is her new salary?$4,560; $61,5603. Tuition Increase The yearly tuition at a college is presently$6,000. Next year it is expected to increase by17%. What will the tuition at this school be next year?$7,0204. Price Increase A market increased the price of cheese sellingfor $4.98 per pound by 3%. What is the new price for apound of cheese? (Round to the nearest cent.)$5.13 per pound5. Car Value In one year a new car decreased in value by20%. If it sold for $16,500 when it was new, what wasit worth after 1 year?$13,2006. Calorie Content A certain light beer has 20% fewer caloriesthan the regular beer. If the regular beer has 120calories per bottle, how many calories are in the samesizedbottle of the light beer?96 calories7. Salary Increase A person earning $3,500 a month gets araise of $350 per month. What is the percent increasein salary?10%8. Rate Increase A student reader is making $6.50 per hourand gets a $0.70 raise. What is the percent increase?(Round to the nearest tenth of a percent.)10.8%9. Shoe Sale Shoes that usually sell for $25 are on sale for$20. What is the percent decrease in price?20%10. Enrollment Decrease The enrollment in a certain elementaryschool was 410 in 2007. In 2008, the enrollment inthe same school was 328. Find the percent decrease inenrollment from 2007 to 2008.20%11. Students to Teachers The chart shows the student toteacher ratio in the United States from 1975 to 2002.What is the percent decrease in student to teacherratio from 1975 to 2002? Round to the nearest percent.21%Student Per Teacher Ratio In the U.S.197519851995200216.2Source: nces.ed.gov20.417.917.812. Health Care The graph shows the rising cost of healthcare. What is the projected percent increase in healthcare costs from 2002 to 2014?130%Health Care Costs on the RiseBillions of Dollars400032002400160080003,585.72,944.22,399.21,936.51,559.02002 2005 2008 2011 2014Source: Centers for Medicare and Medicaid Services
5.5 Problem Set343Calculator ProblemsSet up the following problems the same way you set up Problems 1–22. Then use a calculator to do the calculations.23. Salary Increase A teacher making $43,752 per year getsa 6.5% raise. What is the new salary?$46,595.8824. Utility Increase A homeowner had a $95.90 electric billin December. In January the bill was $107.40. Find thepercent increase in the electric bill from December toJanuary. (Round to the nearest whole number.)12%25. Soccer The rules for soccer state that the playing field must be from 100 to 120 yards long and 55 to 75 yards wide.The 1999 Women’s World Cup was played at the Rose Bowl on a playing field 116 yards long and 72 yards wide. Thediagram below shows the smallest possible soccer field, the largest possible soccer field, and the soccer field at theRose Bowl.Soccer Fields100 yd116 yd 120 yd55 yd72 yd 75 ydSmallest Rose Bowl Largesta. <strong>Percent</strong> Increase A team plays on the smallest field, then plays in the Rose Bowl. What is the percent increase in thearea of the playing field from the smallest field to the Rose Bowl? Round to the nearest tenth of a percent.51.9%b. <strong>Percent</strong> Increase A team plays a soccer game in the Rose Bowl. The next game is on a field with the largest dimensions.What is the percent increase in the area of the playing field from the Rose Bowl to the largest field? Round tothe nearest tenth of a percent.7.8%26. Football The diagrams below show the dimensions of playing fields for the National Football League (NFL), theCanadian Football League (CFL), and Arena Football.100 ydFootball Fields110 yd153 yd365 yd50yd128 yd3NFL Canadian Arenaa. <strong>Percent</strong> Increase In 1999 Kurt Warner made a successful transition from Arena Football to the NFL, winning theMost Valuable Player award. What was the percent increase in the area of the fields he played on in moving fromArena Football to the NFL? Round to the nearest percent.276%b. <strong>Percent</strong> Decrease Doug Flutie played in the Canadian Football League before moving to the NFL. What was the percentdecrease in the area of the fields he played on in moving from the CFL to the NFL? Round to the nearest tenthof a percent.25.4%
344Chapter 5 <strong>Percent</strong>Getting Ready for the Next SectionMultiply. Round to nearest hundredth if necessary.27. 0.07(2,000)14028. 0.12(8,000)96029. 600(0.04) 1 _6 430. 900(0.06) 1 _4 13.531. 10,150(0.06) 1 _4 152.2532. 10,302.25(0.06) 1 _4 154.53Add.33. 3,210 + 224.73,434.734. 900 + 13.50913.5035. 10,000 + 15010,15036. 10,150 + 152.2510,302.2537. 10,302.25 + 154.5310,456.7838. 10,456.78 + 156.8510,613.63Simplify.39. 2,000 + 0.07(2,000)2,14040. 8,000 + 0.12(8,000)8,96041. 3,000 + 0.07(3,000)3,21042. 9,000 + 0.12(9,000)10,080Maintaining Your SkillsThe problems below review some basic concepts of addition of fractions and mixed numbers.Add each of the following and reduce all answers to lowest terms.43. 1 _3 + 2 _3 144. _3 8 + 1 _8 45. _1 2 + 1 _4 46. _1 5 + 3 _10 _ 1 2 _ 3 4 _ 1 2 47. 3 _4 + 2 _3 48. _3 8 + 1 _6 49. 2_11_512 _ 1324 62 + 31 _2 50. 3 1 _4 + 21 _8 5 3 _8
