Section 3.4
PRE-ACTIVITY
PREPARATION
Multiplying and Dividing
Fractions and Mixed Numbers
How much flour will you need to double a recipe that calls
for 3⅛ cups of flour? What is the surface area of your deck
that measures 18¼ feet by 20⅝ feet? How many curtain
panels can you cut from a length of fabric 6⅜ yards long
if each panel is to be 1½ yards long? When the numbers in
daily tasks such as cooking, carpentry, sewing, redecorating,
and home repair are presented in fraction form, knowing
how to multiply and divide such numbers is a practical skill
to possess.
Beyond its relevance to these everyday contexts, having a
thorough understanding of multiplying and dividing fractions
is necessary for any further study of mathematics.
LEARNING OBJECTIVES
•
Master the multiplication of fractions and mixed numbers.
•
Master the division of fractions and mixed numbers.
TERMINOLOGY
PREVIOUSLY USED
NEW TERMS
TO
cancel
improper fraction
invert
common factor
mixed number
of
dividend
product
reciprocal
divisor
reduce
LEARN
factor
BUILDING MATHEMATICAL LANGUAGE
To invert a fraction is to interchange the numerator and denominator of the fraction.
3
8
For example, to invert , write .
8
3
271
Chapter 3 — Fractions
272
The reciprocal of a fraction is the fraction that results from inverting it.
For example, 9 is the reciprocal of 5 .
5
9
When a given fraction is multiplied by its reciprocal, the product will always be 1.
5 9
3 8
For example, × = 1,
× = 1, and so on.
8 3
9 5
The word of after a proper fraction indicates multiplication (read, “times”).
3
3
For example, to calculate
of 52 acres, you would multiply × 52 to get 39 acres.
4
4
The product of two or more fractions is the product of the numerators over the product of their
denominators, as illustrated by the following example.
Example: Find
First, shade in
2
4
of
.
3
7
4
of a whole unit.
7
Then divide the shaded portion into thirds and mark
2
3
4
7
9
9
9
9
2
of the shaded portion with 9 ’s.
3
9
9
9
9
⎛8⎞
Now the whole has been divided into 21 parts, with 8 of them marked ⎜⎜ ⎟⎟ .
⎜⎝ 21⎟⎠
2
4 2× 4
8
That is,
of (×) =
=
3
7 3× 7 21
Section 3.4 — Multiplying and Dividing Fractions and Mixed Numbers
273
METHODOLOGY
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.
Be sure to note the shortcut for canceling in Step 3!
Multiplying Fractions and Mixed Numbers
►
►
7
4
by 4 .
8
5
3 1
Example 2: Multipy: 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.
Example 1
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.
Special
Case:
7 24
×
8 5
Whole number factor(s)
(see page 277, Model 2)
Simplify before multiplying.
Determine the prime factorizations of
both numerators and denominators;
then cancel all common factors.
1
7
1
1
1
2• 2• 2
×
1
???
Quick reduction
(see page 275, Model 1)
Product of more than
Special
two fractions
Case:
(see page 277, Model 3)
Step 4
Multiply
across.
Multiply the remaining numerators and
use the product as the new numerator.
Multiply the remaining denominators, and
use the product as the new denominator.
1
2 • 2 • 2 •3
5
Why can you do this?
Shortcut:
Example 2
7 • 3 21
=
5
5
Chapter 3 — Fractions
274
Steps in the Methodology
Step 5
Convert to a
mixed number
(if necessary).
Step 6
Verify that
the fraction is
reduced.
Step 7
Example 1
If the product is an improper
fraction, convert it to a mixed
number.
Verify that the fraction is fully
reduced.
Note: If you canceled all common
factors in Step 3, it will be fully
reduced. If not, reduce fully now.
21
1
=4
5
5
1
is fully reduced.
5
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.
Example 2
1
5
1
4
4 ÷4
5
5
21 24
=
÷
5
5
21 5
=
×
5 24
1
=
=
3 •7
1
5
1
×
5
1
2•2•2• 3
7
9
8
???
Why can you do Step 3?
The product of two or more fractions is the product of their numerators over the product of their denominators.
It is the same whether you cancel before you multiply the numerators and denominators as indicated in Step 3,
or after you find their products and reduce the result to lowest terms.
