Multiplication of Fractions

Date: 03/22/99 at 09:04:07
From: Kenneth Lee
Subject: Multiplication Of Fractions

I am confused about the multiplication of fractions. My teacher talks 
about cancellations but I cannot understand her point.

Date: 03/22/99 at 12:58:06
From: Doctor Peterson
Subject: Re: Multiplication Of Fractions

Yes, cancellation can be confusing if you do not have a clear idea of 
why you do it. Really, it is not a necessary part of multiplication, 
but a trick to save work simplifying the fraction when you are done. 
Let us look at a sample problem:

     4    15
    --- * --
     5    16

To multiply, all you do is multiply the numerators and the 
denominators:

     4 * 15
    -------
     5 * 16

You could just multiply right now:

     60
    ----
     80

and then simplify:

     60     6     3
    ---- = --- = ---
     80     8     4

But when you do that, you are first multiplying to get the 60 and 80, 
then dividing them to get smaller numbers. That is like building a 
house by first making a solid pile of bricks, then taking out the 
bricks in the doors, windows, and interior - what a waste of work!

So what we do instead is to look BEFORE we multiply and see what 
factors we can cancel - that is, what factors we will be dividing the 
numerator and denominator by in order to simplify the answer. Here's 
what it looks like if I write it out in great detail:

 4 * 15   2*2 * 3*5     1*2*2*3*5    1     2     2     1     3     5
------- = ----------- = --------- = --- * --- * --- * --- * --- * ---
 5 * 16   5 * 2*2*2*2   2*2*2*2*5    2     2     2     2     1     5

           1             1     3         3
        = --- * 1 * 1 * --- * --- * 1 = ---
           2             2     1         4

I factored each number in the numerator and combined the factors into 
one list, then did the same in the denominator; then I paired off the 
factors that matched, which turned into 1's: any factor that is in 
both the numerator and denominator can just be taken out of both 
places, because they cancel each other out. We can write it this way 
to save work, just crossing out the paired factors:

              / /     /
     4 * 15   2*2 * 3*5      3     3
    ------- = ----------- = --- = ---
     5 * 16   5 * 2*2*2*2   2*2    4
              /   / /

Here I just cancelled each pair of identical factors by crossing them 
out. (Those slashes would go through the numbers when you write it.) 
The only factors that are left are the leftover 3 and 4.

Once you are familiar with the process, you can do this more compactly 
by not bothering to write out the factors. Just look at each number in 
the numerator and cancel out any common factors it shares with a 
number in the denominator by dividing them both by that factor. Here, 
I divide 4 and 16 by 4, and I divide 5 and 15 by 5:

     1      3
     /     //
     4     15    3
    --- * --- = ---
     5     16    4
     /     //
     1      4

If all I showed you was this final method, you would wonder where I 
got the idea of canceling; I hope you can see now why we do this.

Now here is another way you can think of it: if both of the fractions 
you multiplied were already in lowest terms (simplified), then you 
know there is nothing to cancel there, so each numerator can only 
cancel with the other denominator. So, you can write the 
multiplication this way:

     4    15    4   15    1     3     3
    --- * -- = -- * -- = --- * --- = ---
     5    16   16    5    4     1     4

Here I just swapped denominators, simplfied each new fraction, and 
then multiplied. If you are comfortable with simplifying fractions and 
find that this makes it easier for you to see where to cancel, use it! 
You will eventually find that you do not have to rewrite the fraction, 
but can just think of it this way and save some lead.

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