Example 1InterestAnyone who has borrowed money from a bank or other lending institution, orwho has invested money in a savings account, is aware of interest. Interest is theamount of money paid for the use of money. If we put $500 in a savings accountthat pays 6% annually, the interest will be 6% of $500, or 0.06(500) = $30. Theamount we invest ($500) is called the principal, the percent (6%) is the interestrate, and the money earned ($30) is the interest.A man invests $2,000 in a savings plan that pays 7% peryear. How much money will be in the account at the end of 1 year?Solution We first find the interest by taking 7% of the principal, $2,000:5.6ObjectivesA Solve simple interest problems.B Solve compound interest problems.Practice Problems1. A man invests $3,000 in a savingsplan that pays 8% per year.How much money will be in theaccount at the end of 1 year?Interest = 0.07($2,000)= $140The interest earned in 1 year is $140. The total amount of money in the accountat the end of a year is the original amount plus the $140 interest:$2,000 Original investment (principal)+ 140 Interest (7% of $2,000)$2,140 Amount after 1 yearThe amount in the account after 1 year is $2,140.Example 2A farmer borrows $8,000 from his local bank at 12%. Howmuch does he pay back to the bank at the end of the year to pay off the loan?Solution The interest he pays on the $8,000 isInterest = 0.12($8,000)= $9602. If a woman borrows $7,500from her local bank at 12%interest, how much does shepay back to the bank if she paysoff the loan in 1 year?At the end of the year, he must pay back the original amount he borrowed($8,000) plus the interest at 12%:$8,000 Amount borrowed (principal)+ 960 Interest at 12%$8,960 Total amount to pay backThe total amount that the farmer pays back is $8,960.A Simple InterestThere are many situations in which interest on a loan is figured on other thana yearly basis. Many short-term loans are for only 30 or 60 days. In these caseswe can use a formula to calculate the interest that has accumulated. This type ofinterest is called simple interest. The formula iswhereI = P ⋅ R ⋅ TI = InterestP = PrincipalR = Interest rate (this is the percent)T = Time (in years, 1 year = 360 days)5.6 InterestAnswers1. $3,240 2. $8,400345
346Chapter 5 <strong>Percent</strong>We could have used this formula to find the interest in Examples 1 and 2. Inthose two cases, T is 1. When the length of time is in days rather than years,it is common practice to use 360 days for 1 year, and we write T as a fraction.Examples 3 and 4 illustrate this procedure.3. Another student takes out aloan like the one in Example 3.This loan is for $700 at 4%. Howmuch interest does this studentpay if the loan is paid back in 90days?Example 3A student takes out an emergency loan for tuition, books,and supplies. The loan is for $600 at an interest rate of 4%. How much interestdoes the student pay if the loan is paid back in 60 days?Solution The principal P is $600, the rate R is 4% = 0.04, and the time T is _60360 .Notice that T must be given in years, and 60 days = _60 year. Applying the formula,we have360I = P ⋅ R ⋅ TI = 600 × 0.04 × 60 _360 I = 600 × 0.04 × _1 6 60 _ = 360 1 _I = 4The interest is $4.6 Multiplication4. Suppose $1,200 is deposited inan account that pays 9.5%interest per year. If all themoney is withdrawn after 120days, how much money iswithdrawn?Example 4A woman deposits $900 in an account that pays 6% annually.If she withdraws all the money in the account after 90 days, how much doesshe withdraw?Solution We have P = $900, R = 0.06, and T = 90 days = _90 year. Using360these numbers in the formula, we haveI = P ⋅ R ⋅ TI = 900 × 0.06 × 90 _360 I = 900 × 0.06 × _1 4 90 _360 = 1 _4 I = 13.5MultiplicationThe interest earned in 90 days is $13.50. If the woman withdraws all the money inher account, she will withdraw$900.00 Original amount (principal)+ 13.50 Interest for 90 days$913.50 Total amount withdrawnThe woman will withdraw $913.50.BCompound InterestA second common kind of interest is compound interest. Compound interestincludes interest paid on interest. We can use what we know about simple interestto help us solve problems involving compound interest.5. If $5,000 is put into an accountthat pays 6% compoundedannually, how much money isin the account at the end of 2years?Answers3. $7 4. $1,238Example 5A homemaker puts $3,000 into a savings account that pays7% compounded annually. How much money is in the account at the end of 2years?Solution Because the account pays 7% annually, the simple interest at the endof 1 year is 7% of $3,000:
