Why Cancel?

Date: 02/25/2003 at 21:28:25
From: Malinda
Subject: Why you have to cancel

Why do you have to cancel numbers? 

Like say 2/10 X 5/10 =

Why do you cancel when you can have the answer when you just multiply 
it out and then reduce?

Date: 02/25/2003 at 23:14:32
From: Doctor Peterson
Subject: Re: Why you have to cancel

Hi, Malinda.

You don't _have_ to cancel, any more than you have to take the short 
route when you walk somewhere, or walk when you can ride. Canceling 
just saves effort, so it's good to know how to do it. And sometimes 
when you forget to cancel first, you feel really stupid afterward 
when you realize how much time you wasted!

When you multiply first and then cancel, you are doing a lot of work 
only to undo it. In a simple problem, that's not hard; but suppose 
you have to multiply 155/346 by 346/754. Why would you bother to 
multiply 155*346 and 346*754 when you know you'll be dividing both 
numbers by 346 afterward? And you might not even notice that you can 
simplify in that case; simplifying is a much harder task than just 
dividing, because you have to look for common factors to divide by, 
and when they are large like this, you may miss them. It's much 
easier to spot opportunities to simplify when the numbers are still 
small. And that's a general rule for life: fix problems as early as 
possible, before they have a chance to grow bigger!

Here are some discussions of canceling in more complex situations:

   Multiplying Fractions by Cancellation
   http://mathforum.org/library/drmath/view/58151.html 

   Multiplying Fractions
   http://mathforum.org/library/drmath/view/58079.html 

If you have any further questions, feel free to write back.

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