Chapter Chapter 77Section Section 55

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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).

11

22

7.57.57.57.5

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Complex Fractions.

The quotient of two mixed numbers in arithmetic, such as can be written as a fraction.

Slide 7.5 - 3

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.

1 12 3

2 4

1 12 21 1 2 22 3

1 12 4 3 34 4

12

21

34

The parts of a complex fraction are named as follows.

Numerator of complex fractionMain fraction barDenominator of complex fraction

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Objective 11

Simplify a complex fraction by writing it as a division problem (Method 1).

Slide 7.5 - 4

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Since the main fraction bar represents division in a complex fraction, one method of simplifying a complex fraction involves division.

Slide 7.5 - 5

Simplify a complex fraction by writing it as a division problem (Method 1).

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.

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EXAMPLE 1

Simplify each complex fraction.Solution:

Simplifying Complex Fractions (Method 1)

Slide 7.5 - 6

2 15 41 12 3

12

6 34

m

mm

4 54 5

2 15 41 12

3 23 23

12

3 2

22

1

4

m

m

m

8 520 20

3 26 6

2 12 2

3 2 1

4

m

m

m

132056

13 5

20 6

13 6

20 5 78

100

39

0

2

52

39

50

2 12

3 2 1

4

m

m

m

3 2 12 1

2 4

mm

m

2 1 2

2 2 1

2

3

mm

m

2

3

m

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Simplify the complex fraction.

EXAMPLE 2

Solution:

Simplifying a Complex Fraction (Method 1)

Slide 7.5 - 7

2 3

4

2

m npm np

2 3 4

2

m n m n

p p

2 3 2

4

m n p

p m n

3 22

4

m

p

n p

nm

2

2

n p

m

m m n pn n p

p m m mm n

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EXAMPLE 3

Solution:

1 21 2

2 12 3

a b

b a

Simplifying a Complex Fraction (Method 1)

Slide 7.5 - 8

Simplify the complex fraction.

2 1

2 11 2

1 2

2 13

3 2

3 22

b a

b a b

b

a

a b

a a b

2 2 2

1 2

2 6 2

2 3

b a

a b

a b

b a

2 2 8

1 2 2 3

a b a b

a b b a

32

1 2 8

2

2

b aa b

a bba

2 3

1 2 8

a b a

a a b

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Objective 22

Simplify a complex fraction by multiplying by the least common denominator (Method 2).

Slide 7.5 - 9

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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.

Simplify a complex fraction by multiplying the least common denominator (Method 2).

Slide 7.5 - 10

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.

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EXAMPLE 4

Solution:2 13 44 19 2

Simplifying Complex Fractions (Method 2)

Slide 7.5 - 11

Simplify each complex fraction.

2 13 4

3

61

6

42

39

24 9

16 18

15

34

62

43

aa

aa

2 6

3 4

a

a

62

43

a

a

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EXAMPLE 5

Simplify the complex fraction.

2 2

2 2

2 3

4 1a b ab

a b ab

Simplifying a Complex Fraction (Method 2)

Slide 7.5 - 12

Solution:

2 2

2

2 2

2 22

2 3

4 1a b ab

a b ab

a b

a b

2 3

4

b a

ab

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Simplify each complex fraction.

EXAMPLE 6

1 21

41

x x

x

Deciding on a Method and Simplifying Complex Fractions

Slide 7.5 - 13

2

2

2 34

4 916

xxxx

2 34

2 3 2 3

4

44

4

44

x x

x x

xx

x x

x x

4

2 3

2 3

2 3

x

x

x

x

4

2 3

x

x

21

41

11

1

x x

xx

x x

x

1 2

4

x x

x

3 1

4

x

x

Remember the same answer is obtained regardless of whether Method 1 or Method 2 is used. Some students prefer one method over the other.