1. Solve for the unknown amount in a percent problem2. Solve for the unknown rate in a percent problem3. Solve for the unknown base in a percent problem
The concept of percent is perhaps the most frequently encountered arithmetic idea that wewill consider in this book. In this section, we will show some of the many applications ofpercent 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 questionsmust she get right?
The rate is 70%. The base is the number of questions on the test, here 50. The amount isthe 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 � 50 � 70
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 150students 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 almostall of us encounter involves interest. When you borrow money, you pay interest. When youplace money in a savings account, you earn interest. Interest is a percent of the whole (inthis case, the principal ), and the percent is called the interest rate.
NOTE A rate, base, andamount will appear in allproblems involving percents.
NOTE Substitute 50 for B and70 for R.
NOTE The money borrowed orsaved is called the principal.
Solving a Problem Involving an Unknown Amount
Find the interest you must pay if you borrow $2000 for 1 year with an interest rate of
You invest $5000 for 1 year at %. How much interest will you earn?812
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 onFriday was the largest tip Simon had ever received. If the bill totaled $250 before the tipwas 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.
250 � R � 45 � 100
250R � 4500
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 $27,500 on her tax return. Of that, she paid$6,600 in taxes. What percent of her income went to taxes?
Now let’s look at an application that requires finding the base.
REMEMBER:
% � 9.5%912
The base (the principal) is $2000, the rate is , and we want to find the interest (theamount). Using the percent proportion gives
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 isthe 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 $210 interest 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 followingexamples, 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 tofind 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
NOTE A commission is theamount that a person is paidfor a sale.
Jenny sells a $36,000 building lot. If her real estate commission rate is 5%, whatcommission will she receive for the sale?
Example 6
Solving a Percent Problem
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 $500 you 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
Solving a Percent Problem
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 $825 in commis-sions, find the total sales that she must make.
Another common application of percents involves tax rates.
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
NOTE In an applicationinvolving taxes, the tax paid isalways the amount.
NOTE This problem can bedone in one step. We’ll look atthat method in the calculatorsection later in this chapter.
NOTE 48 � 5.5 � 264
NOTESelling price � original cost
� markup
C H E C K Y O U R S E L F 8
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?
612
Percents are also used to deal with store markups or discounts. Consider Example 9.
Example 9
Solving a Percent Problem
A store marks up items to make a 30% profit. If an item cost $7.50 from the supplier, whatwill the selling price be?
The base is the cost of the item, $7.50, and the rate is 30%. In the percent proportion, themarkup 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 findthe selling price.
A store wants to discount (or mark down) an item by 25% for a sale. If the originalprice of the item was $45, find the sale price. [Hint: Find the discount (the amountthe 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 severalexamples illustrate.
Example 10
Solving a Percent Problem
The population of a town increased 15% in a 3-year period. If the original population was12,000, 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 550 students the first year, how many students were enrolled the secondyear?
Example 11
Solving a Percent Problem
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.
A virus scanning program is checking every computer file for viruses. It checked 30%of the files in 240 seconds. How long should it take to check all of the files?
9. Test scores. On a test, Alice had 80% of the problems right. If she had 20 problemscorrect, 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 theprice 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 tomake $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 payon 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 , howmuch 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 examhas 120 questions, how many questions must she answer correctly?
17. Unemployment. A study has shown that 102 of the 1200 people in the workforce ofa 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 ofthose surveyed were left-handed?
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. Whatpercent of the parts are faulty?
21. Surveys. In a recent survey, 65% of those responding were in favor of a freewayimprovement project. If 780 people were in favor of the project, how many peopleresponded to the survey?
22. Enrollments. A college finds that 42% of the students taking a foreign language areenrolled in Spanish. If 1512 students are taking Spanish, how many foreign languagestudents are there?
23. Salary. 22% of Samuel’s monthly salary is deducted for withholding. If thosedeductions total $209, what is his salary?
24. Budgets. The Townsend’s budget 36% of their monthly income for food. If theyspend $864 on food, what is their monthly income?
25. Markup. An appliance dealer marks up refrigerators 22% (based on cost). If thecost 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 anenrollment increase of 7% by the beginning of the next year, what is the newenrollment?
