Deitz J.E., Southam J.L. Contemporary Business Mathematics for Colleges
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74 Part 1 Fundamental Review
A numerical equation may have one value that is unknown, but still provide enough
information to complete the sentence. To solve for the unknown value, you can move any
or all of the known values to the other side of the equation by reversing their signs.
EXAMPLE M EXAMPLE N
6 1 2 5 5 1 ? 15 2 3 5 2 1 ?
Change a number Change a number
6 1 2 2 5 5 ? 15 2 3 2 2 5 ?
Therefore, ? 5 3 Therefore, ? 5 10
EXAMPLE O EXAMPLE P
7 1 3 1 6 5 4 1 4 1 ? 20 4 5 5 2 3 ?
Change a number Change a number
7 1 3 1 6 2 4 2 4 5 ? 20 4 5 4 2 5 ?
Therefore, ? 5 8 Therefore, ? 5 2
In business, numerical sentences with equations compare items. Note the following
example.
EXAMPLE Q
Last year a company had sales of $25,000 in Dept. A and $20,000 in Dept. B. If sales this
year were $30,000 in Dept. A, what is the amount needed for Dept. B to equal last year’s
sales?
Last year: Dept. A $25,000 1 Dept. B $20,000 5 $45,000
This year: Dept. A $30,000 1 Dept. B ? 5 $45,000
Dept. B 5 $45,000 2 Dept. A $30,000
Therefore, ? 5 $15,000
Both sides of a true equation are equal. Each side may contain calculations.
7 1 5 5 14 2 2
2 3 9 5 36 4 2
A number may be moved from one side of an equation to the other by reversing its sign.
8 5 6 1 28 2 2 5 67 1 3 5 10 7 5 10 2 3
12 5 4 3 3 12 4 3 5 4 24 4 12 5 2 24 5 2 3 12
✔
CONCEPT CHECK 4.3
Chapter 4 Word Problems and Equations 75
Relationships in a series of numbers may be found by comparing the first three or four
terms in a series and then extrapolating the numbers that would most logically come
next. For example, examining the series 320, 160, 80, 40 indicates that each term is found
by dividing the preceding number by 2. The next two numbers in the series would logi-
cally be 20 and 10—that is, 40 4 2 5 20 and 20 4 2 5 10.
Examining the series 7, 14, 21, 28 suggests the addition of 7 to each preceding num-
ber. The next two numbers in this series would logically be 35 and 42 (28 1 7 5 35 and
35 1 7 5 42).
In the series 5, 15, 35, 75, 155, seeing a relationship is difficult; however, a relationship
does exist. Each number results from multiplying the preceding number by 2 and then
adding 5. In this series, the next number would logically be 315 (155 3 2 1 5 5 315).
Recognizing numerical and series relationships can be important in analyzing, com-
municating, and computing numbers. These relationship series are also used frequently
in initial employment tests.
Numerical Relationships in a Series
In studying relationships in a numerical series, look for patterns. Patterns most commonly fall into categories:
Addition 2, 7, 12, 17, 22, 27 (15, or 32)
Alternating addition/subtraction 12, 24, 18, 30, 24, 36, 30 (112, 26, or 42, 36)
Subtraction 39, 32, 25, 18, 11, 4 (27, or 23)
Alternating subtraction/addition 64, 59, 61, 56, 58, 53, 55 (2 5, 12, or 50, 52)
Multiplication 4, 12, 36, 108, 324, 972 (3 3, or 2,916)
Division 384, 192, 96, 48, 24 (4 2, or 12)
You can also devise patterns such as multiplication with addition or subtraction, division with addition or subtraction,
and many other combinations.
✔
CONCEPT CHECK 4.4
Recognize numerical relationships in
a series.
4
Learning Objective
Making Quick Calculations by Rounding
Numbers
Quick calculations are beneficial when working in business situations. Rounding odd and
difficult-to-compute amounts to even whole numbers that are easier to compute is a
technique often used in business. By rounding, you will be able to get quick and accurate
answers without having to write out the computations.
EXAMPLE R
How much would 5 items at $2.99 each cost?
To make this computation easily, think “$2.99 is $0.01 less than $3.00.” Then think
“5 times $3 equals $15.” Finally, think “$15.00 less $0.05 (5 3 $0.01) is $14.95,” which is
the correct answer.
4.2 Making quick calculations by
rounding numbers is a natural sequel
to estimating and approximating,cov-
ered in Chapter 1.Student review of
and practice in estimating will develop
and enrich this thinking process.
Do quick mental calculations through
a process of rounding numbers.
