Deitz J.E., Southam J.L. Contemporary Business Mathematics for Colleges
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34 Part 1 Fundamental Review
✔
CONCEPT CHECK 2.1
a. Add , , and . b. Add and .
The least common denominator is 45. A common denominator is 6 9 54.
2
1
45
91
45
3
75
54
5 4
21
54
5 4
7
18
5
9
5 5
9 5
25
45
2
5
9
51 2
30
54
2
3
5
2 3 15
3 3 15
5
30
45
1
5
6
5 1
45
54
4
5
5
4 3 9
5 3 9
5
36
45
2
5
9
1
5
6
5
9
2
3
4
5
EXAMPLE G
Add and .
The least common denominator is 24.
3 3
7 7
10 10 1 11
11
24
11
24
35
24
20
24
5
6
15
24
5
8
STEP 1
7
5
6
3
5
8
The procedure for subtracting one fraction from another is almost the same as
the procedure for adding one fraction to another. When you calculate , is
called the minuend and is called the subtrahend, as in the subtraction of whole
numbers.
BORROWING 1
Sometimes, as with , the fraction part of the minuend is smaller than the fraction
part of the subtrahend. To make the fraction part of the minuend larger than the frac-
tion part of the subtrahend, you have to “borrow 1” from the whole-number part of the
minuend. Actually, you’re just rewriting the minuend. Remember that means ,
or the same as 2 1 ,2 , or . These are simply four different ways to
express the same quantity. Figure 2-3 is useful in understanding borrowing.
2
5
4
4
4
1
1
4
1
4
3 1
1
4
3
1
4
3
1
4
2
3
4
3
4
3
1
4
3
1
4
2
3
4
Subtracting Fractions and Mixed Numbers
4
Learning Objective
Subtract fractions and mixed
numbers.
2.5 Point out that one reason for
learning the words subtrahend and
minuend is to be able to write and read
explanations such as the one for
“borrowing.”
STEP 2
STEP 3
EXAMPLE H EXAMPLE I
EXAMPLE J EXAMPLE K
2
11
18
3
5
12
2 1
5
6
521
15
18
521
15
18
521
15
18
2 2
1
3
522
4
12
4
4
9
5 4
8
18
5 3
18
18
1
8
18
5 3
26
18
5
3
4
5 5
9
12
STEP 2STEP 1STEP 1
3
4
2
1
5
5
3 3 5
4 3 5
2
1 3 4
5 3 4
5
15
20
2
4
20
5
11
20
7
8
3
8
7 3
8
4
8
1
2
STEP 3STEP 1STEP 5STEP 3
Chapter 2 Fractions 35
Figure 2-3
a. 3 whole units plus of a unit
1
4
b.2 “whole units” plus of a unit
5
4
Adding and
Subtracting
Fractions and
Mixed Numbers
Video
to Subtract One Fraction or Mixed Number from Another
1. If necessary, change the fractions so that all fractions have a common de-
nominator.The common denominator is the denominator in the fraction
part of the answer.
2. If necessary, “borrow 1”from the whole-number part of the minuend so
that the fraction part of the minuend is at least as large as the fraction
part of the subtrahend.
3. Subtract the numerators in the fractions to make the numerator in the
fraction part of the answer.
4. If there are any whole-number parts, subtract them to make the whole-
number part of the answer.
5. Reduce the fraction part of the answer to lowest terms.
STEPS
STEPS 3 & 4
STEPS 3 & 4
© JOHN WEBER/GRAPHISTOCK/JUPITERIMAGES
2.6 In Step 2,“borrow 1”means to
“subtract 1 from the whole-number
part,convert it to a fraction with the
same denominator as the fraction part,
and add it to the fraction part.”That
will be clear to most students,but it is
a cumbersome and potentially confus-
ing instruction to include in the “steps.”
In fractions, multiplication and division do not require common denominators. This
means that multiplication and division are simpler than addition and subtraction. Recall
that any mixed number can be changed to an improper fraction. Also, a whole number
can be written as an improper fraction by writing the whole number in the numerator
with a denominator of 1. For example, the whole number 5 can be written as the im-
proper fraction .
5
1
36 Part 1 Fundamental Review
a. Subtract from . b. Subtract 2 from 6 .
The least common denominator is 24. The least common denominator is 30.
6 6 5
2 2 2
COMPLETE ASSIGNMENT 2.1.
3
28
30
5 3
14
15
1
24
27
30
27
30
9
10
20
24
5
6
55
30
25
30
5
6
21
24
7
8
5
6
9
10
7
8
5
6
✔
CONCEPT CHECK 2.2
Multiplying Fractions, Mixed Numbers,
and Whole Numbers
5
Learning Objective
Multiply fractions, mixed numbers,
and whole numbers.
to Multiply Fractions, Mixed Numbers, and Whole Numbers
1. If necessary, change any mixed or whole numbers to improper fractions.
2. Multiply all the numerators to get the numerator of the product.
3. Multiply all the denominators to get the denominator of the product.
4. Change the product to a proper fraction or mixed number in lowest terms.
STEPS
Multiplication
and Division of
Mixed Numbers
Video
EXAMPLE L EXAMPLE M
2
3
3
4
5
3
5
6
5
2 3 4 3 5
3 3 5 3 6
5
40
90
5
4
9
1
2
3
3
4
5
5
5
3
3
4
5
5
5 3 4
3 3 5
5
20
15
5 1
5
15
5 1
1
3
STEPS 2 & 3STEP 4STEPS 2 & 3STEP 1
Note: The word of often means multiply when it is used with fractions. For example, you
know that “ of 6 bottles” is 3 bottles. And of 6 . For this reason, multi-
plication may even be the most important arithmetic operation with fractions. In verbal
communication, we will often be saying expressions like “ of 6.”
