Dividing Fractions

Date: 03/16/99 at 12:22:32
From: Alex
Subject: Division of Fractions

I know how to divide fractions. I want to know WHY you cannot divide 
them. Can you please help me?

Date: 03/16/99 at 21:12:40
From: Doctor Peterson
Subject: Re: Division of Fractions

I am not quite sure what you mean. Of course you can divide fractions, 
but I suspect you mean "why don't you divide fractions directly, but 
instead have to do it by multiplying?"

There are several ways to divide fractions, some more direct than 
others. The method of multiplying the reciprocal is usually the 
easiest.

Suppose we want to divide 4/9 by 2/3. We COULD do it just by dividing 
the numerators and denominators:

      4     2    4 / 2    2
     --- / --- = ----- = ---
      9     3    9 / 3    3

You can see that it works. (I am using "/" where you would normally 
use the "bar and two dots" division symbol.)

Why do we usually not do this? Because I chose that problem carefully 
so the divisions would work out. More typically you would have 
trouble:

      3     5    3 / 5   3/5
     --- / --- = ----- = ---
      4     6    4 / 6   2/3

You have not gained anything, since you still have one fraction 
divided by another. We have to fix this up; we can do this by 
multiplying numerator and denominator by the least common denominator 
(LCD) of the two fractions:

      3     5    3/5   3/5 * 15    9
     --- / --- = --- = -------- = --
      4     6    2/3   2/3 * 15   10

Or, we could first convert both fractions to use the same denominator:

      3     5     9   10    9 / 10    9
     --- / --- = -- / -- = ------- = --
      4     6    12   12   12 / 12   10

Since I made the denominators the same, the "denominator" of the 
quotient works out to 1.

But there is really no division in that method, is there? It is still 
multiplication. A variation on this method is:

      3       9
     ---     --
      4      12     9
    ----- = ---- = --
      5      10    10
     ---     --
      6      12

You just multiply numerator and denominator by 12 in the last step to 
eliminate the fractions.

Do you see what I did there? Let us write it out more carefully and 
you will see where "multiplying by the reciprocal" is hidden:

      3      3*6
     ---     ---
      4      4*6    3*6   3*3    9
    ----- = ----- = --- = --- = --
      5      5*4    5*4   5*2   10
     ---     ---
      6      6*4

Here I have ignored LEAST common denominators and just used the 
simplest common denominator, the product of the denominators 4 and 6. 
The product is simply the product of 3/4 and 6/5.

To answer your question simply, we divide fractions by multiplying, 
not because we cannot divide, but because multiplication is easier 
than division, and because division by itself does not always produce 
whole numbers. All these methods just make you multiply the numerator 
of the dividend by the denominator of the divisor, and vice versa, 
with some division to simplify the fraction; the difference is only in 
how much you have to think about what you are doing.

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