Multiplying Fractions

Date: 04/11/99 at 20:53:19
From: Derek Reed
Subject: Fractions

I have trouble multiplying fractions. Could you help me?

Date: 04/12/99 at 12:47:39
From: Doctor Peterson
Subject: Re: Fractions

Hi, Derek.

Rather than just remind you that to multiply two fractions you 
multiply the numerators and the denominators, let's think it through.

Suppose I have half a cake left over from my birthday, and want to 
take 2/3 of it. First I'll cut the half into thirds. If I had cut the 
whole cake into pieces this size, there would have been twice as many 
pieces (6), so the pieces are each one sixth of the whole cake. Now I 
want to take two of these pieces; that make 2/6 of a cake, or 1/3. So 
2/3 of 1/2 is 1/3.

You can see it even more easily if you reverse it: 2/3 of 1/2 is the 
same as 1/2 of 2/3. Half of two thirds is one third.

Here's a picture of how it works. I first cut the cake in two from 
top to bottom, and take one half:

    +--+--+
    |//|  |
    |//|  |
    |//|  |
    |//|  |
    |//|  |
    +--+--+

Now I'll cut that half into three pieces; while I'm at it, I'll cut 
the part I didn't take at the same time, showing that these pieces 
are really sixths:

    +--+--+
    |//|  |
    +--+--+
    |//|  |
    +--+--+
    |//|  |
    +--+--+

Now I'll take two of the three pieces:

    +--+--+
    |XX|  |
    +--+--+
    |XX|  |
    +--+--+
    |//|  |
    +--+--+

The two pieces I've taken are the product of the numerators of the 
fractions 2/3 and 1/2: 2 * 1 = 2.

        1
      +--+
      |XX|
    2 +--+
      |XX|
      +--+

The six pieces the whole cake was cut into are the product of the 
denominators of the fractions: 3 * 2 = 6.

         2
      +--+--+
      |  |  |
      +--+--+
   3  |  |  |
      +--+--+
      |  |  |
      +--+--+

So the pieces I've taken are 2/6 of the cake:

     2     1    2 * 1    2
    --- * --- = ----- = ---
     3     2    3 * 2    6

There's the answer yet again!

Now there's one more thing to add: often you will have to simplify 
the answer (as in this case, which turns it into 1/3), and there's a 
way to make that easier. What we do is to simplify before we actually 
multiply. Just notice that when I wrote out the multiplied fraction 
above,

    2 * 1
    -----
    3 * 2

both numerator and denominator contain a 2. I can rearrange this as

    2 * 1    2     1         1     1
    ----- = --- * --- = 1 * --- = ---
    2 * 3    2     3         3     3

You can do this very easily by just crossing out the 2's, which is 
called cancelling:

    /
    2 * 1    1
    ----- = ---
    3 * 2    3
        /

In more complicated problems, this can save a lot of work, since 
otherwise you would have to multiply, then factor or divide, undoing 
part of your work.

I hope this helps a little. If I haven't helped with whatever is the 
hardest part for you, please write back and give me an example of a 
problem you find hard and how you tried to do it; that will help me 
see where you need the most help.

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