2.4 MULTIPLYING AND DIVIDING FRACTIONS AND MIXED NUMBERS
Greg is building a simple shelf to hold his DVD collection. He has 64 DVDs
in his collection and he knows that each DVD case is 5/8 of an inch wide.
He’s not sure how wide the shelf needs to be, in order for it to hold his
complete collection.
How wide must the shelf be?
40
______________________
inches
Assess your readiness to complete this activity. Rate how well you understand:
Not
ready
Almost
ready
Bring
it on!
• the terminology and notation used when multiplying and dividing fractions
• the process of reducing before multiplying
• how to deal with mixed numbers in multiplication and division
• the division process—how and why division is turned into multiplication
• in what form to present a final answer
• how to validate the answer to a multiplication or division problem involving
fractions or mixed numbers
•
Multiplying any given combinations of fractions •
and mixed numbers correctly
Dividing any given combinations of fractions and
mixed numbers correctly
– presentation of the final answer in lowest terms
– presentation of the final answer in lowest terms
– validation of the answer
– validation of the answer
77
Chapter 2 — Fractions
Simply multiplying the numerators and denominators of two fractions to find their product will often result
in a fraction that must be reduced to lowest terms. The Methodology for Multiplication uses canceling before
finding the product so as not to end up with large numbers to reduce for the final answer. It also addresses how
to efficiently multiply factors that are mixed numbers.
7
4
by 4 .
8
5
3 1
Example 2: Multiply: 3 ×1
4 5
Example 1: Multiply
Try It!
Steps in the Methodology
Step 1
Set up the
problem.
Step 2
Convert
mixed
numbers.
Step 3
Prime factor
and cancel.
Set up the problem horizontally for
ease of calculation.
7
4
×4
8
5
Convert the mixed numbers to
improper fractions and rewrite the
problem.
7 24
×
8 5
Special
Case:
Whole number factor(s)
(see Model 1)
Simplify before multiplying.
Determine the prime factorizations
of both numerators and
denominators, then cancel all
common factors.
Shortcut:
Special
Case:
Step 4
Multiply
across.
Step 5
Convert to
a mixed
number (if
necessary).
78
Example 2
Example 1
1
7
1
1
1
2• 2• 2
×
1
Quick reduction
(see Model 2)
Product of more than two
fractions (see Model 3)
Multiply the remaining numerators
and use the product as the new
numerator. Multiply the remaining
denominators, and use the product
as the new denominator.
If the product is an improper
fraction, convert it to a mixed
number.
1
2 • 2 • 2 •3
5
7 • 3 21
=
5
5
21
1
=4
5
5
Activity 2.4 — Multiplying and Dividing Fractions and Mixed Numbers
Steps in the Methodology
Step 6
Verify that
the fraction is
reduced.
Step 7
Verify that the fraction is fully
reduced.
1
is fully reduced.
5
Note: If you canceled all
common factors in Step 3, it will
be fully reduced. If not, reduce
fully now.
Present your answer.
4
Present the
answer.
Step 8
Validate your
answer.
Validate the final answer by
division, using the original
fractions and/or mixed numbers.
=
=
1
5
1
×
5
1
2•2•2• 3
7
8
5×2
2
3
2
3
In a fraction problem, if a factor is a whole number,
write it in its improper form and proceed from there.
Step 2
5 8
×
1 3
Step 3
5 8
×
no common factors
1 3
Steps 4 & 5
3 •7
Special Case: Whole Number Factor(s)
Multiply: 5 × 2
Step 1
1
5
1
4
4 ÷4
5
5
21 24
=
÷
5
5
21 5
=
×
5 24
1
Model 1
Example 2
Example 1
5 × 8 40
1
=
= 13
3
3
3
Step 6
1
is fully reduced
3
Step 7
Answer : 13
Step 8
Validate:
1
3
1
2
13 ÷ 2
3
3
40 8
=
÷
3
3
5
=
=
40
1
3
1
×1
3
8
5
=5
1
79
Chapter 2 — Fractions
Model 2
Shortcut: Quick Reduction
Multiply 1
7
5
by 2
9
8
Shortcut version (optional)
Step 1
7
5
1 ×2
9
8
Step 1
7
5
1 ×2
9
8
Step 2
16 21
×
9
8
Step 2
16 21
×
9
8
1
Step 3
1
1
2 • 2 • 2 •2
1
3 •3
1
×1
2
3 •7
1
1
2• 2• 2
Step 3
16
3
9
7
×
21
1
8
8 is a factor of
both 8 and 16.
3 is a factor of
both 9 and 21.
Step 8 Validate:
Steps 4 & 5
2 • 7 14
2
=
=4
3
3
3
Step 6
2
is fully reduced
3
Step 7
Answer : 4
2
5
÷2
3
8
14 21
=
÷
3
8
14 8
=
×
3 21
4
1
2• 7 2•2•2
=
×
1
3
3• 7
16
7
=
=1
9
9
80
Cancel the
factors (not
necessarily
prime factors)
you recognize as
common to both
numerator and
denominator.
