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### Multiplying Fractions by Cancellation

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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

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!

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Elementary Fractions
Middle School Fractions

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