Cancelling Fractions

Date: 06/03/2003 at 10:33:34
From: Chris Breeze
Subject: Cancelling

Dear Dr.Math,

 I am reading a book on algebra but there is something I really fail 
to understand concerning cancelling and fractions. The author states 
that cancelling is division too and then shows the following without 
any explanation of how it's done:

                       1   
                       x   x   1
                       2   2   2
                       x   x   x
       3 * 3 * 3 * 3 * 4 * 4 * 4 
           2   2   2   2   2
           x   x   x   x   x
           1   1   1   1   1

Reading through your archives I more or less understand cancelling, 
but I still don't understand this example. I would be very grateful 
if, unlike the author of the textbook, you could explain what is going 
on here.  

Date: 06/03/2003 at 12:31:44
From: Doctor Peterson
Subject: Re: Cancelling

Hi, Chris.

Did you leave out the fraction bar? I think the problem is meant to 
be a simplification of the fraction

       3 * 3 * 3 * 3 * 4 * 4 * 4 
       -------------------------
           2 * 2 * 2 * 2 * 2

The work (corrected slightly from what you wrote) is:

                       1   1
                       x   x
                       2   2   2
                       x   x   x
       3 * 3 * 3 * 3 * 4 * 4 * 4 
       -------------------------
           2 * 2 * 2 * 2 * 2
           x   x   x   x   x
           1   1   1   1   1

The order doesn't matter; each number is crossed off and replaced with 
the quotient. There should be no remainders involved; everything has 
to divide exactly. For example, when we divide 4 by 2, the quotient is 
2, and that is what we write.

Let's take it one step (well, a few steps, to save writing) at a time:

       3 * 3 * 3 * 3 * 4 * 4 * 4 
       -------------------------
           2 * 2 * 2 * 2 * 2

First we see that each of the 4's can be divided evenly by 2; so we 
(three times) divide a 4 and a 2 by 2, replacing them by 2 and 1 
respectively:

                       2   2   2
                       x   x   x
       3 * 3 * 3 * 3 * 4 * 4 * 4 
       -------------------------
           2 * 2 * 2 * 2 * 2
           x   x   x
           1   1   1

Now we still see some 2's on top and some 2's on the bottom, so we 
divide (twice) a 2 on the top and a 2 on the bottom by 2, leaving 1 
in each case:

                       1   1
                       x   x
                       2   2   2
                       x   x   x
       3 * 3 * 3 * 3 * 4 * 4 * 4 
       -------------------------
           2 * 2 * 2 * 2 * 2
           x   x   x   x   x
           1   1   1   1   1

What's left is really just

       3 * 3 * 3 * 3 * 1 * 1 * 2   3 * 3 * 3 * 3 * 2
       ------------------------- = ----------------- = 162
           1 * 1 * 1 * 1 * 1              1

In real life, where the "x"s would be slashes across the numbers 
themselves (which I can't type), the first step would look more like

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

and the second step like

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

(Each slash would have a crossed-out number under it.)

Does that help?

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