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### Canceling and Reducing When Multiplying Several Fractions

```Date: 01/25/2008 at 09:56:07
From: Renee
Subject: cancellation in multiple fractions

I need help when multiplying three or more fractions.  I know how to
cancel diagonally when the fractions are right next to each other.  I
know how to cancel the numerator and denominator of a fraction, too.
I have seen people cancel using the numerators and denominators that
aren't near each other.  How do they do that?

Can you double cancel a number?  How do you keep it all straight then?

Thanks a million for your help!

```

```
Date: 01/25/2008 at 10:26:10
From: Doctor Ian
Subject: Re: cancellation in multiple fractions

Hi Renee,

The key idea is that multiplication is commutative.  Suppose you have
something like

4    3   25
- * -- * --
5   10   30

You can break everything into prime factors,

2*2    3     5*5
--- * --- * -----
5    2*5   2*3*5

Now, to multiply fractions, we just multiply the numerators and
denominators separately, right?  So this is the same as

2*2 *   3  *   5*5
-------------------
5  *  2*5 *  2*3*5

So now, we're just reducing a fraction.  And we can do that by
identifying prime factors that appear in both the numerator and
denominator:

2 *   3  *   5*5
-------------------     Cancel a 2
5  *    5 *  2*3*5

3  *   5*5
-------------------     Cancel another 2
5  *    5 *    3*5

5*5
-------------------     Cancel a 3
5  *    5 *      5

1
-------------------     Cancel two 5's
5

So the result of the multiplications is 1/5.  Does this make sense so
far?  Remember that when you "cancel" two identical numbers you are
making each number into 1.  That's why there is a 1 left in the
numerator after all the numbers that were there have canceled.

Here's how I'd do it in practice:

4    3   25
- * -- * --        Our original problem.
5   10   30

2    3   25
- * -- * --        I canceled a 2 from the 4 and the 10.
5    5   30

1    3   25
- * -- * --        I canceled a 2 from the 2 and the 30.
5    5   15

1    3    5
- * -- * --        I canceled a 5 from the 5 and the 25.
1    5   15

1    3    1
- * -- * --        I canceled the two 5's.
1    1   15

1    1    1
- * -- * --        I canceled a 3 from the 3 and the 15.
1    1    5

1    1    1
- * -- * --        Nothing left to cancel.
1    1    5

As I said, the key idea is that since the numerators and denominators
are going to get multiplied anyway, it doesn't matter whether the
common factors are next to each other, or far away.  What I'm doing
here is realizing that I really have one big fraction that I need to
reduce, rather than three fractions that I need to multiply.

Does that make sense?

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

```

```
Date: 02/06/2008 at 10:18:52
From: Renee
Subject: Thank you (cancellation in multiple fractions)

Thanks a lot Dr. Ian!  I actually understand the concept for the first
time!  Realizing that its all really just one BIG fraction comprised
of prime factors - and you're actually reducing as you go along rather
than just at the end...sure makes sense to me now.  Whoaaa, you're good!
```
Associated Topics:
Elementary Fractions
Elementary Multiplication

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