(Simplifying the Numerator and Denominator)
Simplifying Complex Fractions
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Method 2:
This is one of two methods
for simplifying complex fractions.
Also see Method 1. 

Remember ... a complex fraction is just a fraction within a fraction.

To Simplify a Complex Fraction by Simplifying the Numerator and Denominator
1.  Create one single fraction in the numerator (if necessary).
2.  Create one single fraction in the denominator (if necessary).
3.  Remember the main fraction line means "divide".   Rewrite the fraction using a division symbol .
4.  Follow the normal rules for dividing fractions: Invert the the second term (the denominator of the complex fraction) and multiply (by the numerator of the complex fraction).
5. Simplify if needed.


Let's see Method 2 at work on the following problems:

1.

    

2.

3.

4.

 
5.


 

Complex Fractions are EASY to simplify if you show your work!
Just remember that the
fraction bar means DIVIDE.
Rewrite each fraction using a division symbol in the correct position and solve as you would any other problem involving fractions.

 

1.

Remember:  when dividing fractions, invert (flip over) the second fraction and multiply.  Reduce the final answer if needed (or reduce as you multiply).

 

2.

Remember:  a mixed number should be converted to an improper fraction (heavier on top) before simplifying.

 

3.

Remember:  change the numerator and denominator to single fractions by creating a common denominator.  The least common denominator for the terms in this numerator is 32.  The least common denominator for the terms in this denominator is 16.  (Note:  the two common denominators used to create the single fractions may, or may not, be the same value. Most often they are NOT the same value.)

 

Remember:  invert the second fraction and multiply.

 



   Answer:    

Remember:  it may be necessary to factor to reduce a fraction.

 

 

 

Be sure that your final answer is in simplest reduced form.

 

 

4. Remember:  in this problem, change the denominator to a single fraction before you start to simplify.  The least common denominator for the bottom is 4x.

 
Remember:  after you invert the second fraction, multiply.  Distribute when you multiply the denominators (multiply each term by 3x)

 
 

  Answer:   or 

Remember:  factor the top and the bottom to simplify the answer.

 

5.

Remember:  in this problem, change the numerator to a single fraction before you start to simplify.  The least common denominator for this numerator is x2.  (The second term in the numerator will need to be multiplied by x/x to create the needed denominator.)
 

 

 Remember:  combine the terms in the numerator to form a single fraction in the numerator of this problem.
 

 

 Remember:  after you invert the second fraction, multiply. 

 
  Express the final answer.
(When removing the parentheses in the top, be sure to distribute the 6 ... multiply each term by the 6.)

 

Don't let these fractions trip you up!

Just work carefully through each problem, showing your work as you progress.