What are Compound Fractions?

Date: 11/23/2002 at 12:31:03
From: Eleanor
Subject: Compound fractions

When I look at explanations on your site, I often get confused by 
compound fractions. What are they, and how do they work?

I mean something like

      3 + 3 + 4
  1 + ---------
        976
 ---------------
       21

Date: 11/25/2002 at 22:55:00
From: Doctor Ian
Subject: Re: Compound fractions

Hi Eleanor,

This is a fraction, just like any other fraction, but it's easier to 
think about if you let go of the idea that fractions are somehow about 
dividing things up into pieces, and just accept the truth: a fraction 
is just a division that you haven't done yet.  

So 3/4 is just 3 divided by 4.  And 

   3/4
  -----
    5

is just 3 divided by 4, with the result divided by 5.  It's no
different from the sequence of operations

  (3 / 4 ) / 5

So what about your fraction? 

      3 + 3 + 4
  1 + ---------
        976
 ---------------
       21

This is just the same as the sequence of operations 

  (1 + ((3 + 3 + 4) / 976)) / 21

Remember having to do problems like this when you were learning about
PEMDAS? Well, this is the same thing again, with a slightly different
look. 

Now note that you can simplify these while still keeping them looking
like fractions. To do that, you have to be able to convert mixed
numbers into fractions, and you have to remember that to divide by a
fraction, you multiply by the inverse. So let's look at an example:

             4
         3 + -
             5
   2 + -----------
             3
         2 + -
             8
  ------------------
            5
        1 + -
            6


Each '--------' is like a set of parentheses, so we want to work with
the innermost (smallest) ones first, and work our way out to the
outermost (largest) ones:

             19
             --                 Change to improper fractions
              5
     2 + -----------
             19 
             --
              8
  = ------------------
             11
             --
              6

           19    8
       2 + -- * --              To divide, multiply by the inverse
            5   19
  = ------------------
             11
             --
              6


              8   
          2 + -                Again with the improper fractions
              5    
  = ------------------
             11
             --
              6

             18   
             --                Again, with the division
              5    
  = ------------------
             11
             --
              6

    18    6          
  = -- * --      
     5   11


  = whatever

Does this make sense?  Do they seem a little less mysterious now? 

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

Date: 11/27/2002 at 20:14:56
From: Eleanor
Subject: Thank you (Compound fractions)

Yes, that definitely helps. Thanks a lot.