Dividing Fractions: Why Invert?

Date: 01/24/2001 at 10:15:44
From: MOHD. ZULASYRAF B. MOHD. ZAKI
Subject: Dividing fractions

Why do we have to invert the fraction in dividing fractions?

Date: 01/24/2001 at 11:41:09
From: Doctor Greenie
Subject: Re: Dividing fractions

Hi, Mohd -

Let's first think about the division operation.  When I see the 
division problem

    24
   ---- = ?
     4

I can think of it as asking "how many 4's are there in 24?"

So if I have the division problem

     6
  -------
     1
    ---
     2

I can think of it as asking "how many halves are there in 6?"  The way 
I answer that question is to say that it takes 2 halves to make a 
whole, so in 6 wholes there must be 6*2 = 12 halves. In this case I 
have multiplied 6 by the reciprocal of (1/2), which is (2/1) = 2.

The preceding example was easy to understand, because 1/2 is a very 
simple fraction. When the fractions are more complicated, like

     3
    ---
     4
  -------
     2
    ---
     7

it is very hard to see what is going on. So let's look at the 
mathematical reason for inverting when dividing fractions.

In any fraction

     a
    ---
     b

where a and b are both non-zero, it is true that

     a     b
    --- * --- = 1
     b     a

So if I need to perform the division problem

     3
    ---
     4
  -------
     2
    ---
     7

I simplify the expression by multiplying both numerator and 
denominator by 7/2, so that the denominator becomes 1:

     3         7          3     7         3     7
    ---       ---        --- * ---       --- * ---
     4         2          4     2         4     2
  ------- * -------  =  -----------  = 
     2         7             1
    ---       ---
     7         2

So you can see that the original problem, 3/4 divided by 2/7, is the 
same as the multiplication problem 3/4 times 7/2.

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