Intermediate Algebra
Tutorial 34: Complex Fractions

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Learning Objectives
After completing this tutorial, you should be able to:
 Simplify complex fractions.

Introduction
Tutorial
A complex fraction is a rational expression that has
a fraction
in its numerator, denominator or both.
In other words, there is at least one small fraction
within the overall
fraction.
Some examples of complex fractions are:
and
There are two ways that you can simplify complex
fractions. We
will call them method I and method II. 
Method I
Simplifying a Complex Fraction

In other words, you will be combining all the
parts of the numerator
to form one fraction and all of the parts of the denominator to form
another
fraction. If you need a review on adding and subtracting rational
expressions, go to Tutorial 33:
Adding
and Subtracting Rational Expressions. 
Step 2: Divide the numerator by
the denominator
by multiplying the numerator by the reciprocal of the denominator.
Step 3: If needed, simplify
the rational expression.

Example
1: Simplify . 
Combining only the numerator we get: 

*Rewrite fractions with LCD of
12

Combining only the denominator we get: 

*Rewrite fractions with LCD of
8

Putting these back into the complex fraction we get: 

*Write numerator over denominator

Step 2: Divide the numerator
by the denominator
by multiplying the numerator by the reciprocal of the denominator
AND
Step 3: If needed, simplify
the rational expression. 

*Rewrite div. as mult. of
reciprocal
*Divide out a common factor of 4

Example
2: Simplify . 
Combining only the numerator we get: 

*Rewrite fractions with LCD of ab

The denominator is already written as one fraction: 
Putting these back into the complex fraction we get: 

*Write numerator over denominator

Step 2: Divide the numerator
by the denominator
by multiplying the numerator by the reciprocal of the denominator
AND
Step 3: If needed, simplify
the rational expression. 

*Rewrite div. as mult. of
reciprocal
*Divide out common factors of a and b

Method II
Simplifying a Complex Fraction

Example
3: Simplify . 
The denominator of the numerator’s fraction has the
following two
factors: 
The denominator of the denominator’s fraction
has the following
factor: 
Putting all the different factors together and using
the highest exponent,
we get the following LCD for all the small fractions: 
Multiplying numerator and denominator by the LCD we
get: 

*Mult. num. and den. by (x + 5)(x  5)


*Divide out the common factor
of (2x + 1)

Example
4: Simplify . 
The denominator of the numerator’s fraction has the
following factor: 
The denominator of the denominator’s fraction
has the following
factors: 
y and 

Putting all the different factors together and using
the highest exponent,
we get the following LCD for all the small fractions: 
Multiplying numerator and denominator by the LCD we
get: 

*Mult. num. and den. by y squared


*Divide out the common factor
of (3y + 1) 
Example
5: Simplify . 
At first glance, this does not look like a complex
fraction. However,
once you rewrite it with positive exponents you will see that we really
do have a complex fraction. 

*Rewrite with positive exponents

The two denominators of the numerator’s fractions
have the following
factors: 
The two denominators of the denominator’s
fractions have the
following factors: 
and 

Putting all the different factors together and using
the highest exponent,
we get the following LCD for all the small fractions: 
Multiplying numerator and denominator by the LCD we
get: 

*Mult. num. and den. by a squared b squared

Practice Problems
These are practice problems to help bring you to the
next level.
It will allow you to check and see if you have an understanding of
these
types of problems. Math works just like
anything
else, if you want to get good at it, then you need to practice
it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.
In fact there is no such thing as too much practice.
To get the most out of these, you should work the
problem out on
your own and then check your answer by clicking on the link for the
answer/discussion
for that problem. At the link you will find the answer
as well as any steps that went into finding that answer. 
Practice
Problems 1a  1b: Simplify.
Need Extra Help on these Topics?
Last revised on July 17, 2011 by Kim Seward.
All contents copyright (C) 2001  2011, WTAMU and Kim Seward. All rights reserved.



