Solving Percent Applications

(1)

Solving Percent Applications

6.5

503

6.5 OBJECTIVES

1. Solve for the unknown amount in a percent problem 2. Solve for the unknown rate in a percent problem 3. Solve for the unknown base in a percent problem

The concept of percent is perhaps the most frequently encountered arithmetic idea that we will consider in this book. In this section, we will show some of the many applications of percent and the special terms that are used in these applications.

To use percents in problem solving, you should always read the problem carefully to de-termine the rate, base, and amount in the problem. This is illustrated in our first example.

Example 1

Solving a Problem Involving an Unknown Amount

A student needs 70% to pass an examination containing 50 questions. How many questions must she get right?

The rateis 70%. The baseis the number of questions on the test, here 50. The amountis the number of questions that must be correct.

To find the amount, we will use the percent proportion from Section 6.4.

B

R

so

Dividing by 100 gives

She must answer 35 questions correctly to pass.

A 3500 100 35 100A 5070 A 50 70 100 C H E C K Y O U R S E L F 1

Generally, 72% of the students in a chemistry class pass the course. If there are 150 students in the class, how many can be expected to pass?

As we said earlier, there are many applications of percent to daily life. One that almost all of us encounter involves interest.When you borrow money, you pay interest. When you place money in a savings account, you earn interest. Interest is a percent of the whole (in this case, the principal), and the percent is called the interest rate.

NOTEA rate, base,and

amountwill appear in all

problems involving percents.

NOTESubstitute 50 for Band 70 for R.

NOTEThe money borrowed or saved is called the principal.

Find the interest you must pay if you borrow $2000 for 1 year with an interest rate of .

91 2%

(2)

504 CHAPTER6 PERCENTS

C H E C K Y O U R S E L F 2

You invest $5000for 1year at 81%. How much interest will you earn? 2

Let’s look at an application that requires finding the rate.

Example 3

Solving a Problem Involving an Unknown Rate

Simon works at a restaurant called La Catalana. The $45 tip he received from a family on Friday was the largest tip Simon had ever received. If the bill totaled $250 before the tip was added, what percent of the total was the tip?

The base is the total of the bill, $250. The amount is the $45 tip. To find the percentage, we again use the percent proportion.

250R45100 250R4500

The tip was 18% of the bill.

R 4500 250 18% 45 250 R 100 C H E C K Y O U R S E L F 3

Last year, Xian reported an income of $27,500on her tax return. Of that, she paid $6,600in taxes. What percent of her income went to taxes?

Now let’s look at an application that requires finding the base.

REMEMBER: % 9.5% 91

2

The base (the principal) is $2000, the rate is , and we want to find the interest (the amount). Using the percent proportion gives

so

or

The interest (amount) is $190. A 19,000 100 190 100A 20009.5 A 2000 9.5 100 91 2%

(3)

SOLVINGPERCENTAPPLICATIONS SECTION6.5 505

Example 4

Solving a Problem Involving an Unknown Base

Ms. Hobson agrees to pay 11% interest on a loan for her new automobile. She is charged $2200 interest on a loan for 1 year. How much did she borrow?

The rate is 11%. The amount, or interest, is $2200. We want to find the base, which is the principal, or the size of the loan. To solve the problem, we have

She borrowed $20,000. B 220,000 11 20,000 11B 2200 100 2200 B 11 100 C H E C K Y O U R S E L F 4

Sue pays $210interest for a 1-year loan at 10.5%. What was the size of her loan?

Percents are used in too many ways for us to list. Look at the variety in the following examples, which illustrate some additional situations in which you will find percents.

Example 5

Solving a Percent Problem

A salesman sells a used car for $9500. His commission rate is 4%. What will be his com-mission for the sale?

The base is the total of the sale, in this problem, $9500. The rate is 4%, and we want to find the commission. This is the amount. By the percent proportion

The salesman’s commission is $380. A 38,000 100 380 100A 4 9500 A 9500 4 100

NOTEA commissionis the amount that a person is paid for a sale.

