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### Cancelling Fractions

Date: 05/06/2002 at 09:28:32
From: Stuart
Subject: Cancelling fractions

I am having a little trouble doing the extension task of my maths
I have to fill in the missing numbers :

1. 2/? x 6/7 x ?/12 = 2/3

2. ?/2 x 2/3 x ?/7 = 2/7

3. 1/4 x 4/? x ?/5 = 2/5

4.?/6 x ?/3 x 1/5 =2/3

5. 7/8 x 16/12 x ?/? = 2

I think that there must be a formula for doing this but can not see
it.  I have tried for the last 24 hours.

Stuart Teare

Date: 05/06/2002 at 10:40:15
From: Doctor Ian
Subject: Re: Cancelling fractions

Hi Stuart,

There isn't a 'formula' so much as a 'process'. Let's look at the
first one:

2   6    ?   2
- * - * -- = -
?   7   12   3

Because multiplication is commutative, i.e.,

a *b = b * a

we can rewrite the equation this way:

2 * 6 *  ?   2
---------- = -        (This is how you multiply fractions.)
? * 7 * 12   3

? * 2 *  6   2
---------- = -        (Multiplication is commutative.)
? * 7 * 12   3

?   2 *  6   2
- * ------ = -        (This is how you un-multiply fractions!)
?   7 * 12   3

?   1   2
- * - = -        (Simplify the second fraction.)
?   7   3

Note that if we multiply both sides of the equation by 7/1, we get

?   1   7   2   7
- * - * - = - * -
?   7   1   3   1

?   14
- = --
?    3

We can check that by substituting back into the original equation:

2   6   14   2
- * - * -- = -
3   7   12   3

/   /
2   6   14   2
- * - * -- = -           Cancel
3   7   12   3
/

/
14    2
----- = -          Cancel
3 * 7   3
/

2   2
- = -          True.
3   3

Once you get some practice with this, you can start doing the
cancellations beforehand:

/
1   4   ?   2
- * - + - = -           Cancel
4   ?   5   5
/

1 * ?   2
----- = -
? * 5   5

That can cut down the amount of writing that you have to do, and
sometimes lets you see the answer right away.

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 Fractions

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