Understanding Common Denominators

Date: 07/16/97 at 09:30:53
From: eleanor Turino
Subject: Fractions

What is a common denominator?

Date: 09/01/97 at 11:58:40
From: Doctor Sonya
Subject: Re: Fractions

Dear Eleanor,

Common denominators are the hardest and the most important things to 
learn about when you are studying fractions. They can also be really 
fun, because they allow you to take horribly complicated problems and 
turn them into beautiful, neat fractions.

Let's start with an example in which you would need to use a common 
denominator. If I told you to find 1 + 4, you would know how to add 
them to get 5.  But what if I say:

   What's 4/5 + 1/3 ?

How do we go about adding these two numbers?  The secret to doing this 
is to use a common denominator.

The common denominator is just what it sounds like - a denominator 
that is the same for both fractions (remember that the denominator is 
the number on the bottom of the fraction). For example, what if I 
asked you to do:

   1/5 + 2/5 ?

This is a much easier problem, because there is a common denominator.  
Think of fifths like apples. If you add 1 apple and 2 apples, you get 
three apples.  Similarly, one fifth and two fifths is three fifths. 

Here are some other examples to make this a bit clearer:

   1/6 + 2/6 = 3/6
   2/7 + 4/7 = 6/7

Can you do these ?

   1/5 + 4/5 = ?
   1/4 + 2/4 = ?

The hard part comes in when the two fractions have different 
denominators. Adding four fifths and one third is just like trying to 
add apples and oranges. It can't be done. We need to give these 
fractions the same denominator so we can add them.

The secret to this is realizing that we can always write one fraction 
in many different ways. Let's say we have 1/2 of a pie. Draw a circle 
to represent the pie, divide it in two, and color in half. But what if 
we have 2/4 of a pie?  Again, draw a circle, divide it into four equal 
parts, and color in two of them that are next to each other. This 
should look suspiciously like 1/2, and it is true that 1/2 = 2/4.  
Try the same thing with 4/8, or 8/16. These are all different ways to 
write 1/2. 

If we can write a fraction so many different ways, shouldn't we be 
able to write two of them with a common denominator?

Here's an example that will begin to make things clearer:

   1/4 + 1/2 = ?

Remember that 1/2 = 2/4, so we can replace 1/2 with 2/4 in the 
problem.  Now we have:

   1/4 + 2/4 = ?

These two fractions have the same denominator, and you can now add 
them easily.  

I hope this helps you understand what a common denominator is.  

Finding the common denominator is also hard, but not as hard as just 
understanding what it is. I hope your teacher will explain to you how 
to find common denominators.  If you are still confused, just write us 
back, and we'll do our best to answer your question.

-Doctor Sonya,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/