Multiplying Fractions by Cancellation

Date: 10/09/98 at 22:28:25
From: Patrice 
Subject: Multiplying fractions

In class, we're working on multiplying fractions, where you cross out 
numbers before you multiply. Whenever I look in books it tells me to 
just multiply straight, but that's not how I need to know it. Please 
explain the other way.

Date: 10/10/98 at 15:40:21
From: Doctor Peterson
Subject: Re: Multiplying fractions

Hi, Patrice. I think what you are referring to is what we call 
"canceling." It's not an essential part of multiplying fractions, but 
an extra thing you can do to make the work easier. That's why you 
don't find it when you look up multiplication of fractions.

Here's an example: Let's multiply 2/5 by 15/16. We can just multiply 
straight:

     2    15   2 * 15   30
    --- * -- = ------ = --
     5    16   5 * 16   80

But now we'd like to get it in simplest terms, so we have to find the 
greatest common divisor of 30 and 80. One way to do this is to factor 
each number, and then divide both numerator and denominator by each 
factor they share:

         /       /
    30   2 * 3 * 5               3        3
    -- = ----------------- = --------- = ---
    80   2 * 2 * 2 * 2 * 5   2 * 2 * 2    8
         /               /

Notice that we have multiplied some numbers, then factored them back 
out and divided. Why bother making big numbers when you just have to 
cut them back down to size again? By canceling before you multiply, 
you can save the work:

                    3
               \   //
     2    15   2 * 15    3
    --- * -- = ------ = ---
     5    16   5 * 16    8
               /   \\
                    8

What I did here was to see that the numerator and denominator both 
have a factor of 2, so I divided them by 2 (crossing out the 2 and 
replacing the 16 with 8). Then I saw that both also have a factor of 
5, so I divided by 5 (crossing out the 5 and replacing the 15 with 3). 
That left me with only the 3 on top and the 8 on the bottom, so I 
never had to multiply anything.

So canceling is just a shortcut. We can simplify before we multiply, 
rather than afterward, to save work. You may not think your teacher is 
saving you work by teaching this, but that's what's happening!

For more information, try looking at:

   http://mathforum.org/dr.math/problems/lask1.4.98.html   

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