Deitz J.E., Southam J.L. Contemporary Business Mathematics for Colleges

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Many businesses are organized into divisions or departments. Suppose Cotton’s Clothing

is a retailer of sportswear. It has three departments: women’s clothes, men’s clothes, and

children’s clothes. Management and owners of Cotton’s need to measure the profitability

of each department. Cotton’s also knows the amounts it paid for the merchandise sold

and the salaries of employees in each department. Cotton’s can subtract these depart-

mental costs from the departmental revenues.

But what about rent and other general costs such as electricity? These costs that are

not directly related to the types of merchandise sold are called overhead costs. For ex-

ample, Cotton’s monthly rental expense might be $15,000 for the entire building. How

should that single amount be divided among the three departments? Should each de-

partment be assigned , or $5,000, of the total rent?

Businesses can allocate, or distribute, the total rent based on a measurement related

to the total cost. Rent is a cost of using the building; it could be allocated on the basis of

floor space, since each department occupies some of that space.

94 Part 2 Percentage Applications

A company had sales of $200,000 this month and $160,000 last month (B). Find both the

amount of increase (P) and the rate of increase (R).

The amount of increase is

P 5 $200,000 2 $160,000 5 $40,000

The rate of increase is

R 5 P 4 B 5 $40,000 4 $160,000 5 0.25 5 25%

COMPLETE ASSIGNMENTS 5.2 AND 5.3.

✔

CONCEPT CHECK 5.4

COMPUTING AMOUNTS OF INCREASE AND

DECREASE WITH A CALCULATOR

Review example S. Now consider a variation of example S that says, “Find the total number

of homes that Joslin Realty sold this year.” Last year it sold 135 homes. There was a 40%

increase, which means 54 more homes were sold this year. The total number of homes sold

this year was 135 1 54 5 189 homes. Many calculators allow you to calculate 189 with the

following keystrokes: . The display will show the answer, 189.

If you need to know the actual amount of the increase, it will usually show in the

calculator display immediately after you press the key, but just before you press the

key.

Similarly, suppose the original example had said, “The real estate agency sold 40% fewer

homes this year than it did last year, when it sold 135 homes. Find the total number of

homes that it sold this year.” The amount of the decrease is 54 homes. Therefore, the total

number sold this year is 135 2 54 5 81 homes. On the calculator, you would use the

following keystrokes: . The display will show the answer, 81.

Using Percents to Allocate Overhead Expenses

Learning Objective

Use percents to allocate overhead

expenses.

5.7 Ask students to develop other

bases for allocating overhead in differ-

rent businesses.Several of them will

likely be familiar with various types

of businesses with different attrib-

utes that could be used to allocate

overhead.Even very obvious measure-

ments,such as number of employees

in different departments,might be

suggested.

Chapter 5 Percents 95

to Allocate an Overhead Expense Based on Total Floor Space

(or Some Other Measurement)

1. Find the total square feet of floor space.

2. Divide the floor space of each department by total floor space, then

change the quotients to percents.

3. Multiply each percent (in fractional or decimal form) by the total rent (or

other overhead value).

STEPS

EXAMPLE W

The total rent is $15,000. Using the floor space of each department as shown below, determine the amount of rent to

allocate to each department of Cotton’s.

Department Floor Space Percent of Total Distribution of Rent

Women’s 100 ft 3 50 ft 5 5,000 sq ft 5,000 4 10,000 5 50% 0.5 3 $15,000 5 $ 7,500

Children’s 50 ft 3 60 ft 5 3,000 sq ft 3,000 4 10,000 5 30% 0.3 3 $15,000 5 $ 4,500

Men’s 40 ft 3 50 ft 5 12,000 sq ft 2,000 4 10,000 5 20% 0.2 3 $15,000 5 $ 3,000

10,000 sq ft $15,000

STEP 3STEP 2STEP 1

This same method can be used for many other business expenses, such as utilities,

fire insurance, and salaries of office personnel. Examples of other bases that might be

used for allocation are number of employees, hours worked, and units produced.