(4)

Jenny sells a $36,000building lot. If her real estate commission rate is 5%, what commission will she receive for the sale?

Example 6

A clerk sold $3500 in merchandise during 1 week. If he received a commission of $140, what was the commission rate?

The base is $3500, and the amount is the commission of $140. Using the percent pro-portion we have

The commission rate is 4%.

4 R 14,000 3500 3500R 140 100 140 3500 R 100 C H E C K Y O U R S E L F 6

On a purchase of $500you pay a sales tax of $21. What is the tax rate?

Example 7, involving a commission, shows how to find the total sold.

Example 7

A saleswoman has a commission rate of 3.5%. To earn $280, how much must she sell? The rate is 3.5%. The amount is the commission, $280. We want to find the base. In this case, this is the amount that the saleswoman needs to sell.

By the percent proportion

The saleswoman must sell $8000 to earn $280 in commissions. B 28,000 3.5 8000 280 B 3.5 100 or 3.5B 280 100 C H E C K Y O U R S E L F 7

Kerri works with a commission rate of 5.5%. If she wants to earn $825in commis-sions, find the total sales that she must make.

(5)

Example 8

A state taxes sales at 5.5%. How much sales tax will you pay on a purchase of $48? The tax you pay is the amount (the part of the whole). Here the base is the purchase price, $48, and the rate is the tax rate, 5.5%.

Now

The sales tax paid is $2.64.

A 264 100 2.64 A 48 5.5 100 or 100A 48 5.5 NOTEIn an application involving taxes, the tax paid is always the amount.

NOTEThis problem can be done in one step. We’ll look at that method in the calculator section later in this chapter.

NOTE485.5 264

NOTE

Selling priceoriginal cost markup

Suppose that a state has a sales tax rate of %. If you buy a used car for $1200, how much sales tax must you pay?

61 2

Percents are also used to deal with store markups or discounts. Consider Example 9.

Example 9

A store marks up items to make a 30% profit. If an item cost $7.50 from the supplier, what will the selling price be?

The base is the cost of the item, $7.50, and the rate is 30%. In the percent proportion, the markup is the amount in this application.

Then

The markup is $2.25. Finally we have

Selling price $7.50 $2.25 $9.75 Add the cost and the markup to find the selling price.

A 225 100 2.25 A 7.50 30 100 or 100A 30 7.50

(6)

A store wants to discount (or mark down) an item by 25% for a sale. If the original price of the item was $45, find the sale price. [Hint:Find the discount (the amount the item will be marked down), and subtract that from the original price.]

Increases and decreases are often stated in terms of percents, as our next several examples illustrate.

Example 10

The population of a town increased 15% in a 3-year period. If the original population was 12,000, what was the population at the end of the period?

First we find the increase in the population. That increase is the amount in the problem.

To find the population at the end of the period, we add 12,000 1800 13,800

Original population Increase New population

1800 A 180,000 100 A 12,000 15 100 so 100A 15 12,000 C H E C K Y O U R S E L F 1 0

A school’s enrollment decreased by 8%from a given year to the next. If the enroll-ment was 550students the first year, how many students were enrolled the second year?

Example 11

Enrollment at a school increased from 800 to 888 students from a given year to the next. What was the rate of increase?

First we must subtract to find the amount of the increase. Increase: 88880088 students

(7)

NOTEWe use the original

enrollment, 800, as our base.

C H E C K Y O U R S E L F 1 1

Car sales at a dealership decreased from 350units one year to 322units the next. What was the rate of decrease?

Example 12

A company hired 18 new employees in 1 year. If this was a 15% increase, how many em-ployees did the company have before the increase?

The rate is 15%. The amount is 18, the number of new employees. The base in this prob-lem is the number of employees before the increase. So

The company had 120 employees before the increase. 15B 18 100 or B 1800 15 120 18 B 15 100 C H E C K Y O U R S E L F 1 2

A school had 54 new students in one term. If this was a 12% increase over the previous term, how many students were there before the increase?

There are many computer-related applications that include percentages as either part of the problem or as part of the solution.

Now to find the rate, we have

The enrollment increased at a rate of 11%.

