Cancelling Fractions

Date: 01/04/98 at 21:42:10
From: Lori
Subject: Cancelling fractions

How do you cancel 3   2
                  - x -  ? 
                  5   3

The book I have explains it, but it looks like teachers from Planet X 
are writing it. I totally don't get the 
entire problem. HHHHHEEEEELLLLLPPPPP!

Date: 01/07/98 at 11:10:04
From: Doctor Otavia
Subject: Re: Cancelling fractions

Hi, Lori! I also sometimes find math textbooks hard to understand. If 
I don't understand a part the first time I read it, I usually reread 
it and try some of the examples, and sometimes that helps.  

Let's take a look at a problem that's like your problem. Once you 
understand how to do this kind of problem, you should be able to do 
yours in a jiffy.  Let's use oh... how about

   7     13
  --- * ---  = ?
   3     7

Those seem like good numbers to me. After all, this is just an example 
so you can learn the method for solving problems like these.  

What you want to do is find the product of those two fractions, right? 
And you want to use cancellation to help you do that. So, how do you 
multiply fractions together? You multiply the numerators (the top 
part) and you multiply the denominators (the bottom part), so the 
first step would look like

   7     13      7 * 13
  --- * ---  = ----------.
   15    7       15 * 7

So far, it looks just like regular old fraction multiplication.  So 
where's the cancellation? Well, we know that 3 * 7 = 7 * 3, so we can 
rewrite our fraction, which then looks like


    7 * 13        7 * 13
  ---------- =  ----------.
    15 * 7        7 * 15

Now, we can again rewrite that fraction as a product of two fractions, 
so now we have 

    7 * 13        7 * 13       7     13
  ---------- =  ---------- =  --- * ----- .
    15 * 7        7 * 15       7     15

But wait, 7/7 means 7 divided by 7 which is 1, right?  So what we 
really have is

        13
   1 * ----.
        15

But we know that anything times 1 equals itself, so your answer is 
13/15.

So, that's a step-by-step method for finding the products of fractions 
using cancellation along the way, and now that you've seen it, I'll 
show you a shorter way, but first I had to explain the whole reasoning 
behind it, or else the simpler explanation might not make sense!

Let's say you have some fractions you want to multiply together. 
Imagine that when you do the first step of combining the numerators 
and the denominators, you have a fraction that looks like 

   6 * 11 * 2 * 5
  ---------------- .
   2 * 6 * 10 * 7

An easy way to cancel stuff is to just get rid of any numbers that you 
have in both the numerator and the denominator. The reason this works 
is that what you're really doing is rearranging the numbers and then 
expressing the fraction as a product of **something that's really one 
in disguise** times whatever is left, or in this case, since you have 
a 6 in the numerator and a 6 in the denominator, you can just throw 
out both 6's - and the same goes for the 2.  What you're left with is

    11 * 5
   -------- .
    10 * 7

Looks like you're done, right?  No, not quite yet. Because see, 
10 = 5 * 2, so really the fraction is equal to

    11 * 5
   ----------.
    2 * 5 * 7
  
Now you can again cancel out the 5, and your final result is

   11       11
 ------ = ------.
  2 * 7     14

So I guess another way to describe cancellation is to multiply the 
numerators and the denominators, (in other words, combine all your 
fractions into one big fraction), and then factor every number as much 
as you can. Once you've done this, every number that appears in the 
numerator and in the denominator can be thrown out, or cancelled.

This should help you with all problems of this kind. Now try applying 
this method to your specific problem.  

You start with

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

Now you should be able to go from there. Good luck!  If you have any 
more questions, don't hesitate to ask!

-Doctor Otavia,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/