3.2 Multiplication of Fractions and of Mixed Numbersfaculty.ccbcmd.edu/~lwalte19/Math081C3S2Text.pdfCCBC Math 081 Multiplication of Fractions and of Mixed Numbers Section 3.2 Third

CCBC Math 081 Multiplication of Fractions and of Mixed Numbers Section 3.2

Third Edition 17 pages

160

3.2 Multiplication of Fractions and of Mixed Numbers

Multiplying Fractions

What does multiplication of fractions mean? Let’s review a basic application of multiplication to

illustrate. Remember from chapter 2, the word “of” in a problem signifies multiplication. What

if we were asked to determine 1

2of

2

3. What should we do? This problem is

1 2

2 3 . This says I

want to show 1

2(half) of

2

3.

Here is 2

3

Here is (half)1

2of

2

3

Therefore, we see that (half)1

2of

2

3is

1

3.

In this section we will learn a simple algorithm to make it easy to calculate 1 2 1

2 3 3 .

Whenever we multiply two fractions together, we multiply the two numerators together and the

two denominators together, and simplify the answer (if possible).

So let’s do our previous example using this method.

Example 1: Multiply: 1 2

2 3

1 2 2

2 3 6 Multiply the numerators together and the

denominators together.

2

2 3

Write the numerator and denominator in prime

factorization.

2

2 3

Divide out the common factor of 2.

Note: 21

2

1

3 If all factors divide out, a factor of 1 remains.



161

Practice 1: Multiply: 1 3

3 4 Answer:

1

4

Watch it: http://youtu.be/jtZ9GVfDVjc


3 7

2 5 2 5 10

3 7 3 7 21

Multiply the numerators together and the


Check to see if the answer can be simplified. There are no common factors (other than 1) for 10

and 21, so the final answer is 10

21.


6 9 Answer:

35

54

Watch it: http://youtu.be/oYK5g1M-HvE


11 5

6 3 6 3 18

11 5 11 5 55





55.


7 9 Answer:

20

63

Watch it: http://youtu.be/_Y3alX8lJho

MULTIPLYING FRACTIONS

Symbols Words

a c a c

b d b d

Multiply the numerators together.

Multiply the denominators together

http://youtu.be/jtZ9GVfDVjc

http://youtu.be/oYK5g1M-HvE

http://youtu.be/_Y3alX8lJho



162

Example 4a: Multiply: 2 9

3 14

2 9 2 9 18

3 14 3 14 42



Check to see if the answer can be simplified. There are common factors for 18 and 42, so this

problem’s answer can be simplified. The greatest common factor (GCF), the largest number that

will divide 18 and 42 with no remainder, is 6. To simplify, divide both the numerator and

denominator by 6.

18 18 6 3

42 42 6 7

Maybe you did not see that the GCF of 18 and 42 is 6. It is still possible to end at the simplified

answer by dividing out common factors. Both 18 and 42 are even, so a reasonable common

factor to start with is 2. Divide both the numerator and denominator by 2.

18 18 2 9

42 42 2 21

Divide by 2 in the numerator and denominator.

9 9 3 3

21 21 3 7

There is still a common factor for 9 and 21, so we

must divide by 3 in the numerator and denominator.

It took an extra step, but in the end we still got the final, simplified answer.

Practice 4a: Multiply: 3 8

4 15

Answer:

2

5

Watch it: http://youtu.be/KGxwgUVuymE

We will now show how to do this problem by dividing out factors in the numerator and

denominator before we multiply. Many people prefer to reduce (simplify) before multiplying

because they can see more easily which number can be used to reduce both the numerator and

denominator. Others prefer this method because reducing first means that the numbers to be

multiplied are smaller. Either way, you can reduce before you multiply (if possible) or you can

wait until after you have multiplied the fractions together to reduce. The final result will be the

same. It would be a good idea to compute your answer using one method and then check your

answer using the other method.

http://youtu.be/KGxwgUVuymE



163

Example 4b: Multiply: 2 9

3 14

Divided by 2

2

1

9

3 14

1 9

3 7 Divided by 2

7

Divided by 3

1

3

1

9

1 3 3

7 1 7 7 Divided by 3

3

As you can see, we get the same answer using either method. In both methods we divided the

numerator and denominator by 2 3 6 .