R 8800 800 11 88 800 R 100 so 800R 88 100

Solving a Computer Application

A computer loaded 60% of a new program in 120 seconds. How long should it take to load the entire program?

(8)

C H E C K Y O U R S E L F 1 3

A virus scanning program is checking every computer file for viruses. It checked 30% of the files in 240seconds. How long should it take to check all of the files?

C H E C K Y O U R S E L F A N S W E R S

1. 108 2. $425 3. 24% 4. $2000 5. $1800 6. 4.2%

7. $15,000 8. $78 9. $33.75 10. 506 11. 8% 12. 450

13. 800 s (13 min 20 s)

The rate is 60%, the amount is 120 seconds. The base is the total time taken to load the program.

60B12,000

B200

It will take 200 seconds to load the program. 120

B

60 100

(9)

Exercises

Solve each of the following applications.

1. Interest. What interest will you pay on a $3400 loan for 1 year if the interest rate is 12%?

2. Chemistry. A chemist has 300 milliliters (mL) of solution that is 18% acid. How many milliliters of acid are in the solution?

3. Payroll deductions. Roberto has 26% of his pay withheld for deductions. If he earns $550 per week, what amount is withheld?

4. Commission. A real estate agent’s commission rate is 6%. What will be the amount of the commission on the sale of an $85,000 home?

5. Commission. If a salesman is paid a $140 commission on the sale of a $2800 sailboat, what is his commission rate?

6. Interest. Ms. Jordan has been given a loan of $2500 for 1 year. If the interest charged is $275, what is the interest rate on the loan?

7. Interest. Joan was charged $18 interest for 1 month on a $1200 credit card balance. What was the monthly interest rate?

8. Chemistry. There are 117 milliliters (mL) of acid in 900 mL of a solution of acid and water. What percent of the solution is acid?

6.5

(10)

9. Test scores. On a test, Alice had 80% of the problems right. If she had 20 problems correct, how many questions were on the test?

10. Sales tax. A state sales tax rate is 3.5%. If the tax on a purchase is $7, what was the price of the purchase?

11. Loans. Patty pays $525 interest for a 1-year loan at 10.5%. How much was her loan? 12. Commission. A saleswoman is working on a 5% commission basis. If she wants to

make $1800 in 1 month, how much must she sell?

13. Sales tax. A state sales tax is levied at a rate of 6.4%. How much tax would one pay on a purchase of $260?

14. Down payment. Betty must make a down payment on the purchase of a $2000 motorcycle. How much must she pay down?

15. Commission. If a house sells for $125,000 and the commission rate is , how much will the salesperson make for the sale?

16. Test scores. Marla needs 70% on a final test to receive a C for a course. If the exam has 120 questions, how many questions must she answer correctly?

17. Unemployment. A study has shown that 102 of the 1200 people in the workforce of a small town are unemployed. What is the town’s unemployment rate?

18. Surveys. A survey of 400 people found that 66 were left-handed. What percent of those surveyed were left-handed?

(11)

19. Dropout rate. Of 60 people who start a training program, 45 complete the course. What is the dropout rate?

20. Manufacturing. In a shipment of 250 parts, 40 are found to be defective. What percent of the parts are faulty?

21. Surveys. In a recent survey, 65% of those responding were in favor of a freeway improvement project. If 780 people were in favor of the project, how many people responded to the survey?

22. Enrollments. A college finds that 42% of the students taking a foreign language are enrolled in Spanish. If 1512 students are taking Spanish, how many foreign language students are there?

23. Salary. 22% of Samuel’s monthly salary is deducted for withholding. If those deductions total $209, what is his salary?

24. Budgets. The Townsend’s budget 36% of their monthly income for food. If they spend $864 on food, what is their monthly income?

25. Markup. An appliance dealer marks up refrigerators 22% (based on cost). If the cost of one model was $600, what will its selling price be?

26. Enrollments. A school had 900 students at the start of a school year. If there is an enrollment increase of 7% by the beginning of the next year, what is the new enrollment?

