What Does Cancelling Mean?

Date: 01/22/2003 at 07:35:30
From: P. Bell
Subject: Simplifying rational expressions

My teacher told me that 2k/4n can be simplified to k/2n.  

I'm not sure that this makes sense to me since what he is really 
saying is that (k+k)/(n+n+n+n) = k/n+n. How is it that one k cancels 
out 2 of the n's?

Date: 01/23/2003 at 15:08:05
From: Doctor Edwin
Subject: Re: Simplifying rational expressions

Hello, and thanks for writing.

This is an excellent question, and it shows that you're thinking 
deeply about what you're studying in school. Let's play with some of 
the concepts here and see what we come up with.

One way to look at fractions is as ratios. If I have three grapes and 
6 bananas, then I have half as many grapes as bananas. I could write:

   g = b/2

meaning that the number of grapes is half the number of bananas. I 
could also write:

  g     1
  -  =  -
  b     2

That is, the ratio of grapes to bananas is 1 over 2, or 1:2. Now 
suppose I take away two of the grapes and four of the bananas. I'm 
still left with 1 grape and 2 bananas, for a ratio of 1:2. So I've 
just divided both the top and bottom of my fraction by 3, and the 
fraction has stayed the same.

In other words,

        g + g + g            g
   --------------------- = -----
   b + b + b + b + b + b   b + b

Now, how did 4 b's cancel out 2 g's? We have to look at what we mean 
when we say "canceling out." One meaning for "X cancels out Y" is "I 
can cross out X from one side and Y from the other side and it doesn't 
change the answer." Can you see why that's the case here when X is 
2 g's and Y is 4 b's?

Another way to look at a fraction is as a division problem.

  g     _____
  - = 2 ) 1
  b

Now, if I double both sides of a division problem, what happens to the 
answer? What if I multiply both sides by three, or 137?

Does that help make things clearer?

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