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### Cancelling Fractions

```
Date: 03/06/2002 at 13:58:48
From: KJJ
Subject: Cancelling Fractions

Dear Doctor Math,

I have a question about cancelling fractions. How do I cancel this

Sincerely,
KJJ
```

```
Date: 03/06/2002 at 16:04:21
From: Doctor Ian
Subject: Re: Cancelling Fractions

The thing you have to keep in mind is that

a   c   a * c
- * - = -----
b   d   b * d

Now, usually you use this to go from left to right, to turn multiple
fractions into one fraction:

3   5   3 * 5   15
- * - = ----- = --
4   6   4 * 6   24

But it also works from right to left!

15   3 * 5   3   5       5
-- = ----- = - * - = 1 * - = 5/8
24   3 * 8   3   8       8

Note that each time I can break off a fraction like a/a, it's equal
to 1. And since multiplying by 1 is the same as doing nothing, I can
just ignore it.

So let's look at a case like

1   40   1
- * -- * -
5    1   2

Note that 40 is divisible by 5:

1   5 * 8   1
- * ----- * -
5     1     2

And note that 8 is divisible by 2:

1   5 * 2 * 4    1
- * ---------- * -
5       1        2

So now I can mix the terms up, pair them, and unmix them:

1   5 * 2 * 4    1
- * ---------- * -
5       1        2

1 * 5 * 2 * 4 * 1
=  -------------------
5 * 1 * 2

5 * 2 * 1 * 4 * 1
=  -----------------
5 * 2 * 1

5   2   1
=  - * - * - * 4 * 1
5   2   1

= 1 * 1 * 1 * 4 * 1

= 4

Now, this is _why_ you can cancel.  But in practice, you would do
something like this:

8
x
1   40   1       Note that 40 divided by 5 is 8;
- * -- * -       Get rid of the 5, and replace 40 with 8.
5    1   2
x

Do you see why this works? I'm just saying that if I went to the
trouble to expand 40 into 5 * 8, the 5's would cancel, leaving me with
an 8. The next step would be

4
x
8
x
1   40   1        Note that 8 divided by 2 is 4;
- * -- * -        Get rid of the 2, and replace 8 with 4.
5    1   2
x        x

This is just the same thing again. You keep going until there is
nothing else to cancel. In this case, I'm done, and the only thing
left is a 4 in the numerator, so the answer is 4.

Note that you might have to do the expansions to see the opportunities
for cancellation.  For example,

1    12     1
-- * ---- * --
15     1    28

Now, in this case, nothing divides into anything else.  But if we
break everything into factors,

3*2*2

1    12     1
-- * ---- * --
15     1    28

3*5        2*2*7

Now I can see the things that should cancel:

x x x
3*2*2

1    12     1     1
-- * ---- * -- = ----- = 1/35
15     1    28   5 * 7

3*5        2*2*7
x          x x

Now, note how much easier this is than multiplying 15 by 28, and then
dividing the result by 30. Every pair of operands that you can cancel
is a pair of operations that you don't have to do.

And _this_ is one of the reasons that everyone makes a big deal out of
being able to find the prime factors of numbers. When you've got all
the prime factors out in the open, you can just tick-tick-tick them
off in pairs, leaving you with only the operations that you can't
escape.

I hope this helps.  Write back if you'd like to talk more

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

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