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### 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

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/
```
Associated Topics:
Elementary Division
Elementary Fractions
Elementary Multiplication
Middle School Division
Middle School Fractions

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