Since the fractions are being multiplied, look for any common factors between one numerator

and one denominator. It does not matter if this occurs in a single fraction or between the two; the

important part is that you have to be reducing with a numerator and a denominator.

Practice 4b: Multiply: 3 8

4 15

Answer:

2

5

Watch it: http://youtu.be/jcCaftdaTSA


8 7

In this example, the number 2 can divide both 6 and 8 without a remainder, so we can reduce

before we multiply.

Divided by 2

3

84

6

3 3

7 4 7 Divided by 2

3

If there were any more common factors between a numerator and a denominator we could repeat

this process a second (or third, or fourth, etc.) time. There are no more common factors other

than 1 between any numerator and any denominator, so we can now continue with multiplying.

3 3 3 3 9

4 7 4 7 28

It is always a good idea to make sure the final answer is completely simplified even if you reduce

the fractions before you multiply.

http://youtu.be/jcCaftdaTSA



164

The result for Example 8 is the same if you multiply first and then simplify.

3 6 3 6 18 18 2 9

8 7 8 7 56 56 2 28

As you can see, we arrived at the final answer of 9

28both times.

Note: There may be different ways you can approach simplifying a problem, but as long as you

reduce the fractions properly, the order in which you work does not matter.


7 3 Answer:

4

7

Watch it: http://youtu.be/dPsHxACqOf8

Example 6a: Multiply: 9 30

20 12

Divided by 3

9

3

30

20 12

3 30

20 4 Divided by 3

4

Divided by 10

3

20

2

30

3 3

4 2 4 Divided by 10

3

3 3 9 1

12 4 8 8

Multiply the already reduced fractions. Change to a

mixed number (if necessary).

Practice 6a: Multiply: 12 10

15 9 Answer:

8

9

Watch it: http://youtu.be/ox9qpFhF4J8

http://youtu.be/dPsHxACqOf8

http://youtu.be/ox9qpFhF4J8



165

Here is another way to get the same answer by reducing differently.

Example 6b: Multiply: 9 30

20 12

Divided by 6

9 30

20

12

5

9 5

20 2

Divided by 6

2

Divided by 5

9 =

204

5

9 1

2 4 2 Divided by 5

1

9 1 9 1

14 2 8 8

Multiply the already reduced fractions. Change to a

mixed number (if necessary).

Practice 6b: Multiply: 12 10

15 9 Answer:

8

9

Watch it: http://youtu.be/kp5AHMe5JeI

We can do this problem in yet another way. We will multiply first and then simplify. You will

see. The benefit of the first two methods as we work through this problem

Example 6c: Multiply: 9 30

20 12

9 30 9 30

20 12 20 12



270

240

270 10

240 10

Divide the numerator and denominator by 10.

27

24

27 3

24 3

Divide the numerator and denominator by 3.

9

8

This is the answer as an improper fraction.

11

8

This is the answer as a mixed number.

http://youtu.be/kp5AHMe5JeI



166

Practice 6c: Multiply: 12 10

15 9 Answer:

8

9

Watch it: http://youtu.be/Hz9nyJVVz0k

Example 7: Multiply: 4

83

4 4 88

3 3 1

Since 8 is a whole number, rewrite it as an improper

fraction by writing 8 divided by 1.

4 4 8

83 3 1





3 which can also be written as the mixed number 10

2

3 if you

divide 32 by 3.

Practice 7: Multiply: 7

59 Answer:

35 83

9 9

Watch it: http://youtu.be/h-M2oqakA_M

Note: Remember your rules for dividing signed numbers. We can use these rules when we have

fractions because a fraction is a division problem.

