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

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B (10 points) Solve the following time, rate, distance problems. (5 points for each correct answer)

Wendy leaves St. Paul to travel the 2,000 miles to Los Angeles, driving at a speed of 55 miles per hour.

Mark leaves Los Angeles to travel the same 2,000-mile route to St. Paul, driving at a speed of 45 miles per

hour. How many miles will Mark have traveled when they meet?

900 mi

84 Part 1 Fundamental Review

Assignment 4.2 Continued

2,000 4 100 (55 mph 1 45 mph) 5 20 hr

20 3 45 5 900 mi

10. Car A traveled to a destination 840 miles away at 60 miles per hour. Car B traveled to a destination 660

miles away at 55 miles per hour. How much longer did Car A travel than Car B?

2 hr

A: 840 4 60 5 14 hr

B: 660 4 55 5 12 hr

14 hr 2 12 hr 5 2 hr

Score for B (10)

C (40 points) Solve each of the problems without writing any computations on paper and without using a

calculator or a computer. (2 points for each correct answer)

11. 5 items at $1.99 5 12. 2 items at $7.98 5

13. 4 items at $19.98 5 14. 2 items at $49.96 5

15. 15 items at $0.99 5 16. 10 items at $9.99 5

17. 6 items at $3.95 5 18. 5 items at $1.02 5

19. 19 items at $40 5 20. 3 items at $19.99 5

21. 20 items at $40.05 5 22. 30 items at $1.99 5

23. 20 items at $39.98 5 24. 2 items at $5.99 5

25. 48 items at $5 5 26. 5 items at $1.97 5

27. 7 items at $7.97 5 28. 2 items at $99.98 5

29. 30 items at $2.98 5 30. 99 items at $1.90 5

Score for C (40)

D (10 points) In each of the following equations, rewrite the equation by moving the last number on each

side of the equal sign to the other side and making appropriate sign changes so that the equation is still

true. (Example: Given 13 1 7 1 2 5 10 1 12; Answer 13 1 7 2 12 5 10 2 2) (1 point for each correct

equation)

31. 6 1 4 1 5 5 17 2 2 32. 6 3 2 4 3 5 8 4 4 3 2

$188.10$89.40

$199.96

$55.79

$9.85$240

$11.98$799.60

$59.70$801

$59.97$760

$5.10

$23.70

$99.90$14.85

$99.92

$79.92

$15.96

$9.95

6 1 4 1 2 5 17 2 56 3 2 4 2 5 8 4 4 3 3

33. 9 2 3 2 3 5 2 1 1 34. 8 4 2 3 4 5 24 4 3 3 2

9 2 3 2 1 5 2 1 38 4 2 4 2 5 24 4 3 4 4

35. 20 1 1 2 7 5 16 2 2 36. 3 3 3 3 3 5 18 4 2 3 3

20 1 1 1 2 5 16 1 73 3 3 4 3 5 18 4 2 4 3

37. 12 1 3 2 5 5 7 1 3 38. 7 3 4 4 2 5 28 3 2 4 4

12 1 3 2 3 5 7 1 57 3 4 3 4 5 28 3 2 3 2

39. 64 2 32 2 16 5 8 1 8 40. 63 4 7 3 2 5 3 3 2 3 3

64 2 32 2 8 5 8 1 16 63 4 7 4 3 5 3 3 2 4 2

Score for D (10)

Notes

Chapter 4 Word Problems and Equations 85

Percentage

Applications

Part

5 Percents

6 Commissions

7 Discounts

8 Markup

Percents

Chapter 5 Percents 87

Learning Objectives

By studying this chapter and completing all assignments, you will be able to:

Change percents to decimals.

Change fractions and decimals to percents.

Find Base, Rate, and Percentage.

Use percents to measure increase and decrease.

Use percents to allocate overhead expenses.

Learning Objective

Percents and percentages are used extensively in various business and nonbusiness appli-

cations. Airlines are required to publish the “on-time percentage” for each of their flights.

Every bank publishes its loan rates as percents. The Food and Drug Administration

(FDA) says that packaged foods must contain labels with nutritional information, much

of which is written in percents. Colleges and universities often describe the ethnic diver-

sity of their student bodies and faculty using percents.

88 Part 2 Percentage Applications

Learning Objective

Change percents to decimals.

