Download DEV_2_2

Section 2.2
Multiplication and Division with
Fractions and Mixed Numbers
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Objectives
o Multiply mixed numbers.
o Multiply and reduce with fractions and mixed
numbers.
o Understand the term reciprocal.
o Learn to divide with fractions and mixed numbers.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 1: Multiplication with Mixed Numbers
1 1
Find the product: 1  2
2 5
Solution
Change the mixed numbers to improper fractions, then
multiply the fractions.
3
1 1 3 11 3  11 33
or 3
1 2  


2 5 2 5
2  5 10
10
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 2: Multiplication with Fractions and
Mixed Numbers
3
3
Find of 5 .
5
4
Solution
9
3 3 3 23 69
or 3
5   
5 4 5 4 20
20
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 3: Multiplying and Reducing
with Fractions
15 7
Multiply and reduce to lowest terms:

28 9
Solution
Using prime factors, we have
15 7 15  7
3 5 7
5
5


 .
 
28 9 28  9 2  2  7  3  3 2  2  3 12
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 4: Multiplying and Reducing
with Fractions
9 25 44
Multiply and reduce to lowest terms:
 
10 32 33
Solution
Using prime factors, we have
9 25 44
9  25  44
3  3  5  5  2  2  11
 


10 32 33 10  32  33 2  5  2  2  2  2  2  3  11
3 5
15

 .
2  2  2  2 16
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Completion Example 5: Multiplying and
Reducing with Fractions
55 8 91
Multiply and reduce to lowest terms:   .
26 44 35
Solution
55 8 91 55  8  91
  
26 44 35 26  44  35
5  11  2  2  2  7  13

2  13  2  2  11  5  7

HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
1
 1
1
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6: Multiplying and Reducing
with Mixed Numbers
2 1 1
Multiply and reduce to lowest terms: 4  1  2
3 7 16
Solution
Using prime factors, we have
2 1 1
14 8 33
4 1 2
  
3 7 16
3 7 16
2  7  2  2  2  3  11 11
 11.


3 72222
1
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 7: Multiplying and Reducing
with Fractions
9 25 8
Reduce to lowest terms and multiply:  
10 64 3
Solution
3
5
1
9 25 8 15
  =
10 64 3 16
2
8
1
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
3 divides both 3 and 9.
5 divides both 25 and 10.
8 divides both 8 and 64.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 8: Multiplying and Reducing
with Fractions
36 14 15
Multiply and reduce to lowest terms:
 
49 75 18
Solution
2
2
1
36 14 15 4
  =
49 75 18 35
7
5
1
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
18 divides both 36 and 18.
7 divides both 14 and 49.
15 divides both 15 and 75.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 8: Multiplying and Reducing
with Fractions (cont.)
Another approach is to use factors that are not all
prime:
36 14 15
4  9  2  7  15
4


 
49 75 18 7  7  5  15  2  9 35
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 9: Application of Multiplying
and Reducing with Fractions
5
A study showed that of the members of a public
8
service organization were in favor of a new set of
bylaws. If the organization had a membership of 200
people, how many were in favor of the changes in the
bylaws?
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 9: Application of Multiplying and
Reducing with Fractions (cont.)
Solution
5
We want to find of 200, so we multiply:
8
5
5 200 5  2  2  2  5  5 5  5  5
 125.


 200  
8 1
2  2 2 1
1
8
Thus there are 125 members in favor of the bylaw
changes.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Reciprocals
Reciprocals
a b
The reciprocal of is (a  0 and b  0). The product
b a
of a nonzero number and its reciprocal is always 1.
a b
 1
b a
0
1
Note: 0  , but is undefined.
1
0
That is, the number 0 has no reciprocal.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 10: Reciprocals
2 3
The reciprocal of is .
3 2
2 3 23
1
 
3 2 32
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 11: Reciprocals
1
The reciprocal of 10 is .
10
1 10 1 10  1
1
 

10 
10 1 10 1  10
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Division with Fractions and Mixed Numbers
To Divide Fractions
To divide by any nonzero number, multiply by its
reciprocal. In general,
a c a d
   where b, c , d  0.
b d b c
1 4 1 3 3
For example,     .
2 3 2 4 8
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 12: Dividing with Fractions
3 2
Divide: 
4 3
Solution
2 3
3
The reciprocal of is , so we multiply by .
3 2
2
3 2 3 3 9
   
4 3 4 2 8
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 13: Dividing and Reducing with
Fractions and Mixed Numbers
16
8
Divide and reduce to lowest terms:  2
27
9
Solution
8
First, change the mixed number 2 to the improper
26
26 9 9
9
fraction . The reciprocal of
is , so multiply by
26
9
9
26
and reduce by factoring as follows:
16
8 16 26 16 9
8 29
8
2  
 


27
9 27 9 27 26 3  9  2  13 39
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Completion Example 14: Dividing and
Reducing with Mixed Numbers
1
1
Divide and reduce to lowest terms: 3  19
4
2
Solution
13  2  1
1
1
1 13 39 13 2
3  19  
  

4
2 4 2
4 39 2  2  3  13 6
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 15: Multiplying and Dividing with
Fractions and Mixed Numbers
1
5
If the product of 1 and another number is , what is
2
18
the other number?
Solution
Divide the product by the given number to find the
other number.
5
1
5 3
5 2
52
5
1     

18
2 18 2 18 3 2  9  3 27
5
is the other number.
27
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 15: Multiplying and Dividing with
Fractions and Mixed Numbers (cont.)
Check:
1 5
3 5
3 5
5
1 
 

 .
2 27 2 27 2  3  9 18
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 16: Multiplying and Dividing with
Fractions and Mixed Numbers
3
A box contains 30 pieces of candy. This is of the
5
maximum amount of candy the box can hold.
a. Is the maximum amount of candy the box can hold
more or less than 30 pieces?
Solution
The maximum number of pieces of candy is more than
30.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 16: Multiplying and Dividing with
Fractions and Mixed Numbers (cont.)
3
b. If you want to multiply times 30, would the
5
product be more or less than 30?
Solution
The product would be less than 30.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 16: Multiplying and Dividing with
Fractions and Mixed Numbers (cont.)
c. What is the maximum number of pieces of candy
the box can hold?
Solution
To find the maximum number of pieces, divide:
3 30 5 2  3  5  5 50
 50

30    
1 3
1
5 1 3
The maximum number of pieces the box will hold is 50.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Practice Problems
1. Perform the indicated operations and reduce to
lowest terms.
2
15
17
27
7
5
5
a.
b. 
c.

 2 6
21 22
100 34 9
8 8
5 2
f. 3 
7 7
9
2. If the product of two numbers is
and one of the
10
numbers is 3, what is the other number?
d. 8  2
25 15
e. 6  0
7
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Practice Problem Answers
5
77
d. 12 or 2 2
5
5
3
2.
10
1. a.
b. 1
c. 9 or 2 1
4
4
e. undefined
f. 13
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.