Lesson Page

Math A

Adding and Subtracting Fractions

 

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 with adding and subtracting "regular number fractions").

ALWAYS FIND A COMMON DENOMINATOR BEFORE FACTORING OR REDUCING.

 

Let’s try one:
 

First, choose the least common denominator (lcd).  This is the smallest number that all denominators can divide into.

For the problem above, is the lcd 6, 2, or 12?

6 ?  (yes, 6 is the best answer)
2 ?  (no, 6 and 2 must divide INTO the lcd)
12 ? (no, it’s best to choose the smallest lcd)

Now, if the denominator of the original fraction is also the lcd (the common denominator), the numerator stays the same. However, if the denominator of the original fraction is not the lcd, you must multiply the numerator by the quotient of  the least common denominator divided by the original denominator. 

Just remember:  whenever you multiply the bottom of a fraction by a number to create the new lcd (common denominator), you must multiply the top by that SAME number.


Combining our problem into one fraction, we have:

Now, you may combine like terms in the numerator and reduce the resulting fraction (if needed):              

 

Look at this:

Is the common denominator for this problem 72, 3, 12,  or 24?

72 ? (no, the smallest lcd is best)
 3 ? (no, you need a number which 3, 4 and
       6 GO INTO)
12 ?  (yes, the best answer)
24 ? (no, the smallest lcd is best)

Now, you need to "change " the numerators to match your new common denominator.  Give it a try and then scroll down to see if you did it correctly.

STOP DO THE WORK FIRST
CAUTION DO IT CAREFULLY
GO CHECK YOUR WORK NOW

 

Did you get: 

? 

Remember that in each fraction, you are adjusting the numerator in the same manner that you adjusted the denominator.

Good, now combine like terms in the numerator: 

and reduce:

 

Now, you’re ready for some tougher problems.

The key for these tougher problems is to always use ( ) and to distribute across your parentheses in the numerator.

Here’s one: 

The common denominator is 6.

WRITE THIS OUT:

remember those parentheses!

Now, distribute and combine like terms: 

You need to watch out for "-"(negative signs) as well. If you always write out the quotient and distribute with the sign of the entire fraction, there shouldn’t be any problems. It’s when you take shortcuts, that signs seem to "disappear".

 

In this next problem, even though the common denominator is already there, you must distribute the minus sign to the whole numerator of the second fraction.
Write it out with (  ) to avoid errors!

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

 

 


Lloyd