5.1 Ask students how to say percents

in other languages.For example,in

Italian,the word for 100 is “cento,”the

word for 20 is “venti,”and the words

for percent are “per cento.”So 100% is

“cento per cento,”and 20% is “venti per

cento.”

In Spanish,the word for 100 is

“ciento”(or sometimes “cien”),the

word for 20 is “veinte,”and the words

for percent are “por ciento.”So 100% is

“ciento por ciento,”and 20% is “veinte

por ciento.”

5.2 Remind students again that in per-

cents,when a decimal number has no

preceding whole number,we will write

a zero to the left of the decimal point.

EXAMPLE A

45% 45 .45. 5 0.45

STEP 3STEP 2

EXAMPLE B

175% 175 1.75. 5 1.75

STEP 3STEP 2

EXAMPLE C

4.5% 4.5 .04.5 5 0.045

STEP 3STEP 2

EXAMPLE D

1.75 .01.75 5 0.0175

% 5 1.75%

STEP 3STEP 2STEP 1

(Note: Check the answers to these examples with the percent key on your calculator.)

to Change a Percent to a Decimal

1. If the percent has a fractional part, convert the fraction to its decimal

equivalent.

2. Remove the percent symbol.

3. Move the decimal point two places to the left (insert zeros if needed).

STEPS

Changing Percents to Decimals

We use percents because the word percent makes verbal and written communication

easier. Suppose that we have a 5% sales tax. Which of these phrases sounds better:

(a) “five percent,” (b) “five-hundredths,” (c) “one-twentieth,” or even (d) “point zero

five”? Imagine how complicated the latter three phrases would be if the sales tax rate

were 5.25%. But by using the word percent, we can just say “five point two five percent.”

Percents themselves are actually not used in arithmetic. Before you can do any calcu-

lation with a percent, you must change the percent to a decimal. If you use a calculator

with a percent key , the calculator will first convert the percent to a decimal. Take a

calculator with a percent key and observe the display closely. Enter 75%; that is, press

these three keys: . After pressing the key, the display shows 0.75. There is

no percent symbol and the decimal point has moved two places to the left. The calcula-

tor will use 0.75 in all of its calculations that involve 75%.

Sometimes a percent has a fractional part. For example, we might have a tax rate that

is stated as Even using a calculator, first we must write the fraction as a decimal to

get 5.5%. Using the calculator, press these keys: . After pressing , the dis-

play shows 0.055. Notice that to move two places to the left, the calculator had to insert

an extra zero.

Changing a decimal to a percent is exactly the opposite from changing a percent to a

decimal: Move the decimal point two places to the right, and then write a percent

symbol. If you have a fraction or a mixed number, first change it to a decimal as you did

in Chapter 3. Then change the decimal to a percent. (A decimal point at the extreme

right end of the percent is omitted—examples E, G, and J below.)

Chapter 5 Percents 89

a. Change 250% to a decimal. b. Change to a decimal.

250% 250 2.50. 5 2.50 or 2.5 5 0.25% 0.25 .00.25 5 0.0025

✔

CONCEPT CHECK 5.1

Changing Decimals and Fractions to Percents

to Change a Fraction or a Decimal to a Percent

1. If the number is a fraction, or a mixed number, convert it to its decimal

equivalent.

2. Move the decimal point two places to the right (insert zeros if needed).

3. Write a percent symbol at the right end of the new number.

STEPS

EXAMPLE E EXAMPLE F

0.80. 80% 1.37.5 137.5%1

5 1.375

5 0.8

STEP 3STEP 2STEP 1STEP 3STEP 2STEP 1

EXAMPLE G EXAMPLE H

0.4 0.40. 40% 1.1875 1.18.75 118.75%

STEP 3STEP 2STEP 3STEP 2

EXAMPLE I EXAMPLE J

2.5 2.50. 250% 1 5 1. 1.00. 100%

STEP 3STEP 2STEP 3STEP 2

(Note: To check these examples with your calculator, you can multiply the decimal

number by 100 and write the percent symbol at the right end of the answer.)

Change fractions and decimals to

percents.