2
3
Step 8
Validate:
7 is a factor of
14 and 21.
2
5
÷2
3
8
14 21
=
÷
3
8
4
2
14
8
×3
3
21
16
7
=
=1
9
9
=
Activity 2.4 — Multiplying and Dividing Fractions and Mixed Numbers
Model 3
Special Case: Product of More than Two Fractions
Find the product of
3 4
1
× ×2
10 9
7
Step 1
Step 2
Step 3
3 4
1
, , and 2 .
10 9
7
=
3 4 15
× ×
10 9 7
1
=
2
3
10
4
×3
9
The numerator and denominator in which you recognize a
common factor do not have to be in adjacent fractions.
3
×
15
7
The common factor of 3 and 9 is 3.
The common factor of 15 and 10 is 5.
2 is a factor of 4 and 2. 3 and 3 cancel.
Continue canceling common factors.
Rewrite and cancel again
Steps 4 & 5
Step 6
Step 7
Step 8
=
1
2
4
1
1
3
×1 ×
1
7
2
3
=
OR
12
3
10
×1
2
4
3
9
13
×
15
7
2
, proper fraction
7
2
is fully reduced
7
2
Answer :
7
Validate with two divisions:
2
1 4 2 15 4
÷2 ÷ = ÷
÷
7
7 9 7
7
9
1
=
1
2
7
1
×5
7
15
3
×2
9
4
=
3
10
Dividing Fractions and Mixed Numbers versus
Multiplying Fractions and Mixed Numbers: Critical Differences
It is important to note that the following methodology, Dividing Fractions and Mixed Numbers, is very
similar to the methodology for Multiplying Fractions and Mixed Numbers. There are only two differences
between the methodologies; the most critical is that there is an additional step in the Division Methodology
wherein the divisor is inverted and the operation is changed from division to multiplication. The second
difference is that the validation process for division uses multiplication.
81
Chapter 2 — Fractions
The methodology below converts a given division problem into a multiplication problem to solve. While
Example 1 is worked out, step by step, you are welcome to complete Example 2 as a running problem.
3
1
by 1 .
8
2
3
7
Example 2: Divide: 8 ÷ 1
4
8
Example 1: Divide 6
Try It!
Steps in the Methodology
Step 1
Set up the
problem.
Step 2
Convert mixed
numbers.
Step 3
Invert the
divisor and
multiply.
Set up the problem horizontally with the
dividend first.
51 3
÷
8
2
Invert the divisor (the second fraction) and
change the operation to multiplication.
51 2
×
8 3
Division is defined as the inverse operation
of multiplication. This means that dividing a
number by a second number is the same as
multiplying the first number by the inverse
of the second number. For example:
15
3
15
3
is the same as 15 ×
1
3
Cancel the common factors by prime
factoring first or by using the quick
reduction shortcut.
1
1
4
Multiply
across.
Step 6
Convert to a
mixed number.
82
1
3 • 17
2 •2•2
or
17
Step 5
3
1
÷1
8
2
Special Whole number divisor or
Case: dividend (see Model 1)
and
Cancel.
6
Convert mixed numbers to improper
fractions and rewrite the problem.
15 ÷ 3 can be written as
Step 4
Example 1
51
8
1
×1
×1
2
3
Multiply the remaining numerators and
denominators.
17
17
=
2•2
4
Convert to a mixed number, if necessary.
17
1
=4
4
4
2
3
Example 2
Activity 2.4 — Multiplying and Dividing Fractions and Mixed Numbers
Steps in the Methodology
Step 7
Example 1
Verify that the fraction is fully reduced.
1
is fully reduced
4
Verify the
fraction is
reduced.
Step 8
Present your answer.
4
Present the
answer.
Step 9
Validate your
answer.
Model 1
Divide 10
Validate your final answer by
multiplication, using the original
fractions and/or mixed numbers.
Example 2
1
4
1
1
×1
4
2
17 3 no common
=
×
4 2 factors to cancel
51
3
=
=6
8
8
4
Special Case: Whole Number Divisor or Dividend
2
by 4.
5
2
÷4
5
Step 1
10
Step 2
52 4
÷
5
1
Step 3
52 1
×
5 4
In a fraction problem, if the divisor or dividend is a whole number,
write it as “the whole number” and proceed from there.
1
13
Steps 4, 5 & 6
52
1
13
3
×1 =
=2
5
5
5
4
Step 7
3
is fully reduced
5
Step 8
Answer : 2
Step 9
Validate:
3
5
3
2 ×4
5
13 4 52
3
=
× =
= 10
5 1
5
5
83
Chapter 2 — Fractions
Divide: 3 ÷ 4
5
7
Steps 1 & 2
3÷ 4
Step 3
=
Steps 4 & 5
=
5 3 33
= ÷
7 1
7
3 7
×
1 33
Steps 6 & 7 proper fraction, fully reduced
Step 7
Answer :
7
11
Step 8
Validate:
7
5
×4
11
7
1
3
7
7
× 11
=
1
11
33
1
=
Model 2
Divide
7
3
by 8 .