27. Land value. A home lot purchased for $26,000 increased in value by 25% over 3 years. What was the lot’s value at the end of the period?

28. Depreciation. New cars depreciate an average of 28% in their first year of use. What will a $9000 car be worth after 1 year?

(12)

29. Enrollment. A school’s enrollment was up from 950 students in 1 year to 1064 students in the next. What was the rate of increase?

30. Salary. Under a new contract, the salary for a position increases from $11,000 to $11,935. What rate of increase does this represent?

31. Markdown. A stereo system is marked down from $450 to $382.50. What is the discount rate?

32. Business. The electricity costs of a business decrease from $12,000 one year to $10,920 the next. What is the rate of decrease?

33. Price changes. The price of a new van has increased $2030, which amounts to a 14% increase. What was the price of the van before the increase?

34. Markdown. A television set is marked down $75, to be placed on sale. If this is a 12.5% decrease from the original price, what was the selling price before the sale?

35. Workforce. A company had 66 fewer employees in July 2001 than in July 2000. If this represents a 5.5% decrease, how many employees did the company have in July 2000?

36. Salary. Carlotta received a monthly raise of $162.50. If this represented a 6.5% increase, what was her monthly salary before the raise?

37. Stock. Mr. Hernandez buys stock for $15,000. At the end of 6 months, the stock’s value has decreased 7.5%. What is the stock worth at the end of the period?

38. Population. The population of a town increases 14% in 2 years. If the population was 6000 originally, what is the population after the increase?

(13)

39. Markup. A store marks up merchandise 25% to allow for profit. If an item costs the store $11, what will its selling price be?

40. Payroll deductions. Tranh’s pay is $450 per week. If deductions from his paycheck average 25%, what is the amount of his weekly paycheck (after deductions)?

41. Computers. A computer loaded 80% of a new program in 80 seconds. How long should it take to load the entire program?

42. Computers. A virus scanning program is checking every file for viruses. It has completed checking 40% of the files in 300 seconds. How long should it take to check all the files?

43. Consumer Affairs. The two ads pictured appeared last week and this week in the local paper. Is this week’s ad accurate?

44. At True Grip hardware, you pay $10 in tax for a barbecue grill, which is 6% of the purchase price. At Loose Fit hardware, you pay $10 in tax for the same grill, but it is 8% of the purchase price. At which store do you get the better buy? Why?

45. Retail Sales. A pair of shorts is advertised for $48.75 and as being 25% off the original price. What was the original price?

46. Tipping. If the total bill at a restaurant, including a 15% tip, is $65.32, what was the cost of the meal alone?

(14)

The following chart shows U.S. trade with Mexico from 1992 to 1997. Use this information for exercises 47 to 50.

U.S. Trade with Mexico, 1992–97

(millions of dollars) MEXICO

Year Exports Imports Trade Balance1

1992 . . . $40,592 . . . $35,211 . . . $5,381 1993 . . . 41,581 . . . 39,917 . . . 1,664 19942_{. . . 50,844 . . . 49,494 . . .} _1,350 1995 . . . 46,292 . . . 61,685 . . . 15,393 1996 . . . 56,792 . . . 74,297 . . . 17,506 1997 . . . 71,388 . . . 85,938 . . . 14,549

(1) Totals may not add due to rounding.

(2) NAFTA provisions began to take effect Jan. 1, 1994.

Source:Office of Trade and Economic Analysis. U.S. Dept. of Commerce.

47. What is the rate of increase (to the nearest whole percent) of exports from 1992 to 1997?

48. What is the rate of increase (to the nearest whole percent) of imports from 1992 to 1997?

49. By what percent did exports exceed imports in 1992?

50. By what percent did imports exceed exports in 1997?

Many percent problems involve calculating what is known as compound interest.

Suppose that you invest $1000 at 5% in a savings account for 1 year. For year 1, the interest is 5% of $1000, or 0.05 $1000 $50. At the end of year 1, you will have $1050 in the account.

At 5% $1000 $1050

Start Year 1

Now if you leave that amount in the account for a second year, the interest will be calculated on the original principal, $1000, plus the first year’s interest, $50. This is called compound interest.