5
Learning Objective
76 Part 1 Fundamental Review
EXAMPLE S
The total cost of three equally priced dresses is $119.85. How much does each dress cost?
To figure out this problem easily, think “$119.85 is $0.15 less than $120.00.” Then
think “$120 divided by 3 5 $40, and $40.00 less $0.05 ($0.15 4 3) is $39.95,” the correct
answer.
EXAMPLE T
At 19 miles per gallon, how many miles would a car go on 9 gallons of gas?
To figure out this problem easily, think “19 is just 1 mile less than 20.” Then think
“9 times 20 5 180, and 180 minus 9 (9 3 1) is 171,” the correct answer.
You may have noticed that making quick calculations is quite similar to making estimations,
which you did in Chapter 1. In fact, quick calculation is only an additional step. After estimating
an answer, you determine the degree to which the estimated, or rounded, answer differs from the
actual answer by mentally correcting for the amount of the estimation or rounding.
COMPLETE ASSIGNMENTS 4.1 AND 4.2
✔
CONCEPT CHECK 4.5
equation numerical sentence
Chapter Terms for Review
© CREATAS IMAGES/JUPITERIMAGES
Chapter 4 Word Problems and Equations 77
Notes
78 Part 1 Fundamental Review
THE BOTTOM LINE
Summary of chapter learning objectives:
Learning Objective
4.1
Use a systematic approach to solve word problems
involving basic math processes.
Example
Use the two-step process to solve the word problem.
1. Martha is preparing to make two dresses. One will require 3 yards
of material; the other will require 4 yards of material. The material
for the first dress costs $12.00 per yard; the material for the second
costs $15.00 per yard. Buttons and trimming will cost $8.00 for
each dress. What will be the total cost?
Determine what is being requested.
Determine the processes to be used to solve the problem.
Answer:
Answers: 1. $112 2. 13 hr 3. 360 mi 4. 4 hr 5. 5 3 12 3 2 5 120 6. 5
4.2
Apply formulas to solve rate, time, and distance
problems.
4.3
Solve simple numerical equations.
2. At an average rate of 50 miles per hour, how long would it take to
drive 650 miles?
3. At an average rate of 60 miles per hour, how far could you drive in
6 hours?
4. If you drove 70 miles per hour and covered 280 miles, how much
time did it take?
5. 5 3 12 5 120 4 2 Move the 2 to the opposite
side of the equation.
6. 7 1 8 2 3 5 5 1 2 1 ? Solve for ? amount.
THE BOTTOM LINE
Chapter 4 Word Problems and Equations 79
Summary of chapter learning objectives:
Learning Objective
4.4
Recognize numeric relationships in a series.
Example
Insert the next two numbers.
7. 4, 7, 6, 9, 8, 11, , Pattern:
8. 12, 48, 24, 96, 48, , Pattern:
Answers: 7. (1 3, 2 1) 10, 13 8. (3 4, 4 2) 192, 96 9. (8 3 $4.00) 5 $32.00 2 0.08 5 $31.92
10. (60 3 20) 5 1200 2 20 5 1180 mi
4.5
Do quick mental calculations through a process of
rounding numbers.
9. What is the cost of eight items at $3.99 each?
10. At 59 miles per hour, how far would a car go in 20 hours?
SELF-CHECK
Review Problems for Chapter 4
80 Part 1 Fundamental Review
5 In the first four months of the year, a corporation had monthly earnings of $12,493, $6,007, $3,028, and
$9,728. What were its total earnings in the four months?
6 If the corporation in question 5 had earnings of $74,500 at the end of the year, how much did it earn in the
last eight months of the year?
7 If a tour bus gets 7 miles per gallon of gas and used 61 gallons in a week, how many miles did it travel in the
week?
8 An employer earned $4,000. Half the earnings went into an employee bonus pool. The pool was split among
five employees. How much did each employee receive?
9 A delivery firm bought 21 gallons of gas on Monday, 15 on Tuesday, 24 on Wednesday, 34 on Thursday, and
11 on Friday. If gas cost $2.15 per gallon, how much did the delivery firm pay for the week’s gas?
10 A store owner planned to give away $1,200 at Christmas. The owner gave $150 to each of five full-time em-
ployees and $50 to each of four part-time employees. The remainder was given to a local charity. How much
did the charity receive?
11 How long would it take to travel 1,265 miles at 55 miles per hour?
12 Bob and Mary start traveling toward each other from 1,330 miles apart. Bob is traveling at 30 miles per hour,
Mary at 40 miles per hour. How many hours elapse before they meet?