A landscape maintenance company has two different divisions: commercial and residen-

tial. Employees spend 1,125 hours working on commercial landscapes and 375 hours

working on residential landscapes.The company has a utility expense of $8,000 that it

wants to allocate between the two divisions, based on the percent of employee hours

used by each division.

Total hours worked: 1,125 1 375 5 1,500

Commercial: 1,125 4 1,500 5 0.75, or 75% of employee hours

75% of $8,000 5 0.75 3 $8,000 5 $6,000 of office expense

Residential: 375 4 1,500 5 0.25 or 25% of employee hours

25% of $8,000 5 0.25 3 $8,000 5 $2,000 of office expense

COMPLETE ASSIGNMENT 5.4.

✔

CONCEPT CHECK 5.5

96 Part 2 Percentage Applications

allocate

amount of decrease

amount of increase

Base (B)

“of”

overhead costs

percent

Percentage (P)

Rate (R)

rate of decrease

rate of increase

Chapter Terms for Review

Try Microsoft

Excel

Try working the problems using the Microsoft Excel templates found on your student

CD. Solutions for the problems are also shown on the CD.

THE BOTTOM LINE

Chapter 5 Percents 97

Summary of chapter learning objectives:

Learning Objective

5.1

Change percents to decimals.

Example

1. Change 4.25% to a decimal.

Answers: 1. 0.0425 2. 45% 3. 87.5% 4. 14 5. 20% 6. 275 7. The increase is $1,250; The new salary is $6,250.

8. 20% 9. Store A, $800; Store B, $1,200; Store C, $2,000

5.2

Change fractions and decimals to percents.

5.3

Find Base, Rate, and Percentage.

5.4

Use percents to measure increase and decrease.

5.5

Use percents to allocate overhead expenses.

2. Change 0.45 to a percent.

3. Change to a percent.

4. Find the Percentage: 35% of 40 5 P

5. Find the Rate: R% of 140 5 28

6. Find the Base: 80% of B 5 220

7. Increase a $5,000 salary by 25%.

8. From 300 to 240 is a decrease of what percent?

9. A company has three stores, A, B and C, with 4, 6, and 10 employ-

ees, respectively. Based on the number of employees, allocate a

$4,000 expense among the stores.

1 Change 14.75% to a decimal

2 Change 0.625 to a percent

3 Change 150% to a decimal

4 Change 0.0075 to a percent

5 Change 0.06% to a decimal

6 Change to a percent

7 16% of 70 5

8 250% of 60 5

9 25% of 5 45

10 100% of 5 70

11 % of 40 5 35

12 % of 90 5 144

SELF-CHECK

Review Problems for Chapter 5

98 Part 2 Percentage Applications

13 Sales were $100,000 two months ago and increased by 20% last month. How much were sales last month?

14 Sales were $120,000 last month and decreased by 20% this month. How much were sales this month?

15 Expenses were $200,000 two years ago and $400,000 last year. What was the percent increase last year?

16 Expenses were $400,000 last year and $200,000 this year. What was the percent decrease this year?

17 Peggy Covey owns a nursery. This year she sold 195 more rose bushes than she did last year. This represents

a 12% increase over the previous year. How many rose bushes did Peggy’s nursery sell last year?

18 Jim Dukes manages Internet sales for a company that started selling its product over the Internet two years

ago. Last year, company sales over the Internet were only about $500,000. This year, sales were $1,625,000.

Calculate the company’s percent increase in Internet sales this year.

19 Ken Chard is a bank teller. When he started this morning, his cash drawer had coins worth $86. The coins

represented only 2.5% of all the money that Ken had in his cash drawer. What was the total value of all this

money?

20 Nancy McGraw is an orthopedic surgeon. Last winter, Dr. McGraw performed 50 emergency surgeries.

Thirty-two of those surgeries were the result of ski injuries. What percent of Dr. McGraw’s emergency surg-

eries were the result of ski injuries?

Answers to the Self-Check can be found in Appendix B at the back of the text.