5.6 Interest347Interest after 1 year = 0.07($3,000)= $210Because the interest is paid annually, at the end of 1 year the total amount ofmoney in the account is$3,000 Original amount+ 210 Interest for 1 year$3,210 Total in account after 1 yearThe interest paid for the second year is 7% of this new total, orInterest paid the second year = 0.07($3,210)= $224.70NoteIf the interest earnedin Example 5 werecalculated using theformula for simple interest,I = P ⋅ R ⋅ T, the amount ofmoney in the account at the end oftwo years would be $3,420.00.At the end of 2 years, the total in the account is$3,210.00 Amount at the beginning of year 2+ 224.70 Interest paid for year 2$3,434.70 Account after 2 yearsAt the end of 2 years, the account totals $3,434.70. The total interest earned duringthis 2-year period is $210 (first year) + $224.70 (second year) = $434.70.You may have heard of savings and loan companies that offer interest ratesthat are compounded quarterly. If the interest rate is, say, 6% and it is compoundedquarterly, then after every 90 days (_1 of a year) the interest is added4to the account. If it is compounded semiannually, then the interest is added tothe account every 6 months. Most accounts have interest rates that are compoundeddaily, which means the simple interest is computed daily and added tothe account.Example 6If $10,000 is invested in a savings account that pays 6%compounded quarterly, how much is in the account at the end of a year?Solution The interest for the first quarter (_1 of a year) is calculated using the4formula for simple interest:6. If $20,000 is invested in anaccount that pays 8% compoundedquarterly, how muchis in the account at the end ofa year?I = P ⋅ R ⋅ TI = $10,000 × 0.06 × _1 First quarter4I = $150At the end of the first quarter, this interest is added to the original principal. Thenew principal is $10,000 + $150 = $10,150. Again we apply the formula to calculatethe interest for the second quarter:I = $10,150 × 0.06 × _1 Second quarter4I = $152.25The principal at the end of the second quarter is $10,150 + $152.25 = $10,302.25.The interest earned during the third quarter isI = $10,302.25 × 0.06 × _1 Third quarter4I = $154.53To the nearest centAnswer5. $5,618
348Chapter 5 <strong>Percent</strong>The new principal is $10,302.25 + $154.53 = $10,456.78. Interest for the fourthquarter isI = $10,456.78 × 0.06 × _1 Fourth quarter4I = $156.85To the nearest centThe total amount of money in this account at the end of 1 year isU s i n g$10,456.78 + $156.85 = $10,613.63T e c h n o l o g yCompound Interest from a FormulaWe can summarize the work above with a formula that allows us to calculatecompound interest for any interest rate and any number of compounding periods.If we invest P dollars at an annual interest rate r, compounded n timesa year, then the amount of money in the account after t years is given by theformulaA = P 1 + r _n nt Using numbers from Example 6 to illustrate, we haveP = Principal = $10,000r = annual interest rate = 0.06n = number of compounding periods = 4 (interest is compoundedquarterly)t = number of years = 1Substituting these numbers into the formula above, we haveNoteThe reason that thisanswer is differentfrom the result weobtained in Example 6 is that, inExample 6, we rounded each calculationas we did it. The calculatorwill keep all the digits in all of theintermediate calculations.A = 10,000 1 + _0.064×14 = 10,000(1 + 0.015) 4= 10,000(1.015) 4To simplify this last expression on a calculator, we haveScientific calculator: 10,000 × 1.015 y x 4 =Graphing calculator: 10,000 × 1.015 ^ 4 ENTERIn either case, the answer is $10,613.63551, which rounds to $10,613.64.Answer6. $21,648.64Getting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.1. Suppose the man in Example 1 invested $3,000, instead of $2,000, in thesavings plan. How much more interest would he have earned?2. How much does the student in Example 3 pay back if the loan is paid offafter a year, instead of after 60 days?3. Suppose the homemaker mentioned in Example 5 invests $3,000 in anaccount that pays only 3_1 % compounded annually. How much is in the2account at the end of 2 years?4. In Example 6, how much money would the account contain at the end of1 year if it were compounded annually, instead of quarterly?