For year 2, the interest is 5% of $1050, or 0.05$1050$52.50. At the end of year 2, you will have $1102.50 in the account.

At 5% At 5%

$1000 $1050 $1102.50

Start Year 1 Year 2

In exercises 51 to 54, assume the interest is compounded annually (at the end of each year), and find the amount in an account with the given interest rate and principal.

51. $4000, 6%, 2 years 52. $3000, 7%, 2 years

(15)

Solve the following applications.

55. Automobiles. In 1990, there were an estimated 145.0 million passenger cars registered in the United States. The total number of vehicles registered in the United States for 1990 was estimated at 194.5 million. What percent of the vehicles registered were passenger cars?

56. Gasoline. Gasoline accounts for 85% of the motor fuel consumed in the United States every day. If 8882 thousand barrels (bbl) of motor fuel are consumed each day, how much gasoline is consumed each day in the United States?

57. Petroleum. In 1989, transportation accounted for 63% of U.S. petroleum

consumption. If 10.85 million bbl of petroleum is used each day for transportation in the United States, what is the total daily petroleum consumption by all sources in the United States?

58. Pollution. Each year, 540 million metric tons (t) of carbon dioxide are added to the atmosphere by the United States. Burning gasoline and other transportation fuels is responsible for 35% of the carbon dioxide emissions in the United States. How much carbon dioxide is emitted each year by the burning of transportation fuels in the United States?

59. The progress of the local Lions club is shown below. What percent of the goal has been achieved so far?

(16)

In exercises 60–64, use the following number line.

60. Length AC is what percent of length AB?

61. Length AD is what percent of AB?

62. Length AE is what percent of AB?

63. Length AE is what percent of AD?

64. Length AC is what percent of AE?

Answers

1. $408 3. $143 5. 5% 7. 1.5% 9. 25 questions 11. $5000 13. $16.64 15. $8125 17. 8.5% 19. 25% 21. 1200 people 23. $950 25. $732 27. $32,500 29. 12% 31. 15% 33. $14,500 35. 1200 employees 37. $13,875 39. $13.75 41. 100 s 43. 45. $65 47. 76% 49. 15% 51. $4494.40 53. $4630.50 55. 74.6% 57. 17.22 million bbl 59. 37.5% 61. 75% 63. 50% A C E D B ANSWERS 60. 61. 62. 63. 64. 518 © 2001 McGraw-Hill Companies

(17)

519

Using Your Calculator to Solve

Percent Problems

In many everyday applications of percent, the computations required become quite lengthy, and so your calculator can be a great help. Let’s look at some examples.

Example 1

Solving a Problem Involving an Unknown Rate

In a test, 41 of 720 lightbulbs burn out before their advertised life of 700 hours (h). What percent of the bulbs fail to last the advertised life?

We know the amount and base and want to find the percent (a rate). Let’s use the percent proportion for the solution.

A

B 720R4100

Now use your calculator to divide

4100 720

5.7% of the lightbulbs fail. We round the result to the nearest tenth of a percent. 5.6944444 720R 720 4100 720 41 720 R 100 Example 2

Solving a Problem Involving an Unknown Base

The price of a particular model of sofa has increased $48.20. If this represents an increase of 9.65%, what was the price before the increase?

We want to find the base (the original price). Again, let’s use the percent proportion for the solution. A R 9.65B4820 9.65B 9.65 4820 9.65 $48.20 B 9.65 100 C H E C K Y O U R S E L F 1

Last month, 35of the 475emergency calls received by the local police department were false alarms. What percent of the calls were false alarms?

1

1 1

(18)

Using the calculator gives

4820 9.65 Round to the nearest cent.

The original price was $499.48. 499.48187

The cost for medical insurance increased $136.40 last year. If this represents a 12.35% increase, what was the cost before the increase?

There is an alternative method for solving percent problems when increases or decreases are involved. Example 3 uses this second approach.

Example 3

A store marks up items 22.5% to allow for profit. If an item costs a store $36.40, what will the selling price be?

