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 
problem: 1/5 x 40/1 x 1/2 ? Can you please help me?

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
about this, or anything else.

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