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Intermediate Algebra
Tutorial 34: Complex Fractions
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deskLearning Objectives

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




desk Introduction


Are fractions getting you down?  If so, you are not alone.  Well, you are in luck. We have one more tutorial devoted to rational expressions (fractions).  In this tutorial we will be looking at simplifying complex fractions.  If you need a review on simplifying and dividing rational expressions, feel free to go back to Tutorial 32: Multiplying and Dividing Rational Expressions. If you need a review on finding the LCD of fractions feel free to go back to Tutorial 33: Adding and Subtracting Rational Expressions. It is time to get started with this tutorial.

 

 

desk Tutorial

 

 Complex Fraction

 
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:

complex fraction     and  complex fraction

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.

If you need a review on simplifying rational expressions, go to Tutorial 32: Multiplying and Dividing Rational Expressions.

 
 
notebook Example 1: Simplify example 1a.

 

 
Combining only the numerator we get:

 
example 1b
*Rewrite fractions with LCD of 12
 
 

 

 
Combining only the denominator we get:

 
example 1c
*Rewrite fractions with LCD of 8
 

 

 
Putting these back into the complex fraction we get:

 
example 1d

*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.

 
example 1e

*Rewrite div. as mult. of reciprocal
 
 

*Divide out a common factor of 4
 

 
 
 
 
notebook Example 2: Simplify example 2a.

 
 

 
Combining only the numerator we get:

 
example 2b

*Rewrite fractions with LCD of ab
 
 

 

 
The denominator is already written as one fraction:

 
example 2c

 
Putting these back into the complex fraction we get:

 
example 2d

*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.

 
example 2e
*Rewrite div. as mult. of reciprocal
 
 

*Divide out common factors of a and b

 

 
 
 
 Method II 
Simplifying a Complex Fraction

 
Step 1: Multiply the numerator and denominator of the overall complex fractions by the LCD of the smaller fractions.
 
Recall that when you multiply the exact same thing to the numerator and the denominator, the result is an equivalent fraction.  If you need a review on finding the LCD, go back to Tutorial 33: Adding and Subtracting Rational Expressions.

 

Step 2: If needed, simplify the rational expression.

 
 
notebook Example 3: Simplify example 3a.

 
Step 1: Multiply the numerator and denominator of the overall complex fractions by the LCD of the smaller fractions.

 
The denominator of the numerator’s fraction has the following two factors:

 
example 3b

 
The denominator of the denominator’s fraction  has the following factor:

 
example 3c

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

 
example 3d

 
Multiplying numerator and denominator by the LCD we get:

 
example 3e

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

 

 

 
example 3f
 

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

 

 
 
notebook Example 4: Simplify example 4a.

 
Step 1: Multiply the numerator and denominator of the overall complex fractions by the LCD of the smaller fractions.

 
The denominator of the numerator’s fraction has the following factor:

 
example 4b

 
The denominator of the denominator’s fraction  has the following factors:

 
y and example 4c

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

 
example 4d

 
Multiplying numerator and denominator by the LCD we get:

 
example 4e

*Mult. num. and den. by y squared
 
 
 
 

 

 

 
example 4f
*Num. factors as a difference of two squares
*Den. factors as a perfect square trinomial.
 
 
 

*Divide out the common factor of (3y + 1)

 
 
 
notebook Example 5: Simplify example 5a.

 
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.

 
example 5b
 
 
 

*Rewrite with positive exponents

 

 
Step 1: Multiply the numerator and denominator of the overall complex fractions by the LCD of the smaller fractions.

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

 
a and b

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

 
example 5c  and example 5d

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

 
example 5e

 
Multiplying numerator and denominator by the LCD we get:

 
example 5f

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

 

 

 
example 5g
 

*Factor out the GCF of ab in the num.

 

 

 

 
desk 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.

 

pencil Practice Problems 1a - 1b: Simplify.

 

 

 

desk Need Extra Help on these Topics?


 

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.


 


Last revised on July 17, 2011 by Kim Seward.
All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.