8
4
Divide:
×
33
1
7
=
3
=3
1
3 1
÷
8 14
7
3
÷8
8
4
Steps 1 & 2
3
1
÷
8 14
Step 2
7 35
÷
8
4
Step 3
3 14
×
8
1
Step 3
7
4
×
8 35
Steps 4 & 5
1
2
7
8
1
×5
4
35
Step 5
1×1
1
=
2 ×5 10
Step 6
proper
Step 7
1
is fully reduced
10
Step 8
Answer :
Step 9
Validate:
1
10
3
4
8
7
×
no mixed numbers
to convert
14 3 × 7 21
=
=
1
4
4
1
4
Step 6
=5
Step 7
1
is fully reduced
4
Step 8
Answer : 5
1
4
Step 9
Validate:
1 1
5 ×
4 14
3
=
1
3
×8
10
4
=
7
1
35
=2
×
4
10
7
7
=
=
2× 4 8
84
11
3
Model 3
Step 1
Step 4
1
7
21
1
×2
4
14
3
3
=
4×2 8
Activity 2.4 — Multiplying and Dividing Fractions and Mixed Numbers
Make Your Own Model
Either individually or as a team exercise, create a model demonstrating
how to solve the most difficult problem you can think of.
Answers will vary.
Problem: _________________________________________________________________________
85
Chapter 2 — Fractions
1. What is the first critical step when multiplying or dividing mixed numbers?
The numbers involved must all be made into improper fractions or proper fractions.
2. How are whole numbers converted to fractions for multiplying and dividing?
Whole numbers are converted to fractions by making the denominator a one. The whole number is the numerator
and the denominator is a “1.”
3. How do you convert a division of fractions into a multiplication of fractions?
Replace the divisor with its reciprocal and set up as a multiplication: i.e. invert the divisor and change the operation
to multiplication.
4. What can you do to simplify a multiplication of fractions problem before computing the final answer?
“Canceling” (dividing out) can be done with ANY common factor, not just prime factors.
5. What is the result when all factors in the numerators cancel out?
1
The result is a fraction with 1 in the numerator. Example:
4
6. What is the result when all factors in the denominators cancel out?
The denominator will be one and the result then will be a whole number. Example:
6
=6
1
7. When you multiply a proper fraction by a second number, will the product be greater or less than the
second number? Explain.
It will always be less than the second number. A fractional part of any number is always smaller than the original
number.
8. What aspect of the model you created is the most difficult to explain to someone else? Explain why.
Answers will vary.
86
Activity 2.4 — Multiplying and Dividing Fractions and Mixed Numbers
Solve each problem and validate your answer.
Problem
1)
12 7
×
35 30
2)
2 1 4 3
× × ×
3 8 5 5
3)
3 2
÷
8 7
4) 5
Worked Solution
Validation
5
÷3
8
87
Chapter 2 — Fractions
Problem
5) 6
3
4
×8
4
9
6) 3
1
1
÷5
2
4
7) 6 ÷
1
3
8) Bruno’s share of
the profits from a
land sale is to be
2/7 of $280,000.
Calculate his
share.
88
Worked Solution
Validation
Activity 2.4 — Multiplying and Dividing Fractions and Mixed Numbers
Perform the indicated operations and validate your answers.
2 3
1. 3 ×
9 5
2. 2
3.
1
7
1
×1
8
4
3
19
32
3 2 14 10
× × ×
7 5 15 11
1
7
4. 2 ÷ 4
6
8
1
1
5.. 3 ÷ 1
9
2
14
15
8
55
4
9
6. 2 ÷ 1
2
2
27
1
1
1
2
3
7. 12 × 1
1
1
×4
78
2
3
1 5
8. 12 ÷
2 7
17
1
2
In the second column, identify the error(s) you find in each of the following worked solutions. If the answer
appears to be correct, validate it in the second column and label it “Correct.” If the worked solution is incorrect,
solve the problem correctly in the third column and validate your answer in the last column.
Worked Solution
What is Wrong Here?
1) 12
1
2
×6
6
3
Identify Errors
or Validate
You must change to
improper fractions,
reduce, then
multiply to get the
answer.
Correct Process
Validation
12 1 x 6 2
3
6
10
20
73
x
=
3
3 6
= 730
9
1
Answer 8 1 9
9 730
72
10
9
1
81 1 ÷ 6 2
3
9
730 ÷ 20
3
9
1
73 0
3
=
x
20
3 9
= 73 = 12 1
6
6
)
89
Chapter 2 — Fractions
Worked Solution
What is Wrong Here?
2) 5
3
1
÷3
5
8
Identify Errors
or Validate
You must change
division to
multiplication .
You do this by
multiplying by the
reciprocal of the
second number.
Then reduce.
3) Find the product of
3 4
5
,
, and .
5 15
8
1
1
4) 1 × 5
7
4
90
Reduced incorrectly.
Cannot use 5 twice in
the denominators.
CORRECT
Correct Process
Validation