Reducing Algebraic Fractions by Cancelling

Date: 03/16/2004 at 22:26:49
From: Jennie
Subject: canceling factors or terms

When reducing fractions, how come you can cancel factors but you can't
cancel terms?  Can you please explain, hopefully with examples too? 
Thank you so much! 

Date: 03/17/2004 at 14:34:44
From: Doctor Peterson
Subject: Re: canceling factors or terms

Hi, Jennie.

Good question!  It's important to understand what is really happening 
when you cancel, so you know when you can do it, and why.

Factors are parts of an expression that are multiplied together; you 
can cancel them when you are dividing two expressions.  For example, 
you can simplify this expression by canceling the z's:

   xz     x     z     x         x
  ---- = --- * --- = --- * 1 = ---
   yz     y     z     y         y

The z's "cancel" because the expression can be rearranged to contain 
z/z which is equal to 1.

Terms are parts of an expression that are ADDED together; you can 
cancel them, but only when you are subtracting rather than dividing. 
For example, you can "cancel" the z's here:

  (x + z) - (y + z) = x + z - y - z = x - y + z - z = x - y

This time the z's disappear because when they are brought together we 
have z - z which is zero.  We're subtracting z from both terms, and 
their difference remains the same.

The reason terms can't be canceled when you are dividing is simply 
that addition and division don't work so nicely together.  For 
example, in the expression

  x + z
  -----
  y + z

there is no way to rearrange the expression to give you z/z or z - z.

It may help to give some concrete demonstrations that canceling 
actually doesn't work here. First, we CAN cancel this:

  2*5    2
  --- = ---
  3*5    3

and in fact the original fraction is 10/15 which is equal to 2/3.  And
we CAN "cancel" this:

  (3 + 5) - (2 + 5) = 3 + 5 - 2 - 5 = 3 - 2 + 5 - 5 = 3 - 2 = 1

and the original expression is 8 - 7 which is equal to 1 also.

But this is NOT true:

  2 + 5    2
  ----- = ---
  3 + 5    3

Here the left side is 7/8, which clearly is not equal to 2/3.  In 
terms of a picture, the left side has 7 of 8 parts shaded:

  +---+---+---+---+---+---+---+---+
  |   |xxx|xxx|xxx|xxx|xxx|xxx|xxx|
  +---+---+---+---+---+---+---+---+

If we subtract 5 from both numerator and denominator, we are taking 
away 5 shaded squares, and no unshaded squares:

  +---+---+---+
  |   |xxx|xxx|
  +---+---+---+

Now we have 2 of 3 shaded; taking away only shaded squares changed 
the ratio.  Proper cancellation is balanced:

  2 * 5    2
  ----- = ---
  3 * 5    3

  +---+---+---+---+---+       +---+
  |xxx|xxx|xxx|xxx|xxx|       |xxx|
  +---+---+---+---+---+       +---+
  |xxx|xxx|xxx|xxx|xxx|  -->  |xxx|
  +---+---+---+---+---+       +---+
  |   |   |   |   |   |       |   |
  +---+---+---+---+---+       +---+

Here by dividing numerator and denominator by 5, we take squares away 
in the right proportions so that the fraction is unchanged.

Does that help make things clearer?

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