Transcript Simplifying Complex Fractions (Method 1)

Chapter 7
Section 5
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
7.5
1
2
Complex Fractions
Simplify a complex fraction by writing it as
a division problem (Method 1).
Simplify a complex fraction by multiplying
the least common denominator (Method 2).
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Complex Fractions.
The quotient of two mixed numbers in arithmetic, such as
1
1
2  3 can be written as a fraction.
2
4
1
1
2
1
1
2
2 3  2 
1
2
4 31
3
4
4
2
The last expression is the quotient of expressions that
involve fractions. In algebra, some rational expressions also
have fractions in the numerator, or denominator, or both.
A rational expression with one or more fractions in the
numerator, or denominator, or both is called a complex fraction.
The parts of a complex fraction are named as follows.
1
2
1
3
4
2
Numerator of complex fraction
Main fraction bar
Denominator of complex fraction
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7.5 - 3
Objective 1
Simplify a complex fraction by
writing it as a division problem
(Method 1).
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Slide 7.5 - 4
Simplify a complex fraction by writing it as
a division problem (Method 1).
Since the main fraction bar represents division in a complex
fraction, one method of simplifying a complex fraction involves
division.
Step 1: Write both the numerator and denominator as single
fractions.
Step 2: Change the complex fraction to a division problem.
Step 3: Perform the indicated division.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7.5 - 5
EXAMPLE 1
Simplifying Complex
Fractions (Method 1)
Simplify each complex fraction.
Solution:
2 1
2 4 1 5
8
5
13

  

5 4  5 4 4 5  20 20  20  13  5  13  6  78
5
1 1
1 3 1 2
3 2
20 6 20 5 100

  

6
2 3
2 3 3 2
6 6
39
2  39


50
2  50
2 1
2m 1
2m  1
1
m



m
2m  1 3  2m  1
2 2  2 2 
2
2 


6m  3
3  2m  1
3  2m  1
3  2m  1
2
4m
4m
4m
4m
4m
2m
2m  1 2  2  m



3
2
3  2m  1
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7.5 - 6
EXAMPLE 2
Simplifying a Complex
Fraction (Method 1)
Simplify the complex fraction.
m 2 n3
p
4
mn
p2
Solution:
2 3
4
2 3
2
mn mn
mn p

 2

 4
p
p
p mn
mmnnn
p p


p
mmmmn
2 3
2
mn p

 4
p mn
n2 p
 2
m
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7.5 - 7
EXAMPLE 3
Simplifying a Complex
Fraction (Method 1)
Simplify the complex fraction.
Solution:
 b  2   1  2   a  1
b  2  a  1 b  2  a  1


 a  3  2  1   b  2 
 a  3 b  2 a  3  b  2 
2a  b
2a  b  8


 a  1 b  2  b  2  a  3
1
2

a 1 b  2
2
1

b2 a3
 b  2    2a  2 
a  1 b  2 


 2a  6    b  2 
 b  2  a  3
b  2 a  3

2a  b


a  1 b  2 2a  b  8
2a  b  a  3


 a  1 2a  b  8
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7.5 - 8
Objective 2
Simplify a complex fraction by
multiplying by the least common
denominator (Method 2).
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Slide 7.5 - 9
Simplify a complex fraction by multiplying
the least common denominator (Method 2).
Since any expression can be multiplied by a form of 1 to get
an equivalent expression, we can multiply both the numerator
and denominator of a complex fraction by the same nonzero
expression to get an equivalent rational expression. If we
choose the expression to be the LCD of all the fractions within
the complex fraction, the complex fraction will be simplified.
Step 1: Find the LCD of all fractions within the complex
fraction.
Step 2: Multiply both the numerator and denominator of the
complex fraction by this LCD using the distributive
property as necessary. Write in lowest terms.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7.5 - 10
EXAMPLE 4
Simplifying Complex Fractions
(Method 2)
Simplify each complex fraction.
Solution:
2 1

3 4
4 1

9 2
2 1
36   
24  9
3 4



16  18
4 1
36   
9 2
6
2
a
4
3
a
6

a2 
a
 
4

a3 
a

15

34
2a  6

3a  4
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Slide 7.5 - 11
EXAMPLE 5
Simplifying a Complex Fraction
(Method 2)
Simplify the complex fraction.
Solution:
2
3
 2
2
a b ab
4
1

2 2
a b ab
3 
 2
ab  2  2
a b ab 


4
1 
2 2
ab  2 2 
ab 
a b
2 2
2b  3a

4  ab
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7.5 - 12
EXAMPLE 6
Deciding on a Method and
Simplifying Complex Fractions
Simplify each complex fraction.
1
2

x x 1
4
x 1
2 
1
x  x  1  

x x 1 


 4 
x  x  1 

x

1


2x  3
x4
4 x2  9
x 2  16
 2x  3 
 x  4 x  4 

x  4 2 x  3

x

4




  2 x  3 2 x  3   2 x  3 2 x  3
 x  4 x  4 

  x  4  x  4  
x  1  2 x


4x

3x  1
4x
x  4


 2 x  3
Remember the same answer is obtained regardless of whether Method 1
or Method 2 is used. Some students prefer one method over the other.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7.5 - 13