8 . 1 P e r c e n t , S a l e s Ta x , a n d D i s c o u n t s · 2018. 8. 31. · 1 . 5 . 6 . 7 . 8

8/2/2017 Thinking Mathematically, Sixth Edition

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8 Personal Finance > 8.1 Percent, Sales Tax, and Discounts

8.1 Percent, Sales Tax, and Discounts

What am I Supposed to Learn?After you have read this section, you should be able to:

1 Express a fraction as a percent.

2 Express a decimal as a percent.

3 Express a percent as a decimal.

4 Solve applied problems involving sales tax and discounts.

5 Determine percent increase or decrease.

6 Investigate some of the ways percent can be abused.

“And if elected, it is my solemn pledge to cut your taxes by 10% for each of my first three years in office, for a total cut of 30%.”

PERSONAL FINANCE INCLUDES every area of your life that involves money. It's about what you do with your money and how financial management will affect yourfuture. Because an understanding of percent plays an important role in personal finance, we open the chapter with a discussion on the meaning, uses, and abuses ofpercent.

Basics of PercentPercents are the result of expressing numbers as part of 100. The word percent means per hundred. For example, the circle graph in Figure 8.1 shows that 55 out ofevery 100 college students prefer print textbooks. Thus, indicating that 55% of college students prefer print textbooks. The percent sign, %, is used toindicate the number of parts out of 100 parts.

dFIGURE 8.1 Source: Harris Interactive for Pearson Foundation

A fraction can be expressed as a percent using the following procedure:

Expressing a Fraction as a Percent1. Divide the numerator by the denominator.

2. Multiply the quotient by 100. This is done by moving the decimal point in the quotient two places to the right.

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= 55%,55

100

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3. Add a percent sign.

1 Express a fraction as a percent.

Example 1 Expressing a Fraction as a PercentExpress as a percent.

SOLUTION

Step 1 Divide the numerator by the denominator.

Step 2 Multiply the quotient by 100.

Step 3 Attach a percent sign.

Thus,

Check Point 1Express as a percent.

2 Express a decimal as a percent.

Our work in Example 1 shows that This illustrates the procedure for expressing a decimal number as a percent.

Expressing a Decimal Number as a Percent1. Move the decimal point two places to the right.

2. Attach a percent sign.

Copyright 2017 © Pearson Education, Inc. or its affiliate(s). All rights reserved. Privacy Policy | Terms of Use | Rights and Permissions

5

8

5 ÷ 8 = 0.625

0.625 × 100 = 62.5

62.5%

= 62.5%.5

8

18

0.625 = 62.5%.



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8 Personal Finance > 8.1 Percent, Sales Tax, and Discounts

Great Question!What's the difference between the word percentage and the word percent?

Dictionaries indicate that the word percentage has the same meaning as the word percent. Use the word that sounds better in the circumstance.

Example 2 Expressing a Decimal as a PercentExpress 0.47 as a percent.

SOLUTION

Thus,

Check Point 2Express 0.023 as a percent.

3 Express a percent as a decimal.

We reverse the procedure of Example 2 to express a percent as a decimal number.

Expressing a Percent as a Decimal Number1. Move the decimal point two places to the left.

2. Remove the percent sign.

Example 3 Expressing Percents as DecimalsExpress each percent as a decimal:

a. 19%

b. 180%.

SOLUTION

Use the two steps in the box.

a. d

Thus,

b.

Check Point 3Express each percent as a decimal:

a. 67%

b. 250%.

If a fraction is part of a percent, as in begin by expressing the fraction as a decimal, retaining the percent sign. Then, express the percent as a decimal number.For example,



0.47 = 47%.

19% = 0.19.

180% = 1.80 = 1.80 or 1.8%

%,14











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d

Great Question!Can I expect to have lots of zeros when expressing a small percent as a decimal?

Yes. Be careful with the zeros. For example,

d









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8 Personal Finance > 8.1 Percent, Sales Tax, and Discounts > Percent, Sales Tax, and Discounts

Percent, Sales Tax, and Discounts

4 Solve applied problems involving sales tax and discounts.

Many applications involving percent are based on the following formula:

Note that the word of implies multiplication.