Let’s diagram the problem:

Now the base is $36.40 and the rate is 122.5%, and we want to find the amount (the selling price).

so

The selling price should be $44.59. A 122.5 36.40 100 $44.59 A 36.40 122.5 100 Cost Markup 100% 22.5% $36.40 $? Selling price 122.5%

NOTEThis approach may lead to time-consuming hand calculations, but using a calculator reduces the amount of work involved.

An item costs $75.40. If the markup is 36.2%, what is the selling price?

A similar approach will allow us to solve problems that involve a decrease in one step. Earlier we did a similar

example in two steps, finding the markup and then adding that amount to the cost. This method allows you to do the problem in one step.

(19)

Tom buys a baseball card collection for $750. After 1year, the value has decreased 8.2%. What is the value of the collection after 1year?

Example 4

Paul invests $5250 in a piece of property. At the end of a 6-month period, the value has de-creased 7.5%. What is the property worth at the end of the period?

Again, let’s diagram the problem.

So the amount (ending value) is found as

The ending value is $4856.25. A 92.5 $5250 100 $4856.25 A 5250 92.5 100 Original value 100% or $5250

Decrease Ending value

7.5% 100%7.5%92.5%

NOTEEarlier we did a problem like this by finding the decrease and then subtracting from the original value. Again, using this method requires just one step.

C H E C K Y O U R S E L F A N S W E R S

(20)

Calculator Exercises

Solve each of the following percent problems.

1. What percent is 648 of 8640?

2. 53.1875 is 9.25% of what number?

3. Find 7.65% of 375.

4. 17.4 is what percent (to the nearest tenth) of 81.5?

5. Find the base if 18.2% of the base is 101.01.

6. What is 3.52% of 2450?

7. What percent (to the nearest tenth) of 1625 is 182?

8. 22.5% of what number is 3762?

Solve each of the following applications.

9. Sales. What were Jamal’s total sales in a given month if he earned a commission of $2458 at a commission rate of 1.6%?

10. Payroll. A retirement plan calls for a 3.18% deduction from your salary. What amount (to the nearest cent) will be deducted from your pay if your monthly salary is $1675?

11. Salary. You receive a 9.6% salary increase. If your salary was $1225 per month before the raise, how much will your raise be?

12. Manufacturing. In a shipment of 558 parts, 49 are found to be defective. What percent (to the nearest tenth of a percent) of the parts are faulty?

(21)

13. Dropout rate. Statistics show that an average of 42.4% of the students entering a 2-year program will complete their course work. If 588 students completed the program, how many students started?

14. Interest. A time-deposit savings plan gives an interest rate of 6.42% on deposits. If the interest on an account for 1 year was $545.70, how much was deposited?

15. Taxes. The property taxes on a home increased from $832.10 to $957.70 in 1 year. What was the rate of increase (to the nearest tenth of a percent)?

16. Markdown. A dealer marks down the last year’s model appliances 22.5% for a sale. If the regular price of an air conditioner was $279.95, how much will it be discounted (to the nearest cent)?

17. Population. The population of a town increases 4.2% in 1 year. If the original population was 19,500, what is the population after the increase?

18. Markup. A store marks up items 42.5% to allow for profit. If an item costs a store $24.40, what will its selling price be?

19. Markdown. A jacket that originally sold for $98.50 is marked down by 12.5% for a sale. Find its sale price (to the nearest cent).

(22)

20. Salary. Jerry earned $18,500 one year and then received a 10.5% raise. What is his new yearly salary?

21. Payroll deductions. Carolyn’s salary is $1740 per month. If deductions average 24.6%, what is her take-home pay?

22. Investments. Yi Chen made a $6400 investment at the beginning of a year. By the end of the year, the value of the investment had decreased by 8.2%. What was its value at the end of the year?

Answers

1. 7.5% 3. 28.6875 5. 555 7. 11.2% 9. $153,625 11. $117.60 13. 1387 students 15. 15.1% 17. 20,319 19. $86.19 21. $1311.96 ANSWERS 20. 21. 22. 524 © 2001 McGraw-Hill Companies