Math 40
Prealgebra
Section 4.2 – Multiplying Fractions
4.2 Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers with (b and d not zero), then we rewrite the multiplication
as a single fraction:
a c
ac
 
b d
bd
Then use methods of section 4.1 to reduce the fraction to lowest terms.
5 3
Example 1: Find the product.  
9 10
Solution:
Figure out the sign of the product first! The product of two numbers with different signs is negative.
Write the negative sign at the beginning of your work.
Rewrite the multiplication as a single fraction.
Rewrite each factor using prime factorization.
Divide out common factors.
Multiply the remaining factors.
5 3
53
  
9 10
9  10
53

3 3  2  5
53

3  3 2  5
1 1

3 2
1

6
You Try It 1: Find the product.
a) 
10 3

7 2
b)
4 5

15 7
1
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Math 40
Prealgebra
Section 4.2 – Multiplying Fractions
 4   45 
Example 2: Find the product.      
 3   20 
Solution:
Figure out the sign of the product first! The product of two numbers with the same sign is positive.
4  45
 4  45 
      
3  20
 3  20 
2  2 335

3 2  2 5
2  2  3 3 5

3225
1 1 1  3 1

1 1 1 1
3

1
3
Rewrite the multiplication as a single fraction.
Rewrite each factor using prime factorization.
Divide out common factors.
Multiply the remaining factors.
You Try It 2: Find the product.
a)
22 5

15 11
b)
36  55 
 
77  24 
2
2015 Carreon
Math 40
Prealgebra
Section 4.2 – Multiplying Fractions
CAUTION: In examples 1 and 2, you will notice that when we rewrote the product as a single fraction, we did
not actually find the product of the numerators nor the product of the denominators before we
prime factored.
Why can’t we just multiply the numerators and multiply the denominators at the beginning?
To answer this question, we will work the following problem completing the multiplication first:
9 24

16 25
9 24 9  24


16 25 16  25
216

400
Rewrite the multiplication as a single fraction.
If we multiply the numerators first, we get
Notice that we have to spend time finding 9  24 and 16  25 , which would involve using vertical
multiplication:
1
3
16
3
24
 9 and
216
 25
80
 320
400
We still have to rewrite each factor using prime factorization.
216 2  2  2  3  3  3

400 2  2  2  2  5  5
Notice that this involves undoing the vertical multiplication you just completed!
This is why we don’t actually find the product at the beginning of the problem.
Divide out common factors.
Multiply the remaining factors.
2  2  2 333
2  2  2  255
1 1 1  3  3  3

1 1 1  2  5  5
27

50

With more experience multiplying fractions, you may find that you are able to mentally divide out common
factors. This technique is similar to using common factors to reduce in section 4.1.
In example 3, we show to use this technique to simplify the problems in example 1 and 2.
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Math 40
Prealgebra
Section 4.2 – Multiplying Fractions
Example 3: Find the product by using common factors.
 4   45 
b)      
 3   20 
5 3
a)  
9 10
Solution:
a)
Figure out the sign of the product first!
The product of two numbers with different signs is negative.
Write the negative sign at the beginning of your work.
Rewrite the multiplication as a single fraction.
5 3
53
  
9 10
9  10
Look at each factor in the numerator and determine if they share any common factors with the
denominator. If so, find the greatest common factor.
5 and 10 have a greatest common factor of 5. 3 and 9 have a greatest common factor of 3.
Mentally divide each factor by the corresponding greatest common factor.
55 1
10  5  2
33 1
and
93  3
Make sure you use a slash to show the original factor cancels.
Write the result of the mental division above the factors in the numerator and below the
factors in the denominator.
1
1
5 3

9  10
3
2
1 1
3 2
1

6
Multiply the remaining factors.  
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Math 40
Prealgebra
Section 4.2 – Multiplying Fractions
b) Figure out the sign of the product first!
The product of two numbers with the same sign is positive.
Rewrite the multiplication as a single fraction.
4  45
 4  45 
      