We can use this formula to determine the sales tax collected by states, counties, and cities on sales of items to customers. The sales tax is a percent of the cost of anitem.

Example 4 Percent and Sales TaxSuppose that the local sales tax rate is 7.5% and you purchase a bicycle for $894.

a. How much tax is paid?

b. What is the bicycle's total cost?

SOLUTION

a.

The tax paid is $67.05.

b. The bicycle's total cost is the purchase price, $894, plus the sales tax, $67.05.

The bicycle's total cost is $961.05.

Check Point 4Suppose that the local sales tax rate is 6% and you purchase a computer for $1260.


b. What is the computer's total cost?

None of us is thrilled about sales tax, but we do like buying things that are on sale. Businesses reduce prices, or discount, to attract customers and to reduce inventory.The discount rate is a percent of the original price.

Example 5 Percent and Sales PriceA computer with an original price of $1460 is on sale at 15% off.

a. What is the discount amount?

b. What is the computer's sale price?



.A

A▼

=

is▼

P

P percent▼

⋅of▼

B

B.▼

Sales tax amount = tax rate × item's cost

Sales tax amount

= tax rate × item's cost

= 7.5% × $894 = 0.075 × $894 = $67.05

7.5% of the item's cost,

or 7.5% of $894

▲

Total cost = $894.00 + $67.05 = $961.05

Discount amount = discount rate × original price




















8 Personal Finance > 8.1 Percent, Sales Tax, and Discounts > Percent and Change

SOLUTION

a.

The discount amount is $219.

b. The computer's sale price is the original price, $1460, minus the discount amount, $219.

The computer's sale price is $1241.

TechnologyA calculator is useful, and sometimes essential, in this chapter. The keystroke sequence that gives the sale price in Example 5 is

Press or to display the answer, 1241.

Check Point 5A CD player with an original price of $380 is on sale at 35% off.


b. What is the CD player's sale price?

Great Question!Do I have to determine the discount amount before finding the sale price?

No. For example, in Example 5 the computer is on sale at 15% off. This means that the sale price must be or 85%, of the original price.

Percent and Change

5 Determine percent increase or decrease.

Percents are used for comparing changes, such as increases or decreases in sales, population, prices, and production. If a quantity changes, its percent increase or itspercent decrease can be found as follows:

Finding Percent Increase or Percent Decrease1. Find the fraction for the percent increase or the percent decrease:

2. Find the percent increase or the percent decrease by expressing the fraction in step 1 as a percent.

Example 6 Finding Percent Increase and DecreaseIn 2000, world population was approximately 6 billion. Figure 8.2 shows world population projections through the year 2150. The data are from the United NationsFamily Planning Program and are based on optimistic or pessimistic expectations for successful control of human population growth.



Discount amount

= discount rate × original price

= 15% × $1460 = 0.15 × $1460 = $219

15% of the original price,

or 15% of $1460

▲

Sale price = $1460 − $219 = $1241

1460 .15 1460.− ×

= ENTER

100% − 15%,

Sale price = 85% × $1460 = 0.85 × $1460 = $1241

or .amount of increase

original amount

amount of decrease

original amount












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dFIGURE 8.2 Source: United Nations

a. Find the percent increase in world population from 2000 to 2150 using the high projection data.

b. Find the percent decrease in world population from 2000 to 2150 using the low projection data.









8 Personal Finance > 8.1 Percent, Sales Tax, and Discounts > Percent and Change

FIGURE 8.2 (repeated)

SOLUTION

a. Use the data shown on the blue, high-projection, graph.

The projected percent increase in world population is 400%.

b. Use the data shown on the green, low-projection, graph.

The projected percent decrease in world population is

In Example 6, we expressed the percent decrease as because of the familiar conversion However, in many situations, rounding is needed. Wesuggest that you round to the nearest tenth of a percent. Carry the division in the fraction for percent increase or decrease to four places after the decimal point. Thenround the decimal to three places, or to the nearest thousandth. Expressing this rounded decimal as a percent gives percent increase or decrease to the nearest tenth ofa percent.

Great Question!I know that increasing 2 to 8 is a 300% increase. Does that mean decreasing 8 to 2 is a 300% decrease?

No. Notice the difference between the following examples:

• 2 is increased to 8.