5.6 Problem Set349Problem Set 5.6A These problems are similar to the examples found in this section. They should be set up and solved in the same way.(Problems 1–12 involve simple interest.) [Examples 1–4]1. Savings Account A man invests $2,000 in a savings planthat pays 8% per year. How much money will be in theaccount at the end of 1 year?$2,1602. Savings Account How much simple interest is earned on$5,000 if it is invested for 1 year at 5%?$2503. Savings Account A savings account pays 7% per year.How much interest will $9,500 invested in such anaccount earn in a year?$6654. Savings Account A local bank pays 5.5% annual intereston all savings accounts. If $600 is invested in this type ofaccount, how much will be in the account at the end of ayear?$6335. Bank Loan A farmer borrows $8,000 from his localbank at 7%. How much does he pay back to the bankat the end of the year when he pays off the loan?$8,5606. Bank Loan If $400 is borrowed at a rate of 12% for 1 year,how much is the interest?$487. Bank Loan A bank lends one of its customers $2,000 at8% for 1 year. If the customer pays the loan back at theend of the year, how much does he pay the bank?$2,1608. Bank Loan If a loan of $2,000 at 20% for 1 year is to bepaid back in one payment at the end of the year, howmuch does the borrower pay the bank?$2,4009. Student Loan A student takes out an emergency loanfor tuition, books, and supplies. The loan is for $600with an annual interest rate of 5%. How much interestdoes the student pay if the loan is paid back in 60days?$510. Short-Term Loan If a loan of $1,200 at 9% is paid off in 90days, what is the interest?$2711. Savings Account A woman deposits $800 in a savingsaccount that pays 5%. If she withdraws all the moneyin the account after 120 days, how much does shewithdraw?$813.3312. Savings Account $1,800 is deposited in a savings accountthat pays 6%. If the money is withdrawn at the end of 30days, how much interest is earned?$9
350Chapter 5 <strong>Percent</strong>B The problems that follow involve compound interest. [Examples 5, 6]Compound Interest The chart shows the interest rates for various CD accounts.13. Last year Samuel invested $400 in a 6-month CD. If the interestis compounded quarterly, how much was in the account at theend of 6 months? Round to the nearest cent.$406.34Latest CD Yields (%)4.294.233.164.634.643.324.484.453.344.704.674.1114. If Alice deposited $200 in a 2_1 year CD account earlier this2week, what will the account make at the end of its term if interestis compounded quarterly. Use the compound interest formulaand round to the nearest cent.$223.56This weekLast weekYear agoThis weekLast weekYear agoThis weekLast weekYear agoThis weekLast weekYear ago6-month 1-year 21/2-year 5-yearSource: bankrate.com15. Compound Interest A woman puts $5,000 into a savingsaccount that pays 6% compounded annually. Howmuch money is in the account at the end of 2 years?$5,61816. Compound Interest A savings account pays 5% compoundedannually. If $10,000 is deposited in theaccount, how much is in the account after 2 years?$11,02517. Compound Interest If $8,000 is invested in a savingsaccount that pays 5% compounded quarterly, howmuch is in the account at the end of a year?$8,407.56Some answers may vary in the hundredths column depending onwhether rounding is done in the intermediate steps.18. Compound Interest Suppose $1,200 is invested in a savingsaccount that pays 6% compounded semiannually.How much is in the account at the end of 1_1 2 years?$1,311.27Calculator ProblemsThe following problems should be set up in the same way in which Problems 1–18 have been set up. Then the calculationsshould be done on a calculator.19. Savings Account A woman invests $917.26 in a savingsaccount that pays 6.25% annually. How much is in theaccount at the end of a year?$974.5920. Business Loan The owner of a clothing store borrows$6,210 for 1 year at 11.5% interest. If he pays the loan backat the end of the year, how much does he pay back?$6,924.1521. Compound Interest Suppose $10,000 is invested in each 22. Compound Interest Suppose $5,000 is invested in eachaccount below. In each case find the amount of money in account below. In each case find the amount of money inthe account at the end of 5 years.the account at the end of 10 years.a. Annual interest rate = 6%, compounded quarterly a. Annual interest rate = 5%, compounded quarterly$13,468.55$8,218.10b. Annual interest rate = 6%, compounded monthly b. Annual interest rate = 6%, compounded quarterly$13,488.50$9,070.09c. Annual interest rate = 5%, compounded quarterly c. Annual interest rate = 7%, compounded quarterly$12,820.37$10,007.99d. Annual interest rate = 5%, compounded monthly d. Annual interest rate = 8%, compounded quarterly$12,833.59$11,040.20