For Example 1,
1
1
1
1
1
1
canceling before multiplying:
7 24
7
2 × 2 × 2 ×3 7 ×3 21
× =1
×
=
=
1
1
8 5
5
5
5
2× 2× 2
canceling after multiplying:
7 24 7 × 24 168
2 × 2 × 2 ×3× 7 3× 7 21
× =
=
= 1
=
=
1
1
8 5
8×5
40
5
5
2 × 2 × 2 ×5
Section 3.4 — Multiplying and Dividing Fractions and Mixed Numbers
275
MODELS
Model 1
A
►
Shortcut: Quick Reduction
Multiply 1
5
7
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
THINK
8 is a factor of
both 8 and 16.
3 is a factor of
both 9 and 21.
Step 8
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
Validate: 4
1
2• 7 2•2•2
×
1
3
3• 7
16
7
=
=1 9
9
9
=
Shortcut: Cancel
the factors (not
necessarily prime
factors) you easily
recognize as being
common to both
numerator and
denominator.
2
3
Step 8
Validate:
THINK
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
9
=
Chapter 3 — Fractions
276
B
►
Multiply: 1
Step 1
Step 2
4
19
×10
36
5
1
19
4
×10
36
5
55 54
×
36 5
Step 3
THINK
11
5 is a factor of 5 and 55.
6 is a factor of 36 and 54.
6
55
36
9
×
54
1
5
Continue canceling until there are no
more common factors to divide out.
THINK
3 is a factor of the “new” numerator
9 and the “new” denominator 6.
11
26
55
36
39
×
1
54
5
OR use this optional notation: When you recognize that you can cancel using “new”
numerators and denominators, you may choose to rewrite the problem with its “new”
factors so as not to lose track of them in your notation.
11
For example,
6
9
55
×
36
54
1
Steps 4 & 5
11× 3 33
1
=
= 16
2 ×1
2
2
Step 6
1
is fully reduced
2
Step 7
Answer : 16
Step 8
Validate:
5
⇒
11
2
6
1
2
1
4
÷ 10
2
5
33 54
=
÷
2
5
16
11
=
33
5
× 18
2
54
11×5
2 ×18
55
19
=
=1 9
36
36
=
3
×
9
1
Section 3.4 — Multiplying and Dividing Fractions and Mixed Numbers
Model 2
Special Case: Whole Number Factor(s)
Multiply: 5 × 2
Step 1
5×2
2
3
2
3
In a fraction problem, if a factor is a whole number, write it in
its improper form “the whole number” and proceed from there.
1
Step 2
5 8
×
1 3
Step 3
5 8
×
no common factors
1 3
Step 6
5
=
1
Answer : 13
3
Step 7
Model 3
=
=
3 4 15
× ×
10 9 7
8
5
= 59
1
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.
THINK
Rewrite and cancel again
Steps 4 & 5
Step 8
3
3
3 4
1
, , and 2 .
10 9
7
The common factor of 15 and 10 is 5.
Continue canceling common factors.
Step 7
1
1
×1
3 4
1
× ×2
10 9
7
Step 1
Step 6
40
Special Case: Product of More than Two Fractions
Find the product of
Step 3
1
2
Validate: 13 ÷ 2
3
3
40 8
=
÷
3
3
Step 8
5 × 8 40
1
=
= 13
3
3
3
1
is fully reduced
3
Steps 4 & 5
Step 2
277
=
1
2
4
2 is a factor of 4 and 2. 3 and 3 cancel.
THINK
1
1
3
×1 ×
1
7
2
3
=
OR
12
3
10
×1
2
3
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
=
4
3
9
10
9
13
×
15
7
Chapter 3 — Fractions
278
METHODOLOGY
The methodology below converts a given division problem into a multiplication problem to solve.
Dividing Fractions and Mixed Numbers
►
►
1
3
by 1 .
8
2
7
3
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.
Example 1
Set up the problem horizontally with the
dividend first.
6
Convert mixed numbers to improper
fractions and rewrite the problem.
51 3
÷
8
2
Special Whole number divisor or dividend
Case: (see page 281, Model 3)
Step 3
Invert the
divisor and
multiply.
Step 4
Cancel.
Invert the divisor (the second fraction)
and change the operation to multiplication.
51 2
×
8 3
???
Why do you do this?
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.
Multiply the remaining numerators and
denominators.
Convert to a mixed number, if necessary.
1
3 • 17
2 •2•2
or
17
Step 5
3
1
÷1
8
2
51
8
1
×1
×1
2
3
17
17
=
2•2
4
17
1
=4
4
4
2
3
Example 2
Section 3.4 — Multiplying and Dividing Fractions and Mixed Numbers
279
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.