• 8 is decreased to 2.

Although an increase from 2 to 8 is a 300% increase, a decrease from 8 to 2 is not a 300% decrease. A percent decrease involving nonnegative quantities cannever exceed 100%. When a quantity is decreased by 100%, it is reduced to zero.

Check Point 6a. If 6 is increased to 10, find the percent increase.

b. If 10 is decreased to 6, find the percent decrease.

Example 7 Finding Percent DecreaseA jacket regularly sells for $135.00. The sale price is $60.75. Find the percent decrease of the sale price from the regular price.



Percent increase

=

=

amount of increaseoriginal amount

= = 4 = 400%30−6

6246

Percent decrease

=

=

amount of decreaseoriginal amount

= = = 0.33 = 33 %6−4

626

13

13

13

33 %.13

33 %13

= 0.33 .13

13

Percent increase = = = 3 = 300%amount of increase

original amount

6

2

Percent decrease = = = = 0.75 = 75%amount of decrease

original amount

6

8

3

4












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8 Personal Finance > 8.1 Percent, Sales Tax, and Discounts > Abuses of Percent

SOLUTION

The percent decrease of the sale price from the regular price is 55%. This means that the sale price of the jacket is 55% lower than the regular price.

Check Point 7A television regularly sells for $940. The sale price is $611. Find the percent decrease of the sale price from the regular price.

Abuses of Percent

6 Investigate some of the ways percent can be abused.

In our next examples, we look at a few of the many ways that percent can be used incorrectly. Confusion often arises when percent increase (or decrease) refers to achanging quantity that is itself a percent.

Example 8 Percents of PercentsJohn Tesh, while he was still coanchoring Entertainment Tonight, reported that the PBS series The Civil War had an audience of 13% versus the usual 4% PBSaudience, “an increase of more than 300%.” Did Tesh report the percent increase correctly?

SOLUTION

We begin by finding the percent increase.

The percent increase for PBS was 225%. This is not more than 300%, so Tesh did not report the percent increase correctly.

Check Point 8An episode of a television series had an audience of 12% versus its usual 10%. What was the percent increase for this episode?

Example 9 Promises of a PoliticianA politician states, “If you elect me to office, I promise to cut your taxes for each of my first three years in office by 10% each year, for a total reduction of 30%.”Evaluate the accuracy of the politician's statement.

SOLUTION

To make things simple, let's assume that a taxpayer paid $100 in taxes in the year previous to the politician's election. A 10% reduction during year 1 is 10% of $100.

With a 10% reduction the first year, the taxpayer will pay only or $90, in taxes during the politician's first year in office.




Percent decrease

=

=


= = 0.55 = 55%135.00−60.75

135

74.25

135

Percent increase

=

=

amount of increaseoriginal amount

= = = 2.25 = 225%13%−4%

4%

9%

4%

9

4

10% of previous year tax = 10% of $100 = 0.10 × $100 = $10

$100 − $10,

















8 Personal Finance > 8.1 Percent, Sales Tax, and Discounts > Abuses of Percent

The following table shows how we calculate the new, reduced tax for each of the first three years in office:

Year Tax Paid the Year Before 10% Reduction Taxes Paid This Year1 $1002 $903 $81

Now, we determine the percent decrease in taxes over the three years.

The taxes decline by 27.1%, not by 30%. The politician is ill-informed in saying that three consecutive 10% cuts add up to a total tax cut of 30%. In our calculation,which serves as a counterexample to the promise, the total tax cut is only 27.1%.

Check Point 9Suppose you paid $1200 in taxes. During year 1, taxes decrease by 20%. During year 2, taxes increase by 20%.

a. What do you pay in taxes for year 2?

b. How do your taxes for year 2 compare with what you originally paid, namely $1200? If the taxes are not the same, find the percent increase or decrease.

Blitzer BonusTesting Your Financial Literacy

Scores have been falling on tests that measure financial literacy. Here are four items from a test given to high school seniors. Would you ace this one?