5.6 Problem Set351Getting Ready for the Next SectionChange to percent.23. _75250 30%24. 150 _250 60%25. 400 _2,400 _2,400 26. 20016 _ 2 3 % 8 _ 1 3 %Multiply.27. 0.3(360)10828. 0.4(360)14429. 0.45(360)16230. 0.15(360)54Divide.31. 40 ÷ 5832. 45 ÷ 5933. 15 ÷ 5334. 5 ÷ 51Maintaining Your SkillsThe problems below will allow you to review subtraction of fractions and mixed numbers.35. _3 4 − 1 _4 36. _910 − 7 _10 37. _5 8 − 1 _4 38. _710 − 1 _5 _1 2 _ 1 5 _ 3 8 _ 1 2 39. 2 − _4 3 40. 2 + _4 3 41. 1 + _1 2 42. 1 − _1 2 _2 3 _ 103 = 3 _ 1 3 _ 3 2 = 1 _ 1 2 _ 1 2 43. _1 3 − 1 _4 44. _912 − 1 _5 45. 3_1 4 − 246. 5_1_112 _ 1120 _ 5 4 = 1 _ 1 4 _ 23126 − 31 _4 = 111 _12 47. Find the sum of _815 and 8 _. 16_35 21 48. Find the difference of 8 _15 and 8 _35. 32_105 49. Find the product of _815 and 8 _. 64_35 525 50. Find the quotient of 8 _15 and 8 _35 . 7 _3 = 21 _ 3
352Chapter 5 <strong>Percent</strong>Extending the ConceptsThe following problems are percent problems. Use any of the methods developed in this chapter to solve them.51. Credit Card Debt Student credit-card debt is at an alltimehigh. Consolidated Credit Counseling ServicesInc. reports that 20% of all college freshman got theirfirst credit card in high school and nearly 40% signup for one in their first year at college. Suppose yourcredit card company charges 1.3% in finance chargesper month on the average daily balance in your creditcard account. If your average daily balance for thismonth is $2,367.90 determine the finance charge forthe month.$30.7852. Finding Your Interest Rate In early January, your banksent out a form called a 1099-INT, which summarizesthe amount of interest you have received on a savingsaccount for the previous year. If you received $72 interestfor the year on an account in which you started with$1,200, determine the annual interest rate paid by yourbank.6%53. Movie Making The bar chart below shows the production costs for each of the first four Star Wars movies. Find the percentincrease in production costs from each Star Wars movie to the next. Round your results to the nearest tenth.120115Production costs(millions of dollars)10080604020011Star Wars197718The EmpireStrikes Back198032.5Return ofthe Jedi1983The PhantomMenace1999Douglas Kirkland/Corbis<strong>Percent</strong> increase in production costs: Star Wars 1 to 2, 63.6%; Star Wars 2 to 3, 80.6%; Star Wars 3 to 4, 253.8%54. Movie Making The table below shows how much money each of the first four Star Wars movies brought in during thefirst weekend they were shown. Find the percent increase in opening weekend income from each Star Wars movie tothe next. Round to the nearest percent.Opening Weekend IncomeStar Wars (1977) $1,554,000The Empire Strikes Back (1980) $6,415,000Return of the Jedi (1983) $30,490,000The Phantom Menace (1999) $64,810,000<strong>Percent</strong> increase in opening weekend income: Star Wars 1 to 2, 313%; Star Wars 2 to 3, 375%; Star Wars 3 to 4, 113%
Chapter 5 SummaryThe Meaning of <strong>Percent</strong> [5.1]<strong>Percent</strong> means “per hundred.” It is a way of comparing numbers to the number100.EXAMPLEs1. 42% means 42 per hundred or_42100 .Changing <strong>Percent</strong>s to Decimals [5.1]To change a percent to a decimal, drop the percent symbol (%), and move thedecimal point two places to the left.2. 75% = 0.75Changing Decimals to <strong>Percent</strong>s [5.1]To change a decimal to a percent, move the decimal point two places to the right,and use the % symbol.3. 0.25 = 25%Changing <strong>Percent</strong>s to Fractions [5.1]To change a percent to a fraction, drop the % symbol, and use a denominator of100. Reduce the resulting fraction to lowest terms if necessary.4. 6% = 6 _100 = 3 _50 Changing Fractions to <strong>Percent</strong>s [5.1]To change a fraction to a percent, either write the fraction as a decimal and thenchange the decimal to a percent, or write the fraction as an equivalent fractionwith denominator 100, drop the 100, and use the % symbol.5. _3 = 0.75 = 75%4or_9 = 90_= 90%10 100Basic Word Problems Involving <strong>Percent</strong>s [5.2]There are three basic types of word problems:Type A: What number is 14% of 68?Type B: What percent of 75 is 25?Type C: 25 is 40% of what number?To solve them, we write is as =, of as ⋅ (multiply), and what number or what percentas n. We then solve the resulting equation to find the answer to the originalquestion.6. Translating to equations, wehave:Type A: n = 0.14(68)Type B: 75n = 25Type C: 25 = 0.40nChapter 5Summary353