13 Bob and Mary start traveling toward each other from 960 miles apart. Bob is traveling at 25 miles per hour,
Mary at 55 miles per hour. How far did Bob travel?
14 41 2 6 5 27 1
15 72 1 72 5 300 2
16 10 3 3 5 90 4
17 Four items at $9 each 5 items at $12 each
18 What is the next number in the series 3, 7, 8, 12, ?
19 What is the next number in the series 5, 20, 10, 40, ?
20 To find the price of seven items at $1.99 you would think: 7 times $ less 7 times $ 5
$13.93
1 6 items at $5.99 each 5 2 3 items at $2.48 each 5
3 24 items at $1.99 each 5 4 40 items at $2.02 each 5
Answers to the Self-Check can be found in Appendix B at the back of the text.
B (10 points) Do these problems without using scratch paper or an electronic calculator. (2 points for each
correct answer)
5. How much would you pay for 8 gallons of gasoline selling at $3.95 per gallon?
6. How many items would you have if you had 98 books, 98 cards, and 98 pencils?
7. What is the price of 15 items at $2.99 each?
8. How much would you have if you received $3.99 from one person, $7.99 from a second, $11.99 from a
third, and $1.99 from a fourth?
9. If 27 people were divided into three equal groups and each group added 2 additional members, how
many members would be in each group?
Score for B (10)
C (10 points) Do the steps in the order in which they occur. Do these problems without using scratch
paper or an electronic calculator. (1 point for each correct answer)
10. 12 items at $3 each plus $2 tax 5
11. 15 watches at $30 each less a $50 discount 5
12. 3 lamps at $22 each plus 7 bulbs at $2 each 5
13. 100 belts at $4 each less discounts of $60 and $30 5
14. 3 dozen scissors at $11.20 per dozen plus a $4 shipping charge 5
15. 8 pounds of pears at $3 per pound plus 50¢ per pound for packaging 5
$28
$37.60
$310
$80
$400
$38
11
$25.96
$44.85
294
$31.60
Chapter 4 Word Problems and Equations 81
Assignment 4.1: Word Problems, Equations, and Series
Name
Date Score
A (20 points) Use the three-step process to solve the following word problems. (5 points for each correct
answer)
1. Mayberry Auto, Inc., conducted a direct-mail program. The manager determined that $30,000 in new business
came from the program. If the profit was 40% of sales, how much profit did the program produce?
2. Martha’s Beauty Salon charges $40 for a haircut, $48 for a facial, and $28 for hair coloring. If it had
20 haircut, 22 facial, and 18 coloring customers, what were its total sales?
3. Juan Lopez sold 11 life insurance policies with premiums totaling $24,200. He sold 14 auto policies with pre-
miums totaling $31,920. Which type of policy had the greater premium per sale?
4. The Tulsa Taxi Service had four taxi vehicles. Two got 25 miles per gallon of gas; two got 20 miles per
gallon of gas. The vehicles were each driven 8,000 miles per month. The gas cost $3.80 per gallon. What
was the amount of the gas bill for the month?
Score for A (20)
$5,472
AUTO
$2,360
$12,000
1
Learning Objectives
2 4
$30,000 3 0.40 5 $12,000
(20 3 $40) 1 (22 3 $48) 1 (18 3 $28) 5 $2,360
LIFE: $24,200 4 11 5 $2,200 AUTO: $31,920 4 14 5 $2,280
8,000 4 25 3 2 3 $3.80 5 $2,432 8,000 4 20 3 2 3 $3.80 5 $3,040 $2,432 1 $3,040 5 $5,472
82 Part 1 Fundamental Review
Assignment 4.1 Continued
16. $38 sale price plus $3 tax less an $11 discount plus a $5 delivery charge 5
17. 6 bath towels at $8 each and 4 hand towels at $3 each plus $2.50 tax 5
18. 4 dozen brushes at $25 per dozen plus $5 tax plus $7 shipping charge 5
19. 2 shirts at $30 each, 4 ties at $10 each, and 7 pairs of socks at $2 each 5
Score for C (10)
D (40 points) Complete the following equations by supplying the missing items. (4 points for each correct
answer)
20. 27 1 3 518 21. 13 157 1 28
22. 1 4 5 4 1 16 23. 400 5 17 2 2 1
24. 9 1 17 2 3 5 4 325 25. 160 4 4 1 2 5 7 3 7 2
26. 13 2 11 358 3 8 1 16 27. 3 3 3 3 5 9 4 3 3 9
28. 64 4 32 5 900 4 29. 15 2 9 2 2 5 25 2
Score for D (40)
E (20 points) In each of the following problems, a definite relationship exists among the numbers
in each series. Extend each series two items by following the correct process. (1 point for each correct
line)