Chapter 5 Percents 99

Assignment 5.1: Base, Rate, and Percentage

Name

Date Score

A (20 points) Change the percents to decimals. Change the nonpercents to percents. (1 point for each

correct answer)

1. 31% 5 2. 100% 5 3. % 5

0.0333

10.31

Learning Objectives

2 3

31% 100% %

00.01% 1000%

5 0.50 4.00 0.000025

0.52 245% 0.0825

05.2 224.5% 00.0003%

0.15 0.30 5 1.75

0.875 3.00 %

4. 0.875 5 5. 3 5 6. % 5

0.3367

300%87.5%

7. 0.15 5 8. 0.3 5 9. 5

175%

30%15%

10. 5.2% 5 11. 224.5% 5 12. 0.0003% 5

0.0000032.2450.052

13. 0.52 5 14. 245% 5 15. 5

8.25%

0.08

2.4552%

16. 5 17. 4.0 5 18. 0.000025 5

0.0025%400%50%

19. 0.01% 5 20. 1,000% 5

100.0001

Score for A (20)

B (30 points) In the following problems, find each Percentage amount. (2 points for each correct answer)

21. 0.375% of 56 5 22. 0.25% of 1,600 5 23. 100% of 11.17 5

11.1740.21

0.5 3 $162 0.08 3 200 0.12 3 0.38

0.25 3 $1.16 1.2 3 $45 0.025 3 $66

0.875 3 48 0.40 3 0.85 2.5 3 $66

0.00375 3 56 0.0025 3 1,600 1 3 11.17

24. 87.5% of 48 5 25. 40% of 0.85 5 26. 250% of $66 5

$1650.3442

27. 25% of $1.16 5 28. 120% of $45 5 29. 2.5% of $66 5

$1.65$54$0.29

30. 50% of $162 5 31. 8% of 200 5 32. 12% of 0.38 5

0.045616$81

33. 187.5% of 40 5 34. 1.5% of $86 5 35. 0.2% of 480 5

0.96$1.2975

Score for B (30)

1.875 3 40 0.015 3 $86 0.002 3 480

C (50 points) In each of the following problems, find the Percentage amount, the Rate, or the Base

amount.Write rates as percents. Round dollars to the nearest cent. (2 points for each correct answer)

36.

% of 5 $14 37. of $35 5 $7

38. of 1.12 5 1.4 39. of 14.2 5 28.4

40. of 400 5 14 41. 80% of 5 $0.96

42. 1.25% of 5 1.6 43. of 80 5 120

44. of 0.056 5 0.014 45. 175% of 5 $84

46. 2.5% of $2,820 5 47. 0.25% of 5 $20

48. of $268.50 5 $112.77 49. 62.5% of 5 35

50. 0.025% of $16,400 5 51. 140% of 5 672

52. 120% of 5 $51 53. of 5.4 5 2.16

54. of $1,480 5 $10,064 55. 15% of $140 5

56. 180% of $90 5 57. of 85 5 136

58. 125% of 5 $520 59. 12% of 5 3

60. of 3.2 5 0.704

Score for C (50)

22%

25$416

160%$162

$21.00680%

40%$42.50

480$4.10

5642%

$8,000$70.50

$4825%

150%128

$1.203.5%

200%125%

20%$40.00

100 Part 2 Percentage Applications

Assignment 5.1 Continued

14 4 0.35 $7 4 $35 5 0.20

(R)(B)(P)(R)(B)(P)

1.4 4 1.12 5 1.25 28.4 4 14.2 5 2.00

14 4 400 5 0.035 $0.96 4 0.80

1.6 4 0.0125 120 4 80 5 1.50

0.014 4 0.056 5 0.25 $84 4 1.75

0.025 3 $2,820 $20 4 0.0025

$112.77 4 $268.50 5 0.42 35 4 0.625

0.00025 3 $16,400 672 4 1.4

$51 4 1.20 2.16 4 5.4 5 0.40

$10,064 4 $1,480 5 6.80 0.15 3 $140

1.80 3 $90 136 4 85 5 1.60

$520 4 1.25 3 4 0.12

0.704 4 3.2 5 0.22

Chapter 5 Percents 101

Assignment 5.2: Rate of Increase and Rate of Decrease

Name

Date Score

A (40 points) Calculate the missing values. ( points for each correct answer)2

Learning Objective

1. Decreasing the base value of 280 by 25%

gives the new value .

