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 
homework, cancelling fractions. Please could you give me some advice?
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.

Thank you for your help.
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
about this, or anything else.

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