Example 2
Validate your final answer by
multiplication, using the original
fractions and/or mixed numbers.
1
4
1
1
×1
4
2
17 3 no common
=
×
4 2 factors to cancel
51
3
=
=6 9
8
8
4
???
Why do you do Step 3?
Consider Example 1, 51 ÷ 3 , the number 51 divided by the number 3 .
8
8 2
2
51
You know that you can also write this division as 8 .
3
2
The Identity Property of Multiplication tells you that if you multiply this fraction by
the value of your original number will not change.
2
3 (which equals 1),
2
3
2
51
51
8 = 8 × 3
2
3
3
3
2
2
2
Note that your denominator now equals 1 because 3 times its reciprocal
equals 1.
3
2
⎛13 1 2
⎞⎟
⎜⎜
⎟⎟
×
=
1
⎜⎜ 1
1
⎟⎟
3
⎝ 2
⎠
51 2
×
51 2
51 3 51 2
You are left with 8 3 , or × . That is to say,
÷ = × .
1
8 3
8 2 8 3
The same mathematical reasoning will hold for all dividends and divisors. That is why Step 3 simply says
“invert the divisor and change the operation to multiplication.”
Chapter 3 — Fractions
280
MODELS
Model 1
Model 2
Divide
7
3
by 8 .
8
4
Divide:
3 1
÷
8 14
Step 1
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
Step 4
2
7
8
1
×5
4
35
Step 6
=5
3
4
8
7
×
no mixed numbers
to convert
14 3 × 7 21
=
=
4
1
4
1
4
Step 5
1×1
1
=
2 ×5 10
Step 7
1
is fully reduced
4
Step 6
proper
Step 8
Answer : 5
1
4
Step 7
1
is fully reduced
10
Step 9
Validate:
1 1
5 ×
4 14
Step 8
Answer :
Step 9
Validate:
1
10
3
1
3
×8
10
4
7
1
35
×
2
4
10
7
7
=
= 9
2× 4 8
=
=
21
1
×2
4
14
=
3
3
= 9
4×2 8
Section 3.4 — Multiplying and Dividing Fractions and Mixed Numbers
Model 3
A
►
Divide 10
Special Case: Whole Number Divisor or Dividend
2
by 4.
5
Step 1
2
10 ÷ 4
5
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
B
►
281
52
13
3
1
=2
×1 =
5
5
5
4
Step 7
3
is fully reduced
5
Step 8
Answer : 2
Step 9
Validate:
Divide: 3 ÷ 4
3
5
3
2 ×4
5
13 4 52
3
=
× =
= 10 9
5 1
5
5
5
7
Steps 1 & 2
3÷ 4
Step 3
=
5 3 33
= ÷
7
7 1
3 7
×
1 33
1
3
7
7
× 11
=
1
33 11
Steps 4 & 5
=
Steps 6 & 7
proper fraction, fully reduced
Step 8
Answer :
Step 9
Validate:
7
11
7
5
×4
11
7
1
=
1
7
11
3
×
33
1
7
=
3
=3 9
1
Chapter 3 — Fractions
282
ADDRESSING COMMON ERRORS
Issue
Multiplying or
dividing without
first changing
to improper
fractions
Incorrect
Process
1
4
4
3 × 6 = 18
1
4
5
20
1
= 18
5
Not inverting the
divisor before
multiplying in a
division problem
4 1
4
÷ =
9 5 45
4
Correct
Process
Resolution
Converting mixed
numbers to improper
fractions (Step 2)
must be done prior
to multiplying or
dividing. Using the
equivalent improper
fraction is the most
efficient way to
multiply or divide
mixed numbers.
4
1
3 ×6
5
4
=
Not dividing out
common factors
first and ending
up with large
numerators and
denominators
to reduce, thus
making the
processes of
multiplication
and reduction
more difficult
3
1
5
1
Division is performed
by multiplying the
dividend by the
reciprocal of the
divisor.
÷
5
3
=
7
7
2 14 3 5
× × ×
7 15 4 7
420
=
94
2940
4
420 ÷ 10
=
940 ÷ 10
2940
42
2
=
294
42 ÷ 2
=
294 ÷ 2
21
=
147
Canceling common
factors can only
be done when
the operation is
multiplication or when
reducing a fraction.
If you cancel as many
common factors as
possible, your product
will be much more
easily reduced.
In fact, if all common
factors are divided out
before multiplying,
there is no need to
reduce the fraction.
It is already in lowest
terms.