354Chapter 5 <strong>Percent</strong>Applications of <strong>Percent</strong> [5.3, 5.4, 5.5, 5.6]There are many different kinds of application problems involving percent. Theyinclude problems on income tax, sales tax, commission, discount, percentincrease and decrease, and interest. Generally, to solve these problems, werestate them as an equivalent problem of Type A, B, or C from the previous page.Problems involving simple interest can be solved using the formulaI = P ⋅ R ⋅ Twhere I = interest, P = principal, R = interest rate, and T = time (in years). It isstandard procedure with simple interest problems to use 360 days = 1 year.COMMON MISTAKES1. A common mistake is forgetting to change a percent to a decimal whenworking problems that involve percents in the calculations. We alwayschange percents to decimals before doing any calculations.2. Moving the decimal point in the wrong direction when converting percentsto decimals or decimals to percents is another common mistake.Remember, percent means “per hundred.” Rewriting a number expressedas a percent as a decimal will make the numerical part smaller.25% = 0.25
Chapter 5 ReviewWrite each percent as a decimal. [5.1]1. 35%0.352. 17.8%0.1783. 5%0.054. 0.2%0.002Write each decimal as a percent. [5.1]5. 0.9595%6. 0.880%7. 0.49549.5%8. 1.65165%Write each percent as a fraction or mixed number in lowest terms. [5.1]9. 75%10. 4%11. 145%12. 2.5%_3 4 _ 125 1_ 920 _ 140 Write each fraction or mixed number as a percent. [5.1]13. 3 _10 30%14. 5 _8 62.5%15. _2 3 16. 4_3 4 66_ 2 3 % 475%Solve the following problems. [5.2]17. What number is 60% of 28?16.818. What number is 122% of 55?67.119. What percent of 38 is 19?50%20. What percent of 19 is 38?200%21. 24 is 30% of what number?8022. 16 is 8% of what number?20023. Survey Suppose 45 out of 60 people surveyed believea college education will increase a person’s earningpotential. What percent believe this? [5.3]75%24. Discount A lawnmower that usually sells for $175 ismarked down to $140. What is the discount? What is thediscount rate? [5.5]POWER MOWERREGULARLY $175SALE PRICE$140.00$35; 20%Chapter 5Review355
356Chapter 5 <strong>Percent</strong>25. Total Price A sewing machine that normally sells for$600 is on sale for 25% off. If the sales tax rate is 6%,what is the total price of the sewing machine if it ispurchased during the sale? [5.4, 5.5]$47726. Home Mortgage If the interest rate on a home mortgageis 9%, then each month you pay 0.75% of the unpaid balancein interest. If the unpaid balance on one such loanis $60,000 at the beginning of a month, how much interestmust be paid that month? [5.6]$45027. <strong>Percent</strong> Increase At the beginning of the summer, theprice for a gallon of regular gasoline is $4.25. By theend of summer, the price has increased 16%. What isthe new price of a gallon of regular gasoline? Round tothe nearest cent. [5.5]$4.9328. <strong>Percent</strong> Decrease A gallon of regular gasoline is sellingfor $1.45 in September. If the price decreases 14% inOctober, what is the new price for a gallon of regulargasoline? Round to the nearest cent. [5.5]$1.25JUNE21GAS PRICESREGULAR$4.25UNLEADED $4.30SUPER$4.35AUGUST30GAS PRICESREGULAR$ ?UNLEADED $4.51SUPER$4.5729. Medical Costs The table shows the average yearly costof visits to the doctor, as reported in USA Today. Whatis the percent increase in cost from 1990 to 2000?Round to the nearest tenth of a percent. [5.5]55.4% increase30. Commission A real estate agent gets a commission of 6%on all houses he sells. If his total sales for December are$420,000, how much money does he make? [5.4]$25,200Medical CostsYearAverage AnnualCost1990 $5831995 $7392000 $9062005 $1,17231. Discount A washing machine that usually sells for $300is marked down to $240. What is the discount? What isthe discount rate? [5.5]$60; 20% offSALEWASHING MACHINESALE PRICEREGULARLY $300$240.0032. Total Price A tennis racket that normally sells for $240 ison sale for 25% off. If the sales tax rate is 5%, what is thetotal price of the tennis racket if it is purchased duringthe sale? [5.4]$189.00SALETENNIS RACKETREGULARLY $240SALE DISCOUNT25% OFF
