Associated Topics || Dr. Math Home || Search Dr. Math

### 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

+---+---+---+
|   |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/
```
Associated Topics:
High School Polynomials
Middle School Factoring Expressions
Middle School Fractions

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search