Adding and Subtracting Fractions
 Topic Index | Algebra Index | Regents Exam Prep Center


To ADD and/or SUBTRACT ALGEBRAIC FRACTIONS:

     In order to add or subtract (or "combine"), algebraic fractions, a common denominator is needed.  (This is the same process you use when adding and subtracting "regular number fractions").

Always find a COMMON DENOMINATOR BEFORE factoring or reducing.

 

 Add:                
            

1.  First, choose the least common denominator (lcd).  This is the smallest number that all denominators can divide into.
                 For the problem above, the lcd is 6.
                 While 12 is also a common denominator, arriving at the answer will require
                  less work if we use the smallest common denominator (lcd).

2.  If any denominator in the problem is already the lcd, the numerator of that fraction stays the same (because the adjustment would be a multiplication times 1).  If any denominator in the problem is not the lcd, multiply that numerator and denominator by the same value that was needed to create the lcd. 

The (1) is shown here for illustrative purposes only.  You do not need to show the multiplication times the number 1.

3.  Combine the problem into one fraction:

4.  Combine like terms in the numerator and reduce the resulting fraction (if needed):      

 


 
 This same process applies when the denominators contain a variable.

Look at this problem:

The least common denominator for this problem is 4y.  Adjust the numerators by the same values that were needed to create the new denominators.

The common denominator is 4y.  Adjust each fraction so that it contains this new denominator of 4y.  Remember to adjust each numerator in the same manner that you adjusted that denominator.    

 

 


 

Express as one fraction and combine like terms.

  

Answer:  =

  



Some problems REQUIRE parentheses to arrive at the correct answer.

Consider:

The common denominator is 6.  

Make the adjustments to obtain the common denominator of 6.

 
Do you see the need for the parentheses?  The adjustments must be multiplied times the ENTIRE numerators.

Express as one fraction:  
Distribute across the parentheses and combine like terms:  

 

Be sure to watch out for "-"(negative signs) when distributing.  You should always write out your work.  When you try to take shortcuts and do the work in your head, signs have a tendency to "disappear".

Try this problem: 

In this problem, the common denominator is already set at 8.  The "trouble spot" in this problem is distributing the negative sign to the whole numerator of the second fraction.

Suggestion:  Write the problem using parentheses to avoid the "trouble spot".

                                              

Do you see how easily you could have made a mistake if you didn’t write out the ( )? 
Be careful!

 

 

More challenging: 

In this problem, be careful with all of the different variables.
The common denominator is 4a2b 

Adjust for the common denominator:
 
Express as one fraction.
FACTOR the new numerator.
In this problem, factoring did not
produce a situation that allows for
cancelling.

 

 

Last one: 

In this problem, the common denominator is 2k+1.
We are all set to combine the terms. 

Express as one fraction.  Be sure to use parentheses when subtracting that second term.
 
Combine like terms and reduce.  Be sure to distribute the negative sign in the numerator.

 

 If you are solving multiple choice questions, you can use the calculator to "check" your work with algebraic fractions.
You can use a Numerical Checking process:  Numerical Process
                                  or
You can use an Equation Checking process::  Equation Process