30. Extend each series below through addition.
a. 4, 8, 12, 16, c. 2, 4, 7, 11, 13,
b. 1, 4, 5, 8,
31. Extend each series below through subtraction.
a. 50, 45, 40, 35, c. 100, 90, 81, 73,
b. 50, 45, 43, 38,
32. Extend each series below through multiplication.
a. 4, 8, 16, 32, c. 2, 4, 20, 40,
b. 5, 25, 125,
33. Extend each series below through division.
a. 15,625, 3,125, 625, 125, c. 10,000, 2,000, 1,000, 200,
b. 729, 243, 81, 27,
34. Extend each series below through combinations of the four processes above.
a. 72, 75, 69, 72, e. 7, 4, 8, 5, 9,
b. 200, 100, 300, 150, f. 30, 10, 60, 20,
c. 6, 9, 18, 21, 42, g. 10, 40, 20, 80,
d. 240, 120, 600, 300, 1,500, h. 100, 50, 40, 20,
Score for E (20)
10, 5750, 3,750
40, 16045, 90
120, 40450, 225
6, 1066, 69
9, 3
100, 2025, 5
625, 3,125
200, 40064, 128
36, 31
66, 6030, 25
9, 12
16, 2020, 24
21450
340
77
385
16
22
22
$114
$112
$62.50
$35
Chapter 4 Word Problems and Equations 83
Assignment 4.2: Word Problems, Formulas, and Equations
Name
Date Score
A (40 points) Solve the following word problems. (5 points for each correct answer)
1. A store regularly sold 2 cans of soup for $1.28. It advertised a special sale of 6 cans for $3.12. A customer
bought 12 cans at the sale. How much did the customer save over the regular price?
$1.44
1
Learning Objectives
2 3 4 5
$1.28 4 2 5 $0.64 each $0.64 2 $0.52 5 $0.12
$3.12 4 6 5 $0.52 each $0.12 3 12 5 $1.44 saved
2. A sales representative’s car gets 18 miles to a gallon of gas. It was driven 120 miles each day for 30 days. Gas
cost an average of $2.27 per gallon. What was the sales representative’s total 30-day cost for gas?
$454
120 3 30 5 3,600 mi
3,600 4 18 5 200 gal
200 3 $2.27 5 $454.00
3. A store clerk sold a customer a ruler for $1.67, three pencils for $0.29 each, notebook paper for $0.99, and
an eraser for $0.35 and was given $10.00 in payment. How much change did the clerk give the customer
from the $10.00? (All prices include tax.)
$6.12
$1.67 1 (3 3 $0.29) 1 $0.99 1 $0.35 5 $3.88
$10.00 2 $3.88 5 $6.12
4. A college student worked at a local store for $9.00 per hour, as his class schedule permitted. The student
worked 3 hours each Monday, Tuesday, Wednesday, and Thursday. He also worked 2 hours each Friday and
8 hours each Saturday. How many weeks did the student have to work to earn $792 for a new bicycle?
4 weeks
(3 3 4) 1 2 1 8 5 22 hrs per week
$9.00 3 22 5 $198 per week
$792 4 $198 5 4 weeks
5. A box, a crate, and a trunk weigh a total of 370 pounds. The crate weighs 160 pounds. The trunk weighs
4 pounds more than the box. What does the box weigh?
103 lb
370 2 160 2 4 4 2 5 103 lb
6. A hotel has 12 floors. Each floor has 30 single-person rooms and 40 two-person rooms. What is the total
guest capacity of the hotel?
1,320
30 1 (40 3 2) 3 12 5 1,320 guest capacity
7. A department store offers its customers socks for $1.50 per pair or $15.00 per dozen. If two customers buy
1 dozen together and each pays half the cost, how much will each customer save by paying the quantity
price?
$1.50
$1.50 3 12 5 $18.00
$18.00 2 $15.00 5 $3.00
$3.00 4 2 5 $1.50 each saved
A: (5 3 $9) 1 (2 3 $3) 5 $51
B: $8.50 1 $39.95 5 $48.45
C: $51 2 $48.45 5 $2.55
Score for A (40)
8. Supply Clerk A ordered five staplers at $9 each and two large boxes of staples for $3 each. Supply Clerk B
ordered a box of computer disks for $8.50 and a box of computer paper for $39.95. How much less did
Clerk B spend than Clerk A? (All prices include tax.)
$2.55