1. Which of the following is true about sales taxes?

A. The national sales-tax percentage rate is 6%.

B. The Federal Government will deduct it from your paycheck.

C. You don't have to pay the tax if your income is very low.

D. It makes things more expensive for you to buy.

2. If you have caused an accident, which type of automobile insurance would cover damage to your own car?

A. Comprehensive

B. Liability

C. Term

D. Collision

3. Which of the following types of investment would best protect the purchasing power of a family's savings in the event of a sudden increase in inflation?

A. A 10-year bond issued by a corporation

B. A certificate of deposit at a bank

C. A 25-year corporate bond

D. A house financed with a fixed-rate mortgage



0.10 × $100 = $10 $100 − $10 = $90

0.10 × $90 = $9 $90 − $9 = $81

0.10 × $81 = $8.10 $81 − $8.10 = $72.90

Percent decrease

=

=


= = = 0.271 = 27.1%$100−$72.90

$100

$27.10

$100

27.1

100

58% of high

school seniors

answered incorrectly.

63% of high

school seniors


64% of high

school seniors













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4. Sara and Joshua just had a baby. They received money as baby gifts and want to put it away for the baby's education. Which of the following tends to have thehighest growth over periods of time as long as 18 years?

A. A checking account

B. Stocks

C. A U.S. government savings bond

D. A savings account

Source: The Jump$tart Coalition's 2008 Personal Financial Survey

Answers: 1. D; 2. D; 3. D; 4. B


83% of high

school seniors









8 Personal Finance > 8.1 Percent, Sales Tax, and Discounts > Concept and Vocabulary Check

Concept and Vocabulary CheckFill in each blank so that the resulting statement is true.

1. Percents are the result of expressing numbers as part of ___________________.

2. To express as a percent, divide ___________________ by ___________________, multiply the quotient by ___________________, and attach______________________.

3. To express 0.1 as a percent, move the decimal point ___________________ places to the ___________________ and attach ______________________.

4. To express 7.5% as a decimal, move the decimal point ___________________ places to the _________________ and remove ________________________.

5. To find the sales tax amount, multiply the ___________________ and the ___________________.

6. To find the discount amount, multiply the _______________________ and the ________________________.

7. The numerator of the fraction for percent increase is _______________________________ and the denominator of the fraction for percent increase is____________________________.

8. The numerator of the fraction for percent decrease is ______________________________ and the denominator of the fraction for percent decrease is________________________________.

Exercises 9–10 are based on items from a financial literacy survey from the Center for Economic and Entrepreneurial Literacy. Determine whether each statement is trueor false. If the statement is false, make the necessary change(s) to produce a true statement.

9. Santa had to lay off 25% of his eight reindeer because of the bad economy, so only seven reindeer remained. (65% answered this question incorrectly. Santa mightconsider leaving Thinking Mathematically in stockings across the country.) ______

10. You spent 1% of your $50,000-per-year salary on gifts, so you spent $5000 on gifts for the year. ______

Exercise Set 8.1Practice ExercisesIn Exercises 1–10, express each fraction as a percent.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

In Exercises 11–20, express each decimal as a percent.

11. 0.59

12. 0.96

13. 0.3844

14. 0.003

15. 2.87

16. 9.83

17. 14.87

18. 19.63

19. 100

20. 95



7

8

2

5

3

5

1

4

3

4

3

8

7

8

1

40

3

40

9

80

13

80












In Exercises 21–34, express each percent as a decimal.

21. 72%

22. 38%

23. 43.6%

24. 6.25%

25. 130%

26. 260%

27. 2%

28. 6%

29.

30.

31.

32.

33.

34.

Use the percent formula, A is P percent of B, to solve Exercises 35–38.

35. What is 3% of 200?

36. What is 8% of 300?

37. What is 18% of 40?

38. What is 16% of 90?

Practice PlusThree basic types of percent problems can be solved using the percent formula

Question Given Percent FormulaWhat is P percent of B? P and B Solve for A.A is P percent of what? A and P Solve for B.A is what percent of B? A and B Solve for P.

Exercises 35–38 involved using the formula to answer the first question. In Exercises 39–46, use the percent formula to answer the second or third question.

39. 3 is 60% of what?

40. 8 is 40% of what?

41. 24% of what number is 40.8?

42. 32% of what number is 51.2?

43. 3 is what percent of 15?

44. 18 is what percent of 90?