27. Land value. A home lot purchased for $26,000 increased in value by 25% over3 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. Whatwill a $9000 car be worth after 1 year?
29. Enrollment. A school’s enrollment was up from 950 students in 1 year to 1064students 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 thediscount 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 a14% 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 a12.5% 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. Ifthis represents a 5.5% decrease, how many employees did the company have in July2000?
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’svalue 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 populationwas 6000 originally, what is the population after the increase?
39. Markup. A store marks up merchandise 25% to allow for profit. If an item costs thestore $11, what will its selling price be?
40. Payroll deductions. Tranh’s pay is $450 per week. If deductions from his paycheckaverage 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 longshould it take to load the entire program?
42. Computers. A virus scanning program is checking every file for viruses. It hascompleted checking 40% of the files in 300 seconds. How long should it take tocheck all the files?
43. Consumer Affairs. The two ads pictured appeared last week and this week in thelocal 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 thepurchase price. At Loose Fit hardware, you pay $10 in tax for the same grill, but it is8% 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 theoriginal price. What was the original price?
46. Tipping. If the total bill at a restaurant, including a 15% tip, is $65.32, what was thecost of the meal alone?
(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 to1997?
48. What is the rate of increase (to the nearest whole percent) of imports from 1992 to1997?
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 becalculated on the original principal, $1000, plus the first year’s interest, $50. This is calledcompound interest.
For year 2, the interest is 5% of $1050, or 0.05 � $1050 � $52.50. At the end ofyear 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 eachyear), and find the amount in an account with the given interest rate and principal.
51. $4000, 6%, 2 years 52. $3000, 7%, 2 years
53. $4000, 5%, 3 years 54. $5000, 6%, 3 years
Solve the following applications.
55. Automobiles. In 1990, there were an estimated 145.0 million passenger carsregistered in the United States. The total number of vehicles registered in the UnitedStates for 1990 was estimated at 194.5 million. What percent of the vehiclesregistered were passenger cars?
56. Gasoline. Gasoline accounts for 85% of the motor fuel consumed in the UnitedStates 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. petroleumconsumption. If 10.85 million bbl of petroleum is used each day for transportation inthe United States, what is the total daily petroleum consumption by all sources in theUnited States?
58. Pollution. Each year, 540 million metric tons (t) of carbon dioxide are added to theatmosphere by the United States. Burning gasoline and other transportation fuels isresponsible for 35% of the carbon dioxide emissions in the United States. How muchcarbon dioxide is emitted each year by the burning of transportation fuels in theUnited States?
59. The progress of the local Lions club is shown below. What percent of the goal hasbeen achieved so far?
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). Whatpercent 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 percentproportion for the solution.
A
B 720R � 4100
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 increaseof 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 forthe solution.A R
9.65B � 4820
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, 35 of the 475 emergency calls received by the local police departmentwere false alarms. What percent of the calls were false alarms?
Tom buys a baseball card collection for $750. After 1 year, the value has decreased8.2%. What is the value of the collection after 1 year?
Example 4
Solving a Problem Involving an Unknown Amount
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%
NOTE Earlier we did a problemlike this by finding the decreaseand then subtracting from theoriginal value. Again, using thismethod requires just one step.
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. Whatamount (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 monthbefore the raise, how much will your raise be?
12. Manufacturing. In a shipment of 558 parts, 49 are found to be defective. Whatpercent (to the nearest tenth of a percent) of the parts are faulty?
13. Dropout rate. Statistics show that an average of 42.4% of the students entering a2-year program will complete their course work. If 588 students completed theprogram, how many students started?
14. Interest. A time-deposit savings plan gives an interest rate of 6.42% on deposits. Ifthe 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 originalpopulation 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 asale. Find its sale price (to the nearest cent).
20. Salary. Jerry earned $18,500 one year and then received a 10.5% raise. What is hisnew yearly salary?
21. Payroll deductions. Carolyn’s salary is $1740 per month. If deductions average24.6%, what is her take-home pay?
22. Investments. Yi Chen made a $6400 investment at the beginning of a year. By theend of the year, the value of the investment had decreased by 8.2%. What was itsvalue at the end of the year?