2. Increasing the base value of 240 by 40% gives the

new value .

336210

3. Start with 75, decrease it by 60%, and end up

with .

4. Start with 80, increase it by 14%, and end up

with .

91.230

5. Sales were $8,000 last month and increased

by 4% this month. Sales were

this month.

6. Profits were $67,500 last month, but decreased

by 3% this month. Profits were this

month.

$65,475$8,320

13. $300 is what percent less than $400? 14. 480 is what percent greater than 160?

200%

25%

15. Sales were $500,000 in June but only $400,000

in July. The rate of decrease was .

16. Profits were $11,000 last month and $10,000

the previous month. The rate of increase was

10%

20%

7. Base value 5 272; increase 5 100%; new

(final) value 5

8. Base value 5 250; decrease 5 100%; new (final)

value 5

0544

9. A $17 increase is 10% of the base value

of .

10. A decrease of 45 units is 15% of the base value

of units.

300$170

11. The price decreased from $450 to $378; the

percent decrease was .

12. Production increased from 8,000 units to 10,000

units; the percent increase was .

25%16%

0.25 3 280 5 70

280 2 70 5 210

0.60 3 75 5 45

75 2 45 5 30

0.04 3 $8,000 5 $320

$8,000 1 $320 5 $8,320

1.00 3 272 5 272

272 1 272 5 544

$17 4 0.10 5 $170

$450 2 $378 5 $72

$72 4 $450 5 0.16

$400 2 $300 5 $100

$100 4 $400 5 0.25

$500,000 2 $400,000 5 $100,000

$100,000 4 $500,000 5 0.20

0.40 3 240 5 96

240 1 96 5 336

0.14 3 80 5 11.20

80 1 11.2 5 91.2

0.03 3 $67,500 5 $2,025

$67,500

2 $2,025 5 $65,475

1.00 3 250 5 250

250 2 250 5 0

45 4 0.15 5 300

480 2 160 5 320

320 4 160 5 2.00

10,000 2 8,000 5 2,000 increase

2,000 4 8,000 5 0.25

$11,000 2 $10,000 5 $1,000

$1,000 4 $10,000 5 0.10

Score for A (40)

B (30 points) The following table shows the volumes of various items sold by Thrift’s Speed Shop during

the past two years. Compute the amount of change and the rate of change between this year and last

year. Compute the rates to the nearest tenth of a percent. If the amount and rate are increases, write a

1 in front of them; if they are decreases, enclose them in parentheses (). (1 point for each correct

amount; 2 points for each correct rate)

Thrift’s Speed Shop

Volume Sold (number of units)

Amount of Rate of

Description of Item This Year Last Year Change Change

17. Batteries 516 541

18. Brake fluid (pints) 1,781 1,602

19. Coolant (gallons) 2,045 1,815

20. Headlight lamps 5,291 5,687

21. Oil (quarts) 13,428 14,746

22. Mufflers 639 585

23. Shock absorbers 895 1,084

24. Tires, auto 6,742 5,866

25. Tires, truck 2,115 1,805

26. Wiper blades 1,927 2,342

Score for B (30)

C (30 points) During May and June, Kalman’s Paint Store had sales in the amounts shown in the following

table. Compute the amount of change and the rate of change between May and June. Compute the rates

of change to the nearest tenth of a percent. If the amount and rate are increases, write a 1 in front of

them; if they are decreases, enclose them in parentheses (). (1 point for each correct amount; 2 points for

each correct rate)

Kalman’s Paint Store

Volume Sold (in dollars)

Amount of Rate of

Description of Item June May Change Change

27. Brush, 20 wide $ 611.14 $ 674.67

28. Brush, 30 wide 564.20 512.51

29. Brush, 40 wide 429.87 374.27

30. Drop cloth, 9 3 12 143.50 175.66

31. Drop cloth, 12 3 15 174.29 151.55

32. Paint, latex (gal) 46,921.64 49,075.29

33. Paint, latex (qt) 5,072.35 4,878.96

34. Paint, oil (gal) 7,308.44 7,564.27

35. Paint, oil (qt) 4,358.35 4,574.96

36. Paint scraper 274.10 238.82

Score for C (30)

