Renaming Fractions

Date: 05/22/2000 at 11:24:53
From: Alyssa Bombard
Subject: Fractions

I don't understand how you rename fractions. Can you help me? Thank 
you.

Alyssa

Date: 05/22/2000 at 12:52:31
From: Doctor Peterson
Subject: Re: Fractions

Hi, Alyssa.

Let's start with a simple fraction, 1/2. That means we've taken 
something, say a block, and cut it into two (the denominator) equal 
pieces, and then taken one (the numerator) of them:

     +---------+---------+
     |XXXXXXXXX|         |
     |XXXXXXXXX|         |
     |XXXXXXXXX|         |
     |XXXXXXXXX|         |
     |XXXXXXXXX|         |
     |XXXXXXXXX|         |
     +---------+---------+

Now, if I cut each of those pieces in half, I'll have a total of 4 
pieces, of which I've taken 2:

     +---------+---------+
     |XXXXXXXXX|         |
     |XXXXXXXXX|         |
     |XXXXXXXXX|         |
     +---------+---------+
     |XXXXXXXXX|         |
     |XXXXXXXXX|         |
     |XXXXXXXXX|         |
     +---------+---------+

In other words, the fraction 1/2 is the same as 2/4; they are two 
names for the same amount. We call these equivalent fractions. Do you 
see how I changed 1/2 to 2/4? I just had to multiply both the 
numerator and the denominator by 2, since there are twice as many 
pieces in all, and I took twice as many of them. I could just as well 
have divided the pieces in thirds, to make a different fraction:

     +---------+---------+
     |XXXXXXXXX|         |
     |XXXXXXXXX|         |
     +---------+---------+
     |XXXXXXXXX|         |
     |XXXXXXXXX|         |
     +---------+---------+
     |XXXXXXXXX|         |
     |XXXXXXXXX|         |
     +---------+---------+

Now I've rewritten 1/2 as 3/6, by multiplying numerator and 
denominator by 3, so that I have 3 out of 6 pieces. Here's a list of 
equivalent fractions:

      1     2     3     4     5     6     7     8
     --- = --- = --- = --- = --- = --- = --- = --- = ...
      2     4     6     8    10    12    14    16

Now, we can also rename fractions in the opposite direction. If I 
DIVIDE both the numerator and the denominator of a fraction by the 
same number, then I get another name for the same fraction. For 
example, if you gave me the fraction 3/6, I would see that I could 
divide both 3 and 6 by 3, giving 1/2. By doing this, I have SIMPLIFIED 
(or REDUCED) the fraction; the smaller numbers are easier to work 
with, so this is a good thing to be able to do. 

You can always make the numbers bigger by multiplying them by 
something; but you can't always make them smaller, since two numbers 
are not always divisible by the same thing. If I can't divide them by 
anything, we say the fraction is "in lowest terms" - that is, we can't 
make the numerator and denominator any smaller. Our original 1/2 is in 
lowest terms, but 2/4 is not.

You can find more help with this process by going to the Dr. Math 
search page at http://mathforum.org/mathgrepform.html    and entering 
words like  reduce fractions  or  lowest terms  - be sure to select 
the button for "that exact phrase."

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/