3  20
 3  20 
Look at each factor in the numerator and determine if they share any common factors with the
denominator. If so, find the greatest common factor.
4 and 20 have a greatest common factor of 4. 45 and 3 have a greatest common factor of 3.
Mentally divide each factor by the corresponding greatest common factor.
44 1
20  4  5
45  3  15
and
33 1
Make sure you use a slash to show the original factor cancels.
Write the result of the mental division above the factors in the numerator and below the
factors in the denominator.
1
15
1
5
4  45

3  20

115
1 5
Notice that some of the remaining factors still have a common factor.
15 and 5 have a greatest common factor of 5. Mentally divide out the greatest common factor.
3
1  15

1 5
1

Multiply the remaining factors.
1 3
1 1
3
5
2015 Carreon
Math 40
Prealgebra
Section 4.2 – Multiplying Fractions
1
Note that the last three steps can be condensed: 
3
15
4  45
3  20
1
5
1
3
However, this takes a great deal of keeping track of the numbers and you may be more
likely to make a careless error.
You Try It 3: Find the product by using common factors.
a)
22 5

15 11
b)
36  55 
 
77  24 
So which is better? Using prime factorization or common factors to multiply?
This is up to you! Both techniques (if carried out correctly) lead to the same answer. Some problems you will
find that using common factors is easier; some problems you will find it easier to use prime factorization.
We suggest starting off by using prime factorization because it leads to less errors by allowing you to see
exactly which factors cancel out. (It may also lead to more partial credit on exam, should you make a minor
error.)
Once you get more comfortable with multiplying fractions, you may want to try using common factors or a
combination of both techniques. Just make sure you carefully track your steps to avoid making a minor errors.
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Math 40
Prealgebra
Section 4.2 – Multiplying Fractions
Example 4: Find the product.
6 x 35 y 3

15 y 10 x 4
Solution:
Figure out the sign of the product first! The product of two numbers with different signs is negative.
Write the negative sign at the beginning of your work.
6 x 35 y 3
6 x  35 y 3



15 y 10 x 4
15 y 10 x 4
Rewrite the multiplication as a single fraction.
Rewrite each factor using prime factorization.
Don’t forget to write the variables in expanded notation.

Divide out common factors.

2 2 x 57  y  y  y
35 y  2 5 x  x  x  x
2 2 x  5 7 y  y  y
3 5  y  2 5 x  x  x  x
1  2 1 1  7 1  y  y
3 1 1 1  5 1  x  x  x
14 y 2

15 x 3

Multiply the remaining factors.
You Try It 4: Find the product.
a)
20 x  y 6

9 y 3 4 x3
b)
11a 6 c 6

12c 6 4a 2
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2015 Carreon
Math 40
Prealgebra
Section 4.2 – Multiplying Fractions
Example 5: Find
3
of 84.
4
Solution:
When the key word “of” follows a fraction, it indicates we are supposed to multiply.
Replace “of” with a multiplication symbol.
Write 84 as a fraction 84 
84
.
1
3
of 84
4
3
 84

4
3
84


4
1
Rewrite the multiplication as a single fraction. 
3  84
4 1
Rewrite each factor using prime factorization. 
3 2  2 3 7
2  2 1
Divide out common factors.

3 2  2  3  7
2  2 1
3 1 1  3  7
1 1 1
63

1
 63

Multiply the remaining factors.
You Try It 5: Find
2
of 96.
3
8
2015 Carreon
Math 40
Prealgebra
Section 4.2 – Multiplying Fractions
Example 6: Some financial advisors suggest that you should spend no more than
7
of your pre-tax income
20
on your monthly mortgage payment.
If your pre-tax income is $3600, what is the highest monthly mortgage payment you can afford?
Solution:
The problem is asking for the highest monthly mortgage payment you can afford.
The problem states that you are only able to spend
7
of your pre-tax income on your monthly
20
mortgage payment.
Translate
7
of your pre-tax income into mathematical expression.
20
7
 $3600
20
“of” follows a fraction so “of” means multiply
Your pre-tax income is $3600.
7

 3600
20
7  3600

20
 1260
Rewriting multiplication as a single fraction
Using prime factorization or dividing out common factors
The highest monthly mortgage payment you can afford is $1260.
You Try It 6: Some financial advisors suggest that you should spend no more than
7
of your pre-tax income
20
on your monthly mortgage payment.
If your pre-tax income is $5600, what is the highest monthly mortgage payment you can afford?
9
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