FRACTIONS AND NEGATIVE

a a a

b b b

If only the numerator is negative or only the

denominator is negative, the fraction is negative.

a a

b b

If both the numerator and the denominator are

negative, the fraction is positive.

http://youtu.be/Hz9nyJVVz0k

http://youtu.be/h-M2oqakA_M



167


2 4

There are no common factors in the numerator and denominator. Therefore, we can multiply the

numerators together and the denominators together.

3 5 3 5 15

2 4 2 ( 4) 8

We can think of this answer as a negative divided by a negative because the numerator and the

denominator are both negative. Therefore the fraction is a positive.

15 15

8 8

This is the answer as an improper fraction.

7

18

This is the answer as a mixed number.


11 7

Answer:

12

77

Watch it: http://youtu.be/pIw1oPoE6nA

Multiplying Mixed Numbers

To multiply mixed numbers you must first rewrite each mixed number as an improper fraction

and then follow the steps for multiplying fractions.

MULTIPLYING MIXED NUMBERS

First change mixed numbers to improper fractions.

Then, multiply as you would fractions.

http://youtu.be/pIw1oPoE6nA



168


37 3

2 2 (7 3) 2 2

37 3 7 3

Change the mixed number to an improper fraction.

23 2

7 3

Simplify.

23 2 46

7 3 21

Now that we have fractions, multiply the





4

21 if you

divide 46 by 21.


25 4 Answer:

13

20

Watch it: http://youtu.be/N-rtc4rCl04


2 311 2

5 1 (11 2) 5 (2 3) 1

2 311 2 11 2

Change each mixed number to an improper fraction.

27 7

11 2

Simplify.

27 7

11 2

There are no common factors in the numerator and

denominator. Therefore, we multiply the


189

22

Check to see if the answer can be simplified. There are no common factors (other than 1) for

189 and 22, so the final answer is 189


13

22 if

you divide 189 by 22.


2 39 3 Answer:

250 79

27 27

Watch it: http://youtu.be/OPfxovjRBoU

http://youtu.be/N-rtc4rCl04

http://youtu.be/OPfxovjRBoU



169

The following problem consists of a negative mixed number. Recall that we change a negative

mixed number into an improper fraction by first determining the improper fraction in the

parentheses and then making the improper fraction negative.

Example 11: Multiply: -5 5

1 49 7

5 5 (9 1) 5 (7 4) 5

1 49 7 9 7

Change each mixed number to an improper fraction.

14 33

9 7

Simplify.

14

2

33

9 7

2 33

9 1

1

Divide out a 7 in the numerator and denominator.

2

9

3

33

2 11

1 3 1

11

Divide out a 3 in the numerator and denominator.

2 11 22

3 1 3



1

73

This is the answer written as a mixed number.


4 23 5

Answer: 182 2

1215 15

Watch it: http://youtu.be/ueIi30b42V0


4 46 5

1 4 (6 4) 1 (5 4) 44 4

6 5 6 5

Recall that mixed numbers must be rewritten as

improper fractions in order to multiply.

25 24

6 5

25

5

24

6 5

5 24

6 1

1

Divide the first numerator and second denominator

by 5.

5

6

1

24

1

4

Divide the first denominator and second numerator

by 6.

5 4 20

201 1 1

Multiply the already reduced fractions.

http://youtu.be/ueIi30b42V0



170


5 23 4 Answer: 12

Watch it: http://youtu.be/iWotOoaM3qs

Exponents and Fractions

In Chapter 1 you learned that exponents are a way of expressing a repeated multiplication.

Exponents can be used with fractions as well.

Whenever fractions are raised to a power, the fraction is placed in parentheses and the exponent

is written on the outside.

Example 13: Evaluate:

32

5

To evaluate this expression, use the fact that the exponent tells you to multiply the entire number

inside the parentheses by itself three times.

Then the expression

32

5

is the fraction 2

5 raised to the third power.

3 3

3

2 2 2 2 2 8

5 5 5 5 5 125

Practice 13: Evaluate:

33

4

Answer: 27

64

Watch it: http://youtu.be/SRVUUSMvLIE

EXPONENTS AND FRACTIONS

n n

n

a a

b b

http://youtu.be/iWotOoaM3qs

http://youtu.be/SRVUUSMvLIE



171


23

4

23 3 3 ( 3)( 3) 9

4 4 4 4 4 16


22

3

Answer: 4

9

Watch it: http://youtu.be/ocRhIVgD1xQ

When you are raising a mixed number to a power, you must first rewrite that mixed number as

an improper fraction before you can multiply. We will do this in Example 15.