Chapter 5 Cumulative ReviewSimplify.1. 6,8015392. 5,038− 2,769+ 3742,2697,714_______4. 1,023 ÷ 15 5. 4.73) 156.09 68.2337. _5 36 _ 125 8. 4.551 + 3.082167.63110. 1.2(0.21)0.25211. 7 _15 ⋅ 5 _14 1 _3. 52(867)45,0846 12. 8 _276. 7 _8 − 5 _8 1 _4 9. 5 − 3.6781.322÷ 20_63 14 _15 13. _3 8 + 7 _12 23 _24 14. 81 _5 − 57 _10 2 _ 1 2 15. 9 ⋅ 42 _3 4216. Subtract 5_3 from 10.375.8517. Find the quotient of 1_1 2 and 1 _4 .618. Translate into symbols, and then simplify: Twice thesum of 2 and 9. 2(2 + 9) = 2219. Write the ratio of 3 to 12 as a fraction in lowest terms._1 4 20. If 1 mile is 5,280 feet, how many feet are there in 2.5miles? 13,200 feet21. If 1 square yard is 1,296 square inches, how manysquare inches are in _1 square yard? 648 in2222. Write _1 as a percent.812.5%23. Convert 46% to a fraction._2350 24. Solve the equation _2 x = 5 _8 3.225. 3 ⋅ 5 2 + 2 ⋅ 4 2 − 5 ⋅ 2 36726. What number is 5% of 32?1.627. 55 is what percent of 275?20%28. 8.8 is 15% of what number?58_2 3 29. Unit Pricing If a six-pack of Coke costs $2.79, what isthe price per can to the nearest cent?47¢30. Unit Pricing A quart of 2% reduced-fat milk containsfour 1-cup servings. If the quart costs $1.61, find theprice per serving to the nearest cent.40¢5(F − 32)31. Temperature Use the formula C = _to find the9temperature in degrees Celsius when the Fahrenheittemperature is 212°F.100°C32. <strong>Percent</strong> Increase Kendra is earning $1,600 a monthwhen she receives a raise to $1,800 a month. What isthe percent increase in her monthly salary?12.5%33. Driving Distance If Ethan drives his car 230 miles in 4hours, how far will he drive in 6 hours if he drives atthe same rate?345 miles34. Movie Tickets A movie theater has a total of 250 seats.If they have a sellout crowd for a matinee and eachticket costs $7.25, how much money will ticket salesbring in that afternoon? $1,812.5035. Geometry Find the perimeter and area of a square withside 8.5 inches.P = 34 in., A = 72.25 in 236. Average If a basketball team has scores of 64, 76, 98,55, and 102 in their first five games, find the meanscore.7937. Hourly Pay Jean tutors in the math lab and earns $56 inone week. If she works 8 hours that week, what is herhourly pay?$7 per hourUse the illustration to answer problem 38.Half Time Gives a Musical BoostAlbum increases after Superbowl halftime performances:2005 Paul McCartneyBack in the U.S.542%2004 Justin TimberlakeJustified2004 Janet JacksonAll For You160%160%2003 No DoubtTragic Kingdom58%2002 U2All That You Can't Leave Behind142%2001 AerosmithGreatest Hits2000 Phil CollinsSerious Hits100%57%0 100 200 300 400 500 600Source: ACNielsen38. If Paul McCartney had sold 1.7 million albums before theSuperbowl, what was the total numbers of albums soldbefore and after the Superbowl? 10.914 million albumsChapter 5 Cumulative Review 357
Chapter 5 TestWrite each percent as a decimal.1. 18%2. 4%0.180.04Write each decimal as a percent.3. 0.5%0.00519. Total Price A tennis racket that normally sells for $280is on sale for 25% off. If the sales tax rate is 5%, whatis the total price of the tennis racket if it is purchasedduring the sale?$220.504. 0.4545%5. 0.770%6. 1.35135%Write each percent as a fraction or a mixed number in lowestterms.7. 65%8. 146%9. 3.5%_1320 1_ 2350 _ 7200 Write each number as a percent.10. _720 35%11. _3 8 37.5%12. 1_3 4 175%13. What number is 75% of 60?4514. What percent of 40 is 18?45%15. 16 is 20% of what number?8016. Driver’s Test On a 25-question driver’s test, a studentanswered 23 questions correctly. What percent of thequestions did the student answer correctly?92%17. Commission A salesperson gets an 8% commission rateon all computers she sells. If she sells $12,000 in computersin 1 day, what is her commission?$96018. Discount A washing machine that usually sells for $250is marked down to $210. What is the discount? What isthe discount rate?$40; 16% offSALETENNIS RACKETREGULARLY $280SALE DISCOUNT20. Simple Interest If $5,000 is invested at 8% simple interestfor 3 months, how much interest is earned?$10021. Compound Interest How much interest will be earnedon a savings account that pays 10% compoundedannually, if $12,000 is invested for 2 years?$2,52022. <strong>Percent</strong> Increase A driver gets into a car accident andhis insurance increases by 12%. If he paid $950 beforethe accident, how much is he paying now?$1,064Use the illustration to answer problem 23.Stuck in the WebSource: Forrester Research, 200525% OFFThe activities that keep consumersonline the longest:Use e-mail: 99%Research products for purchase: 70%Purchase products: 65%Send or recieve photos: 56%Play games alone: 49%Visit daily newspaper sites: 43%23. How many people said they like to play games if 6,000people were surveyed. 2,940 peopleSALEWASHING MACHINEREGULARLY $250SALE PRICE$210.00358Chapter 5 <strong>Percent</strong>
