Introduction | Addition of Fractions | Common Denominators & Equivalent Fractions | Subtraction of Fractions 

Common Denominators & Equivalent Fractions

We now know that in order to add fractions, the denominators have to be the same.  If they are not, we have to use equivalent fractions to make them the same.  Let's look at that in greater detail so we can get real good at doing that.

Let's add .  The denominators are not the same.  But if we multiply the numerator of  by 2, we get the equivalent fraction of  and now we can add:

Now you try it.  Add:

  1.  
     
  2.  
     
  3.  

You got them right if your answers were:  

  1.  
     
  2.  
     
  3.  

Was that pretty easy?  You only had to change one of the two fractions to get the denominators the same.  All you had to do was, in....

  1. multiply    by  
     
  2. multiply  by
     
  3. multiply  by
But suppose we want to add   +  .  Well, you can't multiple 3 by any whole number to get 4, nor multiple 4 by any number to get 3.  But we do know the denominators have to be the same in order to add the fractions.  Well, we know that 3 x 4 = 12, and that 4 x 3 = 12.  That means that we can change each fraction to equivalent fractions with a common denominator of 12.  Here's how we do it:

 

 

With the denominators the same, we can add the numerators.  

Can you see that we can always get a common denominator by multiplying the two denominators together and arrange for the equivalent fractions to have that denominator?  In fact, if we are adding 3 fractions, the common denominator can be the product of the 3 denominators.

Let's do a few problems together and then you can try a few on your own.

Add .

The common denominator  is:  4 x 7 = 28

 

Sum = 

Now you try some.  Add:

  1.    = _____
     
  2.    = _____
     
  3.   = _____

The trick to doing this is first, find the common denominator by multiplying the two denominators together and second, change each fraction to an equivalent fraction with the new denominator.  Then add the new numerators and place them over the common denominator.  When you get these answers, you are ready to continue:

  1.  
     
  2.  
     
  3.  
Suppose we want to add .  We could get the common denominator in the usual way: 6 x 8 = 48 and change each fraction to these equivalent fractions:

Sum =

But is the fraction in lowest terms?  No.  It is not because both terms can be divided by 2.  When we divide both the denominator and the numerator by 2, we reduce the fraction to .  This is now in the lowest terms, but is still equivalent to .

Could we have done that problem differently so that the sum ended up in lowest terms?  Yes.  Instead of multiplying 6 x 8 = 48, we need to ask "what is the smallest number that 6 and 8 divide into evenly?"  Well, both go into 24 evenly and so we can use 24 as the lowest common denominator since 6 goes into 24  4 times.  

 can be used as the multiplier for getting the equivalent fraction:

 

 

Sum =

Both 24 and 48 are common denominators, but 24 is called the lowest or least common denominator.

For additional instructional material and practice exercises on this subject, go to: www.aaamath.com/fra66j-lcd.html,  www.quia.com/jg/65839.html, and www.aaamath.com/fra42a-idequivfract.html.
 

Subtraction of Fractions  (top)
CA GR5 NS 2.3,  CA GR6 NS 2.1

Subtraction works the same way.  When the denominators are the same, subtract the smaller from the larger numerator.

Subtract:

 

For additional instructional material and practice exercises on this subject, go to: www.aaamath.com/fra57b-subfractld.html.

 

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