19
1
5
×
3
1 95 25
÷6 =
÷
4
4
4
4
23
19
25
4
=
5
95
3
=
= 23
4
4
1
95
4
1
×5
4
25
19
4
=
=3 9
5
5
4 1 4 5
÷ = ×
9 5 9 1
2•2 5
×
=
3•3 1
20
2
=
=2
9
9
In other words,
invert the divisor
(the second fraction)
before multiplying.
Not inverting
the divisor
before canceling
common factors
in a division
problem
Validation
2
2 1
×
9 5
4
=
3 5 3 7 21
÷ = × =
5 7 5 5 25
20
1
4
×1 = 9
9
9
5
21 5
×
25 7
3
=
5
21
25
1
×1
5
7
=
3
9
5
21
3•7
=
147 3 • 7 • 7
so it can be
reduced further;
therefore, it is not
the correct lowest
terms answer.
2 14 3 5
× × ×
7 15 4 7
1
=1
2
7
1
=
7
1
×1
2
3
14
15
1
×1
2
3
4
1
×
5
7
1 5 3 14
÷ ÷ ÷
7 7 4 15
1
1
2
4
13
15
×1 × 1 × 2
7
5
3
14
2
= 9
7
=1
7
Section 3.4 — Multiplying and Dividing Fractions and Mixed Numbers
Incorrect
Process
Issue
Improperly
dividing out
common factors
(improper
cancellation)
Incorrectly
representing a
whole number
as a fraction
Cancel any one factor
in the numerator with
1
2
3
2
14 only one matching
= 1 ×3 ×5
denominator factor.
7
9
15
any number
2•2
4
=1
=
=
that same number
3 • 5 15
2
=
1
4
2
4
1
1
2
× ×
3 5
2
=
15
Not fully
reducing the
final answer
1
1
4 ×2
2
6
9 13
= ×
2 6
17
7
9
117
=9
=
2
12
12
Correct
Process
Resolution
3 2 14
× ×
7 9 15
1 2
4× ×
3 5
283
In a fraction problem,
write a whole number
as
the whole number
1
and proceed.
Before presenting
your final answer,
always verify that the
proper fraction portion
is fully reduced, in
case you missed a
possible cancellation.
(There was a common
factor of 3 that could have
been canceled before the
multiplication.)
3 2 14
× ×
7 9 15
1
3
2
2
14
×3 ×
7
9 15
2•2
4
=
=
3 • 15 45
=1
1 2
4× ×
3 5
4 1 2
= × ×
1 3 5
8
=
15
1
1 9 13
4 ×2 = ×
2
6 2 6
9
117
=
=9
12
12
9
Is
reduced?
12
9÷3
3
=
Not yet.
12 ÷ 3 4
3
Final answer: 9
4
Validation
4 14 2
÷
÷
45 15 9
12
=1
how to deal with mixed numbers in multiplication and division
the division process—how and why division is turned into multiplication
how to present a final answer
how to validate the answer to a multiplication problem involving fractions
how to validate the answer to a division problem involving fractions
14
3
×1
9
2
8 2 1
÷ ÷
15 5 3
4
=1
3
8
15
1
×1
5
2
1
×
3
1
=
4
=4 9
1
9
3
1 39 13
÷2 =
÷
4
6
4
6
3
=
=
Before proceeding, you should have an understanding of each of the following:
the value and process of reducing before multiplying
45
×7
15
3
9
7
=
PREPARATION INVENTORY
the terminology and notation associated with multiplying and dividing fractions
3
1
4
39
2
4
3
×1
6
13
9
1
=4 9
2
2
Section 3.4
ACTIVITY
Multiplying and Dividing
Fractions and Mixed Numbers
PERFORMANCE CRITERIA
• Multiplying any given combinations of fractions and mixed numbers correctly
– presentation of the final answer in lowest terms
– validation of the answer
• Dividing any given combinations of fractions and mixed numbers correctly
– presentation of the final answer in lowest terms
– validation of the answer
CRITICAL THINKING QUESTIONS
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.
284
Section 3.4 — Multiplying and Dividing Fractions and Mixed Numbers
285
5. What property(ies) permit you to cancel a factor in a numerator with a factor in another denominator when
multiplying fractions?
When multiplying a combination of fractions, reduce the fractions through the use of the Identity Property
of Multiplication by dividing common factors from the numerator and denominator.
6. What is the result when all factors in the numerators cancel out?
The result is a fraction with 1 in the numerator.
Example:
1
4
7. 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
8. How do you validate that your final answer is both properly presented and correct?
When you multiply fractions, validate using division and when dividing using multiplication for validation.