1
2
1
2
3
6
1
5
6
2
5 3
1
2
1
2
STEP 4
As the last step in example M, we reduced the fraction to its lowest terms, . Recall that
reducing this fraction means that we divide both the numerator and the denominator by
10. As an option, we can do that division in advance, before doing any multiplication.
Examining the three numerators and three denominators, we discover that they have
common factors of 2 and 5. Divide out, or cancel, both the 2 and the 5 in both the
numerators and denominators as shown in example N. This division of the common
factors is often called cancellation. Canceling common factors is an option; it is not
required to calculate the correct product.
EXAMPLE N
Multiply the three fractions, using cancellation.
EXAMPLE O
Multiply the fraction and the whole number, using cancellation.
EXAMPLE P
Multiply the fraction and the mixed number, using cancellation.
2
5
2
3
4
2
5
11
4
2
1
5
11
4
2
1 11
5 2
11
10
1
1
10
2
5
of 2
3
4
5
STEP 4STEPS 2 & 3STEP 1
3
4
of 12
3
4
12
3
4
12
1
3
4
1
12
3
1
3 3
1 1
9
1
9
2
3
3
4
5
3
5
6
5
2
3
3
4
5
1
3
5
1
6
5
2
3
3
4
2
5
1
3
5
1
6
3
5
2 3 2 3 1
3 3 1 3 3
5
4
9
4
9
40
90
Chapter 2 Fractions 37
Canceling Common Factors in Numerators
and Denominators
Multiply the fraction, whole number, and mixed number, using cancellation.
1
8
3 4 3 2
1
3
5
1
8
3
4
1
3
7
3
5
1
8
2
3
4
1
1
3
7
3
5
1 3 1 3 7
2 3 1 3 3
5
7
6
5 1
1
6
✔
CONCEPT CHECK 2.3
2.7 Point out in example N that it is
more efficient to do the cancellation
in one step, as in
It is shown with two steps in the exam-
ple for clarity.
2
3
3
4
2
5
1
3
5
1
6
3
STEPS 2 & 3STEP 1 STEP 4
STEPS 2 & 3STEP 1 STEP 4
38 Part 1 Fundamental Review
Dividing Fractions, Mixed Numbers,
and Whole Numbers
6
Learning Objective
Divide fractions, mixed numbers, and
whole numbers.
Recall that with whole numbers, division is the inverse of multiplication. You can check
a multiplication problem with division. With fractions, you actually perform a division
problem by doing multiplication. That is, you invert the divisor and multiply.
to Divide Fractions, Mixed Numbers, and Whole Numbers
1. If necessary, change the dividend and/or the divisor from mixed or whole
numbers to improper fractions.
2. Invert the divisor (that is, exchange the numerator and denominator).
3. Change the division symbol to a multiplication symbol.
4. Multiply the two factors (canceling where possible, if desired).
5. Write the result as a proper fraction or mixed number in lowest terms.
STEPS
Multiplication
and Division of
Mixed Numbers
VIDEO
EXAMPLE Q
3
10
4
2
5
5
3
10
3
5
2
5
3
10
2
3
5
1
2
5
3 3 1
2 3 2
5
3
4
STEP 4STEPS 2 & 3
EXAMPLE R
6 4 1
3
5
5
6
1
4
8
5
5
6
1
3
5
8
5
6
3
1
3
5
8
4
5
3 3 5
1 3 4
5
15
4
5 3
3
4
STEP 5STEP 4STEPS 2 & 3STEP 1
Divide .
Change both mixed numbers to improper fractions: 4 .
Invert the divisor to and multiply:
COMPLETE ASSIGNMENT 2.2.
15
4
3
2
3
5
15
5
4
2
3
2
1
3
1
5
5 3 1
2 3 1
5
5
2
5 2
1
2
2
3
3
2
3
2
15
4
3
3
4
by 1
1
2
✔
CONCEPT CHECK 2.4
© R. ALCORN/CENGAGE LEARNING
Chapter 2 Fractions 39
cancel higher terms lowest terms
cancellation improper fraction mixed number
common denominator least common denominator numerator
denominator lower terms proper fraction
fractions
Chapter Terms for Review
40 Part 1 Fundamental Review
THE BOTTOM LINE
Summary of chapter learning objectives:
Learning Objective
2.1
Change improper fractions and mixed numbers.
Example
1(a). Change to a mixed number.
1(b). Change 2 to an improper fraction.