Learning Objective

Suppose that you have $5 and spend $4 for breakfast. Example E showed that the frac-

tion equals 80%. You can say that “you spent 80% of your money ($5) for your break-

fast ($4).” Without the context of your breakfast, you have simply “80% of $5 5 $4.” In

this book we call 80% the Rate (R), $5 the Base (B) amount, and $4 the Percentage (P)

amount. The Base and the Percentage amounts will always have the same units (e.g.,

dollars, feet, or pounds). The Rate is the percent. (The word rate comes from the word

ratio—in this case, .) It may make sense for you to think of the Base amount as the

denominator in the rate (that is, ratio 5 ) because the denominator is the “base” (i.e.,

bottom) of the fraction.

Note: In practice, the terms percent and percentage are often used interchangeably. Some-

times, you will see the word percentage used to mean a rate and the word percent used to

mean an amount. You will even see the two words percentage rate to mean the rate. In

this book, however, we use only one meaning for each word.

90 Part 2 Percentage Applications

a. Change to a percent. b. Change 0.075 to a percent.

2.70. 270% 0.075 0.07.5 7.5%2

5 2.7

✔

CONCEPT CHECK 5.2

Finding Base, Rate, and Percentage

Learning Objective

Find Base, Rate, and Percentage.

5.3 Students may understandably be

confused by the way in which we

distinguish between percent and

percentage because the terms often are

not used consistently.Ask students to

look up the definitions and then find

examples on television or in the

newspaper.They will quickly discover

the inconsistent usage.

EXAMPLE K EXAMPLE L EXAMPLE M

80% of $5 5 $4 25% of 20 ft 5 5 ft 50% of 60 gal 5 30 gal

80% is the Rate 25% is the Rate 50% is the Rate

$5 is the Base 20 ft is the Base 60 gal is the Base

$4 is the Percentage 5 ft is the Percentage 30 gal is the Percentage

Figure 5-1

100%

80%

80% of $5 is $4

100%

25%

25% of 20 ft is 5 ft

20 ft

5 ft

0 ft

100%

50%

50% of 60 gal is 30 gal

60 gal

30 gal

0 gal

Figure 5-1 shows three diagrams, one each for examples K, L, and M. In each diagram,

the Rate (or percent) is shown in the left-hand column. Each Percentage is represented

by the shaded portion of the right-hand column. Each Base is represented by the entire

height of the right-hand column.

The word of often appears in problems that involve percents. Recall from Chapter 2

that with fractions, of means multiply. We just showed that 80% 5 . Also recall that you

can “check” a division problem by multiplication. We would get 80% 3 $5 5 $4. In

words, we say that “80% of $5 is $4.”

Rule: The number that follows the word of is the Base (and is the denominator in the

fraction); the number that follows the word is is the Percentage amount.

The preceding examples illustrate the basic relationship among the Rate, Base, and

Percentage: Rate 3 Base 5 Percentage. As a formula, it is written as R 3 B 5 P or as

P 5 R 3 B.

When you know any two of these three numbers, you can calculate the third by

changing the formula:

If you want to find B, the formula becomes B 5 P 4 R or P 4 R 5 B.

If you want to find R, the formula becomes R 5 P 4 B or P 4 B 5 R.

Chapter 5 Percents 91

Figure 5-2

100%

75%

75% of $B is $45

$45

100%

R% of 30 lb is 6 lb

30 lb

6 lb

0 lb

100%

50%

50% of 300 yd is P yd

300 yd

P yd

0 yd

5.4 Looking at examples L and M,we

could say that “ of 20 feet is 5 feet”

and “ of 60 gallons is 30 gallons.”The

mathematics are 3 20 5 5 and

3 60 5 30.

EXAMPLE N EXAMPLE O EXAMPLE P

Find P when Find R when Find B when

R 5 50% and B 5 300 yd B 5 30 lb and P 5 6 lb P 5 $45 and R 5 75%

Figure 5-2 illustrates these relationships, which are calculated as follows:

P 5 R 3 BR5 P 4 BB5 P 4 R

P 5 50% 3 300 yd R 5 6 lb 4 30 lb B 5 $45 4 75%

P 5 0.50 3 300 yd R 5 0.20 B 5 $45 4 0.75

P 5 150 yd R 5 20% B 5 $60

Note that in example O, the calculator cannot automatically “move” the decimal point

two places to the right. If you want the calculator to do it, you “multiply by 100.” It is

faster to just move the decimal point places without a calculator.

5.5 For many students,it is useful to

use a diagram to estimate the answers

in problems (see Figure 5-2).Most,

however,will decide it is much

faster/easier to use a calculator.