Chapter 5 Projects<strong>Percent</strong>group PROJECTGroup ProjectNumber of PeopleTime NeededEquipmentBackgroundProcedure25 minutesPencil, paper, and calculator.All of us spend time buying clothes and eatingmeals at restaurants. In all of these situations,it is good practice to check receipts. This projectis intended to give you practice creatingreceipts of your own.Fill in the missing parts of each receipt.Sales ReceiptSales ReceiptJeans 29.99Sales Tax (7.75%)Total2 Buffet Dinners @ 9.99 19.98Discount (10%)TotalSales ReceiptSales ReceiptComputer 400.00Discount: 30% offDiscounted PriceCouchSales Tax (7%)Total 588.50Sales Tax (6%)TotalChapter 5Projects359
RESEARCH PROJECTCredit-Card DebtCredit-card companies are now offering creditcardsto college students who would not beable to get a card under normal credit-cardcriteria (due to lack of credit history and lowincome). The credit-card industry sees youngpeople as a valuable market because researchshows that they remain loyal to their first cardsas they grow older. Nellie Mae, the student loanagency, found that 78% of college students hadcredit cards in 2000. For many of these students,lack of financial experience or educationleads to serious debt. According to Nellie Mae,undergraduates with credit-cards carried anaverage balance of $2,748 in 2000. Half ofcredit-card-carrying college students don’t paytheir balances in full every month. Choose acredit-card and find out the minimum monthlypayment and the APR (annual percentage rate).Compute the minimum monthly payment andinterest charges for a balance of $2,748.Stockbyte/SuperStock360Chapter 5 <strong>Percent</strong>
A Glimpse of AlgebraThere is really no direct extension of percent to algebra. Because that is the case,we will go back to some of the algebraic expressions we have encountered previouslyand evaluate them.To evaluate an expression, such as 5x + 4, when we know that x is 7, we simplysubstitute 7 for x in the expression 5x + 4 and then simplify the result.When x = 7the expression 5x + 4becomes 5(7) + 4or 35 + 4= 39Here are some examples.Example 1Find the value of the expression 4x + 3x − 8 when x is 2.SolutionSubstituting 2 for x in the expression, we have:Practice Problems1. Find the value of the expression6x + 3x − 10 when x is 3.4(2) + 3(2) − 8 = 8 + 6 − 8= 14 − 8= 6We say that 4x + 3x − 8 becomes 6 when x is 2.Example 2Find the value of the following expression when a is 5:Solution_4a + 83a − 8 Replacing a with 5 in the expression, we have:_4(5) + 8 = 20_ + 83(5) − 8 15 − 8 = _287 = 42. Find the value of the followingexpression when a is 10:4a + 20_5a − 20 Example 3Find the value of x 2 + 5x + 6 when x is 4.SolutionWhen x is 4, the expression x 2 + 5x + 6 becomes3. Find the value of x 2 + 6x + 8when x is 3.(4) 2 + 5(4) + 6 = 16 + 20 + 6= 42Answers1. 17 2. 2 3. 35A Glimpse of Algebra361
362Chapter 5 <strong>Percent</strong>4. Find the value of the followingexpression when x is 10 andy is 4:_3x + y3x − y Example 4Find the value of 4x + ySolutiony with 2 to get_when x is 5 and y is 2:4x − yThis time we have two different variables. We replace x with 5 and_4(5) + 2 = 20_ + 24(5) − 2 20 − 2 = 22 _18 = 11 _9 5. Find the value of(4x + 1)(4x − 1) when x is 2.Example 5Find the value of (2x + 3)(2x − 3) when x is 4.SolutionReplacing x with 4 in the expression, we have:(2 ⋅ 4 + 3)(2 ⋅ 4 − 3) = (8 + 3)(8 − 3)= (11)(5)= 556. Find the value of the followingexpression when x is 5._x3 + 8x + 2 Example 6Find the value of x3 − 8Solution when x is 5:x − 2_We substitute 5 for x and then simplify:_53 − 8= 125_ − 85 − 2 3= 117 _3 = 39Answers4. _17 5. 63 6. 1913
Problem A Glimpse Set of 5.6 Algebra ProblemsFind the value of each of the following expressions for the given values of the variables.1. 6x + 2x − 7 when x is 293. 4x + 6x + 8x when x is 101804a + 205. _ when a is 55a − 2082a + 3a + 17. __ when a is 38. 7a + a + 44a + 5a + 3_815 9. x 2 + 5x + 6 when x is 22011. x 2 + 10x + 25 when x is 13613. _3x + y when x is 5 and y is 214. 5x − y3x − y_1713 = 14 _13 A Glimpse of Algebra Problems2. 8x + 10x − 5 when x is 3494. 9x + 2x + 20x when x is 51556. _4a + 8 when a is 83a − 82_ 1 2 __ when a is 106a + 2a + 3_ 8483 = 1 _ 183 10. x 2 + 6x + 8 when x is 68012. x 2 + 10x + 25 when x is 025_ when x is 10 and y is 55x + y_ 911 363
364Chapter 5 <strong>Percent</strong>4x + 6y15. _ when x is 5 and y is 46x + 4y16. _8x − 3y when x is 5 and y is 103x + 8y_2223 _ 219 17. (3x + 2)(3x − 2) when x is 414018. (5x + 4)(5x − 4) when x is 28419. (2x + 3) 2 when x is 12520. (2x + 3) 3 when x is 234321. _x3 + 1 when x is 2x + 1322. _x3 − 1 when x is 4x − 121x23. __3 − 8 when x is 3x 2 + 2x + 41x24. __3 + 8 when x is 3x 2 − 2x + 4525. _x4 − 16 when x is 5x 2 + 42126. _x4 − 16 when x is 3x + 213
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