31

24

3 3 31 (4 2) 1 9

24 4 4

First, convert the mixed number to improper form:

9 9 9

4 4 4 Expand the exponent by multiplying

9

4 by itself 3

times.

729 25

1164 64

Multiply. Change to a mixed number if necessary.


32

33

Answer: 1331 8

4927 27

Watch it: http://youtu.be/14PmX2ISfG0

http://youtu.be/ocRhIVgD1xQ

http://youtu.be/14PmX2ISfG0



172

AREA OF A RECTANGLE

Area of a Rectangle = l w

Example 16: Determine the area of the rectangle.

When we determine the area of a rectangle we multiply the length times the width.

24 2 8in or square inches

5 3 15A

Practice 16: Determine the area of the rectangle. Answer: 25 in

18

Watch it: http://youtu.be/vvlveLY20uM

http://youtu.be/vvlveLY20uM



173

Example 16: Determine the area of the triangle.

1

2

1 1 23 2

2 2 7

1 7

2 7

2

7

16

2

1

A b h

2 7

16

2

16

4

4 cm

Use the formula for area of a triangle.

Substitute the values for base and height.

Change mixed numbers to improper fractions.

Divided out a 7 in the numerator and denominator.

Multiply numerators together.

Divide 16 by 4.

Practice 17: Determine the area of the triangle. Answer: 153 33

340 40

Watch it: http://youtu.be/rArRImYHRhk

Watch All: http://youtu.be/xzvJa-bIqyE

AREA OF A TRIANGLE

Area of a Triangle = 1

2b h

b = base

h = height

http://youtu.be/rArRImYHRhk

http://youtu.be/xzvJa-bIqyE



174

3.2 Multiplication of Fractions and of Mixed Numbers Exercises

1. Multiply: 2 7

5 9

2. Multiply:

3 5

4 7

3. Multiply: 5 8

6 15

4. Evaluate:

31

22

5. Multiply: 1 1

4 52 6

6. Multiply: 2 15

37 10

7. Multiply: 3 2

5 24 5 8. Evaluate:

42

3

9. Multiply:

4 21 2

5 3 10. Multiply:

54 3

7

11. Multiply: 7 4

29 5 12. Multiply:

1 258

5 82

13. Multiply: 2 7

63 9

14. Multiply: 1 9

38 10

15. Multiply: 5 5

3 8 16. Evaluate:

33

5

17. Write this problem in reduced form, but do not multiply: 5 12

8 17


7 15


15 26

20. Multiply: 1 2

2 3

21. Multiply: 1 2

25 3

22. Multiply: 1

3 46

23. Evaluate:

21

4



175

24. Determine the area of the rectangle:

25. Determine the area of the triangle:



176

3.2 Multiplication of Fractions and of Mixed Number Exercises Answers

1. 14

45 14.

45

16= 2

13

16

2. 15

28 15.

25

24= 1

1

24

3. 4

9 16.

27

125

4. 125 5

158 8

17.

5 3

2 17

5. 93 1

234 4

18.

1 2

1 5

6. 69

14= 4

13

14 19.

2 5

1 13

7. 69

5= 13

4

5 20.

1

3

8. 16

81 21.

71

15

9. 24

5= 4

4

5 22.

112

2

10. 104

7= 14

6

7 23.

1

16

11. 98

45= 2

8

45 24.

225

3 ft

2

12. 5

2= 2

1

2 25.

1

7mm

2

13. 140 5

527 27

Documents

3.2 Multiplication of Fractions and of Mixed Numbersfaculty.ccbcmd.edu/~lwalte19/Math081C3S2Text.pdfCCBC Math 081 Multiplication of Fractions and of Mixed Numbers Section 3.2 Third