Canceling Instead of Multiplying and Dividing

Date: 09/25/2003 at 23:54:31
From: Chris
Subject: multiplying fractions

Find the value of the product

  3/5 x 5/7 x 7/9 x 9/11 x 11/13 x 13/15 = ?

Date: 09/26/2003 at 11:27:41
From: Doctor Ian
Subject: Re: multiplying fractions

Hi Chris,

The first thing is to make sure you know how to multiply two
fractions, e.g., 

  3/5 * 5/7

If you're not sure how to do that, see 

  Multiplying and Dividing Fractions
  http://mathforum.org/library/drmath/view/58080.html 

Next, you would multiply the first two, then multiply the product by
the third, and multiply _that_ product by the fourth, and so on, just
as you would with integers:

   2 * 3 * 4 * 5 * 6
   ......
      .
      .............................. = 6


       6 * 4 * 5 * 6
       .....
         .
         ........................... = 24


          24 * 5 * 6
          ......
             .
             ....................... = 120 

             120 * 6


                 720

However, having said all that, in this particular case, there is an
interesting pattern that becomes more apparent when you write the
fractions using horizontal bars:

  3   5   7    9   11   13
  - * - * - * -- * -- * --
  5   7   9   11   13   15

Let's look at the first pair:

  3   5 
  - * - 
  5   7 

To multiply two fractions, you multiply the numerators, and multiply
the denominators:

  3   5   3 * 5
  - * - = -----
  5   7   5 * 7

But we get the same result if we switch denominators:

  3   5   3 * 5
  - * - = -----
  7   5   5 * 7

Does that make sense?  But now one of the terms is just 1!  So 

  3   5   3   5   3
  - * - = - * - = -
  5   7   7   5   7

The quick way to do this is to notice that in 

  3   5 
  - * - 
  5   7 

we're going to be multiplying by 5 (in the numerator), and dividing by
5 (in the denominator).  So we can just save ourselves some time and
trouble by forgetting about the two operations, since they just cancel
each other out.  So when we see something like

  3   5 
  - * - 
  5   7 

we can just 'cancel' the 5's to get 

      x
  3   5   3
  - * - = - 
  5   7   7
  x

Can you see how this will allow you to find the product of all those
fractions without actually doing any multiplication? 

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