Always use the original problem so you also capture transcription errors. Additionally, check that the
validation answer matches the original problem number.
Chapter 3 — Fractions
286
TIPS
FOR
SUCCESS
• Write fractions using a horizontal fraction bar rather than a slash ( 2 rather than 2/3). Using a slash can
3
interfere with proper alignment of the problem.
• Use neat and consistent notation when dividing out common factors so that you do not cancel too many or
too few of them.
• Replace each completely canceled factor with a 1.
• For multiplication, if you cancel all common factors within the problem, your result will be a fully reduced
answer.
• Always verify that your final answer is fully reduced by prime factoring your answer.
• Because a fraction problem has intermediate steps, it is especially important to validate the final answer
using the original fractions and/or mixed numbers.
DEMONSTRATE YOUR UNDERSTANDING
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
Worked Solution
Validation
Section 3.4 — Multiplying and Dividing Fractions and Mixed Numbers
Problem
4)
5
3
×6
9
7
5)
6
3
4
×8
4
9
6)
3
1
1
÷5
2
4
7)
6÷
1
3
Worked Solution
287
Validation
Chapter 3 — Fractions
288
Problem
Worked Solution
Validation
5
÷3
8
8)
5
9)
5
2
×4 ×6
16
5
10) Bruno’s share
of the profits
from a land sale
is to be 2/7
of $280,000.
Calculate his
share.
MENTAL MATH
Try to do these “in your head.”
1
a) What is
of 42?
21
2
c)
2
of 90 is what number? 60
3
e) 1 of what number is 20? 80
4
b) What is
1
of 80?
4
d) 12 is what part of 36?
f)
20
1
3
1
of what number is 12? 60
5
Section 3.4 — Multiplying and Dividing Fractions and Mixed Numbers
TEAM EXERCISES
Discuss and circle the correct answer to each of the following.
1. When you multiply a proper fraction by a proper fraction, your answer will always be:
a) a mixed number
b) an even smaller proper fraction
c) a larger proper fraction
2. When you multiply a proper fraction and a mixed number, your answer will always be:
a) less than the mixed number
b) greater than the mixed number
3. When you divide a mixed number by a proper fraction, your answer will always be:
a) less than the mixed number
b) greater than the mixed number
4. When you divide a mixed number by a larger mixed number, your answer will always be:
a) less than one
b) greater than one
289
Chapter 3 — Fractions
290
IDENTIFY
AND
CORRECT
THE
ERRORS
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)
7 4
×
8 5
Identify Errors
or Validate
The final answer
must be reduced to
lowest terms.
14 is equivalent to
20
the correct answer,
but it is not fully
reduced:
14
2 ×7
=
20
2 × 2 ×5
Correct Process
7
2
8
1
×
4
7
=
5
10
OR
7 4
28
× =
8 5
40
28 ÷ 2 14 ÷ 2
=
40 ÷ 2 20 ÷ 2
7
=
10
Answer:
2)
5
3
1
÷3
5
8
Validation
7
4
÷
10 5
=
7
2
1
×
10
7
=
9
8
5
4
7
10
You must change
division to
multiplication .
You do this by
multiplying by the
reciprocal of the
second number.
Then reduce.
3)
5 3 8 3
•
• •
6 10 9 4
Reduction
(canceling) was
done incorrectly.
9
Section 3.4 — Multiplying and Dividing Fractions and Mixed Numbers
Worked Solution
What is Wrong Here?
4)
Identify Errors
or Validate
291
Correct Process
1
1
1 ×5
7
4
CORRECT
5)
6)
Find the product of
3 4
5
, and .
,
5 15
8
12
1
2
×6
6
3
Reduced incorrectly.
Cannot use 5 twice
in the denominators.
You must change to
improper fractions,
reduce, then
multiply to get the
answer.
Validation
Chapter 3 — Fractions
292
ADDITIONAL EXERCISES
Perform the indicated operations and validate your answers.
2 3
1. 3 ×
9 5
2. 2
3.
7
1
×1
8
4
1
14
15
3
19
32
8
55
3 2 14 10
× × ×
7 5 15 11
4
9
1
7
4. 2 ÷ 4
6
8
1
1
5. 3 ÷ 1
9
2
6. 2 ÷ 1
1
2
7. 12 × 1
1
1
×4
2
3
1 5
8. 12 ÷
2 7
3
9. 18 ÷ 20
4
2
2
27
1
1
3
78
17
1
2
15
16