Dividing Fractions

Date: 09/08/99 at 22:29:38
From: Olivia Baker
Subject: Dividing Fractions

How do you divide fractions? I love math and I am good at it. This is 
the only thing slowing me down. I've tried and cannot find a way.

Thanks.

Date: 09/09/99 at 09:29:02
From: Doctor Rick
Subject: Re: Dividing Fractions

Hi, Olivia.

Let's think about a few simple examples. Once you understand them, you 
will be able to put them together to make harder problems.

What is 2 divided by 1/5? Here are 2 blocks:

     +---------+---------+
     |         |         |
     |         |         |
     |         |         |
     |         |         |
     |         |         |
     |         |         |
     |         |         |
     |         |         |
     |         |         |
     +---------+---------+

Here is 1/5 of a block (the shaded part):

     +---------+
     |XXXXXXXXX|
     +.........+
     |         |
     +.........+
     |         |
     +.........+
     |         |
     +.........+
     |         |
     +---------+

To divide 2 by 1/5, we divide 2 blocks into pieces the size of the 1/5 
block.

     +---------+---------+
     |         |         |
     +.........+.........+
     |         |         |
     +.........+.........+
     |         |         |
     +.........+.........+
     |         |         |
     +.........+.........+
     |         |         |
     +---------+---------+

There are 10 pieces this size in the 2 blocks. In other words,

     2 divided by 1/5 = 10

That's the same as

     2 x 5 = 10

Do you see why? To make 1/5 size blocks, we had to make each whole 
block into 5 pieces. In other words, we multiplied the number of 
pieces by 5.

Next problem: 

    2 divided by 2/5 = ?

Here is 2/5 of a block (the shaded part):

     +---------+
     |XXXXXXXXX|
     +.........+
     |XXXXXXXXX|
     +.........+
     |         |
     +.........+
     |         |
     +.........+
     |         |
     +---------+

Let's divide the 2 blocks into pieces this size.

     +---------+---------+
     |         |         |
     +         +         +
     |         |         |
     +.........+.........+
     |         |         |
     +         +         +
     |         |         |
     +.........+.........+
     |                   |
     +---------+---------+

The bottom block is a different shape, but it's the same size, because 
it's made of 2 1/5-size pieces. Since the pieces we want are twice as 
big, there are only 1/2 as many of them. The answer is

     2 divided by 2/5 = 10/2 = 5

This is the same as

     2 x 5/2 = 5

Do you see how the two equations are related? You take the fraction, 
2/5, and "turn it upside down" (in math words, we say "invert the 
fraction," or "take its reciprocal"). Then replace the division sign 
by a multiplication sign.

Now we can tackle harder problems. The same method always works, even 
if the answer isn't a whole number as in the simple examples. For 
instance,

     5 divided by 3/8 = ?

Invert the fraction 3/8; you get 8/3. Then multiply by this fraction:

     5 x 8/3 = 40/3 = 13 1/3

We can even replace the whole number in the problem with a fraction, 
like this:

     5/7 divided by 3/8 = ?

     5/7 x 8/3 = ?

      5     8     40       19
     --- x --- = ---- = 1 ----
      7     3     21       21

If you are comfortable with multiplying fractions, then you will have 
no trouble dividing fractions. Just remember: invert the fraction you 
are dividing by (the divisor), and multiply instead of dividing.

If you still have trouble, write back and tell me just what you don't 
understand; if you can do some problems but you have trouble with 
others, give some examples of the ones you have trouble with. I want 
to give you just the help you need, so the more you can tell me, the 
better.

You can also read the answers to other students' questions about 
dividing fractions. Go to our Search Dr. Math page:

   http://mathforum.org/mathgrepform.html   

Type in "divide fractions" (no quotes), select "that exact phrase" 
and click on "Search." You will see half a dozen or more answers.

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