114.8%135.28

(4.7%)(216.61)

(3.4%)(255.83)

14.0%1193.39

(4.4%)(2,153.65)

115%122.74

(18.3%)(32.16)

114.9%155.60

110.1%151.69

(9.4%)($63.53)

(17.7%)(415)

117.2%1310

114.9%1876

(17.4%)(189)

19.2%154

(8.9%)(1,318)

(7.0%)(396)

112.7%1230

111.2%1179

(4.6%)(25)

102 Part 2 Percentage Applications

Assignment 5.2 Continued

Chapter 5 Percents 103

Assignment 5.3: Business Applications

Name

Date Score

A (50 points) Solve the following problems. Round dollar amounts to the nearest cent. Round other

amounts to the nearest tenth.Write rates as percents to the nearest tenth of a percent. (5 points for

each correct answer)

1. Walter Electric shipped 5,500 capacitors in May. Clients eventually returned 4% of the capacitors. How

many of the capacitors shipped in May were eventually returned?

2. Jim Walter, CEO of Walter Electric, wants the company to reduce the percent of capacitors that cus-

tomers return. In June, the company shipped 5,000 capacitors, and 150 were eventually returned. What

percent of the June shipment was eventually returned?

3. By July of the following year, Walter Electric had reduced the percent of capacitors returned to 2% of the

number shipped. If 130 capacitors were returned from that month’s shipment, how many had been

shipped?

4. A European food importer, Fontaine’s Food Expo, imports 35% of its vinegars from France, 40% from

Italy, and the rest from Spain. The total value of all the vinegars that it imports is $1,216,000. What is the

value of the vinegars that are imported from Spain?

5. Next year, Fontaine’s is planning to import $462,000 worth of vinegars from France, $532,000 worth of

vinegars from Italy, and $406,000 worth of vinegars from Spain. If next year’s imports occur as currently

being planned, what percent of the total imports will be from Italy?

6. Rigik Parka Products, Inc., manufactures only parkas for adults and children. Last year, Rigik manufactured

all its children’s parkas in Asia. Those children’s parkas represented 35% of all the Rigik production. If the

company made a total of 240,000 parkas, how many children’s parkas did it produce?

7. This year, Rigik again plans to manufacture all its children’s parkas in Asia, and Rigik will expand the

children’s product line to 40% of the total number of parkas produced. If Rigik plans to produce 112,000

children’s parkas, how many parkas does the company plan to produce in total?

8. Next year, Rigik plans to keep the percent of children’s parkas at 40% but increase the number of chil-

dren’s parkas produced to 125,000. How many parkas does the company plan to produce for adults?

(Hint: First you need to calculate the total number of all parkas to be produced next year.)

9. Manuel Sosa is a single father. He tries to save 15% of his monthly salary for his son’s education. In

August, Manuel’s salary was $4,800. How much should he save to meet his objective?

10. In September, Manuel Sosa got a promotion and a raise. Because his monthly expenses did not increase

very much, Manuel was able to save more dollars. He saved $1,350, which was 25% of his new salary.

How much was Manuel’s new salary?

Score for A (50)

$5,400

$720

187,500

280,000

84,000

38%

$304,000

6,500

220

Learning Objectives

P 5 R 3 B 5 0.04 3 5,500 5 220

R 5 P 4 B 5 150 4 5,000 5 0.03, or 3%

B 5 P 4 R 5 130 4 0.02 5 6,500

P 5 R 3 B 5 0.35 3 240,000 5 84,000

B 5 P 4 R 5 112,000 4 0.40 5 280,000

35% 1 40% 5 75%; 100% 2 75% 5 25%

P 5 R 3 B 5 0.25 3 $1,216,000 5 $304,000

$462,000 1 $532,000 1 $406,000 5 $1,400,000

R 5 P 4 B 5 $532,000 4 $1,400,000 5 0.38, or 38%

B 5 P 4 R 5 125,000 4

0.40 5 312,500 total

312,500 2 125,000 5 187,500 for adults

B 5 P 4 R 5 $1,350 4 0.25 5 $5,400

P 5 R 3 B 5 0.15 3 $4,800 5 $720