5
6
18
5
Answers: 1(a). 3 1(b). 2(a). 2(b). 3. 4. 2 5. 6. 2
2
5
4
21
13
20
4
11
24
35
60
2
5
17
6
3
5
2.2
Change fractions to lower and higher terms.
2.3
Add fractions and mixed numbers.
2.4
Subtract fractions and mixed numbers.
2.5
Multiply fractions, mixed numbers, and whole numbers.
2.6
Divide fractions, mixed numbers, and whole numbers.
2(a). Reduce to lowest terms.
2(b). Raise to sixtieths; that is, .
?
60
7
12
7
12
12
30
3. Add , , and .2
3
4
5
6
7
8
4. Subtract 1 from 4 .
2
5
3
4
5. Multiply: .
2
9
3
6
7
6. Divide: .1
4
5
4
3
4
Chapter 2 Fractions 41
Write all answers as proper fractions or mixed numbers in lowest terms.
Answers to the Self-Check can be found in Appendix B at the back of the text.
SELF-CHECK
Review Problems for Chapter 2
1 Change to an improper fraction
2 Change to a mixed number
3 Reduce to lowest terms
4 Raise to 56ths
5 Add
6 Add
7 Add
8 Subtract
9 Subtract
10 Subtract
11 Multiply
12 Multiply
13 Multiply
14 Divide
15 Divide
16 Divide
1
2
7
by 1
1
14
1
11
25
by
24
35
15
16
by
5
12
2
1
10
,
8
15
, and 2
1
12
9
16
by 1
13
15
5
6
by
12
35
2
4
9
from 4
1
5
8
9
from 2
5
6
1
3
from
4
5
3
4
, 2
4
5
, and 4
5
8
and 1
1
6
2
3
,
3
5
, and
3
10
10
14
54
63
90
12
2
5
6
17 JoAnn Brandt decided to use an expensive, but effective, herbicide to kill weeds and brush on a client’s land. For
one part of the land, she needed quarts of herbicide; for a second part, she needed quarts; and for the third
part, she needed quarts. In total, how many quarts of herbicide did JoAnn need for this client?
18 An electrician had a piece of electrical conduit that was feet long. She cut off two pieces that were feet and
feet long, respectively. How many feet of conduit did she have left over?
19 The Central Hotel just hired a new chef. This chef makes a hot sauce that uses tablespoons of chili powder, but
he needs to increase the recipe by times. How many tablespoons of chili powder should he use?
20 How many whole pieces of copper inches long can be cut out of one piece that is inches long?
How long is the shorter piece that is left over?
24
1
2
2
5
8
3
1
2
1
3
4
5
2
3
4
3
4
12
1
2
1
5
6
2
3
4
3
2
3
42 Part 1 Fundamental Review
Notes
6
58
45
5 7
13
45
8
59
30
5 9
29
30
7
12
6
5 95
43
24
5 6
19
24
1 1
9
45
1 5
10
30
1
5
6
1 4
22
24
3
24
45
3
25
30
3
4
6
15
24
2
25
45
24
30
4
3
6
1
6
24
28. 29. 30. 31.
32. 33. 34. 35.
Score for C (24)
1 1
1
5
51 5
1
3
51
5
6
51 4
11
12
5
3
8
15
5 3
5
6
5 3
2
3
5
5
8
5
2
5
9
5
4
5
5 4
1
2
51
1
4
5
1 2
3
4
51 2
11
16
1
3
10
1
3
8
1
2
3
5
9
16
7
10
1
8
Chapter 2 Fractions 43
Assignment 2.1: Addition and Subtraction of Fractions
Name
Date Score
A (12 points) Change the improper fractions to whole numbers or to mixed numbers. Change the mixed
numbers to improper fractions. (1 point for each correct answer)
1
Learning Objectives
2 3 4
1. 2. 3.
4.
25
15
18
6
32
10
13
6
5.
6. 7.
8.
2
11
12
3
7
10
25
8
23
13
9.
10. 11.
12.
4
5
6
6
3
5
3
3
4
2
5
8
Score for A (12)
B (15 points) In problems 13–20, reduce each fraction to lowest terms. In problems 21–27, raise each
fraction to higher terms, as indicated. (1 point for each correct answer)
13. 14.
15. 16.
12
16
18
30
9
24
10
25
17. 18. 19. 20.
16
32
42
60
24
42
32
48
21.
22.
23. 24.
7
12
5
36
5
8
5
24
3
4
5
20
5
6
5
18
25. 26. 27.
4
5
5
45
2
3
5
15
3
7
5
35
Score for B (15)
C (24 points) Add the following fractions and mixed numbers.Write the answers as fractions or mixed
numbers, with fractions in lowest terms. (3 points for each correct answer)
21
8
1
10
13
2
1
6
2
3
2
5
4
7
3
8
7
10
3
5
4
8
5
1
2
3
17
12
5 4
5
12
2
20
16
5 3
1
4
10
10
5 1
1
2
3
4
15
15
15
15
10
36
21
15
4
3
1
8
3
1
5
29
6
35
12
1
2
3
33
5
37
10
1 2
9
12
1
8
12
3