USING PERCENTS IN BUSINESS

Percent problems occur frequently in business. Examples Q and R are typical fundamen-

tal applications, in which we solve for the Base (B) amount and the Rate (R), respectively.

92 Part 2 Percentage Applications

EXAMPLE Q

Lena Hoover is a financial analyst. In December, she received a $600 bonus, which

equaled 15% of her monthly salary. What was her monthly salary?

P 5 amount of bonus 5 $600

R 5 rate of bonus 5 15%

B 5 monthly salary 5 ?

As P 4 R 5 B,

P 4 R 5 $600 4 15% 5 $600 4 0.15 5 $4,000 monthly salary

EXAMPLE R

Last year Bayside Coffee Shop had total expenses of $300,000. Of that total, $210,000 was

the expense for employee salaries. Last year at Bayside, the employee salary expense was

what percent of total expenses?

P 5 employee salaries 5 $210,000

R 5 ?

B 5 total expenses 5 $300,000

Since P 4 B 5 R,

P 4 B 5 $210,000 4 $300,000 5 0.70 5 70%

a. Find the Base when the Rate is 40% and the Percentage amount is 50 ft.

B 5 P 4 R 5 50ft 4 40% 5 50ft 4 0.40 5 125 ft

b. Find the Rate when the Base is 12 oz and the Percentage amount is 3 oz.

R 5 P 4 B 5 3 oz 4 12 oz 5 0.25 5 25%

COMPLETE ASSIGNMENT 5.1.

✔

CONCEPT CHECK 5.3

Using Percents to Measure Increase

and Decrease

In business, percents are used to measure change from one year to the next or from one

month to the next. Real estate firms compare the number of homes sold this year with

the number of homes sold last year. Read and carefully compare the following four state-

ments about home sales last year and this year:

Joslin Realty sold 40% more homes this year than it did last year, when it sold 135

homes.

Learning Objective

Use percents to measure increase

and decrease.

Rossi & Shanley Real Estate sold 25 more homes this year than last year, which

represents 20% more homes sold this year than last year.

Real estate agent Nancy Lo sold 5 fewer homes this year than she did last year, when

she sold 40 homes.

Charles Peterson, a real estate broker, sold 30 homes last year; this year he sold 36

homes.

The number of homes sold last year is the Base (B) amount (last year is called the

base year). The change in homes sold can be reported as a number, which would be the

Percentage amount (P), or as a percent, which would be the Rate (R). If any two of the

three values are given, the third can be determined using one of the three formulas in

this chapter.

Chapter 5 Percents 93

5.6 Point out that,if you increase by an

amount and then decrease by the

same amount,the percents are differ-

ent.Suppose that you start with 100

and increase by 25 to 125.That is a

25% increase.If you then decrease by

25,back to 100,that is only a 20% de-

crease.Similarly,suppose that you start

with 100 and increase by 10% to 110.If

you then decrease by 10%,you go to

99 instead of 100.A similar difference

will occur in Chapter 8 when we

encounter markup based on cost and

markup based on selling price.

EXAMPLE S

Find the number of additional homes (P) that Joslin Realty sold this year.

B 5 135 and R 5 40%. Since P 5 R 3 B,

P 5 40% 3 135 5 0.40 3 135 5 54 more homes this year

EXAMPLE T

Find the number of homes that Rossi & Shanley Real Estate sold last year (B).

P 5 25 and R 5 20%. Since B 5 P 4 R,

B 5 25 4 20% 5 25 4 0.20 5 125 homes sold last year

EXAMPLE U

Find Nancy Lo’s rate of decrease (R) from last year’s sales.

P 5 5 and B 5 40. Since R 5 P 4 B,

R 5 5 4 40 5 0.125 5 12.5% decrease

To find the percent change when the only numbers reported are the amounts (B) for

last year and this year, the first step is to find the amount of increase or the amount of

decrease. P is the difference between the amounts for the two years. Then use R 5 P 4 B

to find the rate of increase or the rate of decrease.

EXAMPLE V

Find Charles Peterson’s rate of change (R).

Charles sold 30 homes last year (B) and 36 this year. The amount of change is

P 5 36 2 30 5 6 more homes this year

The rate of change is

R 5 P 4 B 5 6 4 30 5 0.20 5 20% increase