45. What percent of 2.5 is 0.3?

46. What percent of 7.5 is 0.6?

Application Exercises47. Suppose that the local sales tax rate is 6% and you purchase a car for $32,800.


b. What is the car's total cost?

48. Suppose that the local sales tax rate is 7% and you purchase a graphing calculator for $96.


b. What is the calculator's total cost?

49. An exercise machine with an original price of $860 is on sale at 12% off.


b. What is the exercise machine's sale price?

50. A dictionary that normally sells for $16.50 is on sale at 40% off.


b. What is the dictionary's sale price?

%1

2

%3

4

%5

8

%1

8

62 %1

2

87 %1

2

A = PB:

A = PB.



8 Personal Finance > 8.1 Percent, Sales Tax, and Discounts > Concept and Vocabulary Check

The circle graph shows a breakdown of spending for the average U.S. household using 365 days worked as a basis of comparison. Use this information to solveExercises 51–52. Round answers to the nearest tenth of a percent.

d

Source: The Tax Foundation

51. What percentage of work time does the average U.S. household spend paying for federal taxes?

52. What percentage of work time does the average U.S. household spend paying for state and local taxes?

Although you want to choose a career that fits your interests and abilities, it is good to have an idea of what jobs pay when looking at career options. The bar graphshows the average yearly earnings of full-time employed college graduates with only a bachelor's degree based on their college major. Use this information to solveExercises 53–54. Round all answers to the nearest tenth of a percent.

d

Source: Arthur J. Keown, Personal Finance, Fourth Edition, Pearson, 2007.

53. Find the percent increase in the average yearly earnings from students majoring in social work to students majoring in engineering.

54. Find the percent increase for the average yearly earnings from students majoring in philosophy to students majoring in accounting.

55. A sofa regularly sells for $840. The sale price is $714. Find the percent decrease of the sale price from the regular price.

56. A FAX machine regularly sells for $380. The sale price is $266. Find the percent decrease of the sale price from the regular price.

57. Suppose that you have $10,000 in a rather risky investment recommended by your financial advisor. During the first year, your investment decreases by 30% of itsoriginal value. During the second year, your investment increases by 40% of its first-year value. Your advisor tells you that there must have been a 10% overallincrease of your original $10,000 investment. Is your financial advisor using percentages properly? If not, what is your actual percent gain or loss of your original$10,000 investment?

58. The price of a color printer is reduced by 30% of its original price. When it still does not sell, its price is reduced by 20% of the reduced price. The salespersoninforms you that there has been a total reduction of 50%. Is the salesperson using percentages properly? If not, what is the actual percent reduction from the originalprice?

Writing in Mathematics59. What is a percent?

60. Describe how to express a decimal number as a percent and give an example.

61. Describe how to express a percent as a decimal number and give an example.














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62. Explain how to use the sales tax rate to determine an item's total cost.

63. Describe how to find percent increase and give an example.

64. Describe how to find percent decrease and give an example.

Critical Thinking ExercisesMake Sense? In Exercises 65–68, determine whether each statement makes sense or does not make sense, and explain your reasoning.

65. I have $100 and my restaurant bill comes to $80, which is not enough to leave a 20% tip.

66. I found the percent decrease in a jacket's price to be 120%.

67. My weight increased by 1% in January and 1% in February, so my increase in weight over the two months is 2%.

68. My rent increased from 20% to 30% of my income, so the percent increase is 10%.

69. What is the total cost of a $720 iPad that is on sale at 15% off if the local sales tax rate is 6%?

70. A condominium is taxed based on its $78,500 value. The tax rate is $3.40 for every $100 of value. If the tax is paid before March 1, 3% of the normal tax is givenas a discount. How much tax is paid if the condominium owner takes advantage of the discount?

71. In January, each of 60 people purchased a $500 washing machine. In February, 10% fewer customers purchased the same washing machine that had increasedin price by 20%. What was the change in sales from January to February?

72. When you buy something, it actually costs more than you may think—at least in terms of how much money you must earn to buy it. For example, if you pay 28%of your income in taxes, how much money would you have to earn to buy a used car for $7200?







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8 . 1 P e r c e n t , S a l e s Ta x , a n d D i s c o u n t s · 2018. 8. 31. · 1 . 5 . 6 . 7 . 8