Multiplying Mixed Numbers

Date: 04/06/2004 at 12:55:59
From: Jane
Subject: Multiplying mixed numbers

Why do you need to convert mixed numbers to improper fractions before 
multiplying them together?

I teach 5th grade mathematics, and the children asked me about this. 
We understand that you have to do it, but what is the mathmatical
theory behind this operation?  I have looked in numerous math 
resources and cannot find an explanation.

Date: 04/06/2004 at 13:50:53
From: Doctor Peterson
Subject: Re: Multiplying mixed numbers

Hi, Jane.

This is an excellent question!

Like many things in math, converting to improper fractions isn't 
really NECESSARY, just CONVENIENT (though you might not think so until 
you finish reading this!).

You can multiply mixed numbers directly if you want, just the same way 
you multiply whole numbers.  The key is the distributive property: to 
multiply a sum, you can multiply each part of the sum and then add.  
With whole numbers, this looks like

  25 * 32 = (20 + 5)*32
          = 20*32 + 5*32
          = 20*(30 + 2) + 5*(30 + 2)
          = 20*30 + 20*2 + 5*30 + 5*2
          = 600 + 40 + 150 + 10
          = 800

This is really what we are doing when we write

    32
  * 25
  ----
   160 <-  5*32
   64  <- 20*32
  ----
   800

With mixed numbers, you can do the same thing if you want:

  (2 1/2)*(1 1/5) = (2 + 1/2)*(1 + 1/5)
                  = 2*(1 + 1/5) + 1/2*(1 + 1/5)
                  = 2*1 + 2*1/5 + 1/2*1 + 1/2*1/5
                  = 2 + 2/5 + 1/2 + 1/10
                  = 2 + 4/10 + 5/10 + 1/10
                  = 2 + 10/10
                  = 3

You could write this something like this:

     2 1/2
  *  1 1/5
  --------
      1/10 = 1/10
      2/5  = 4/10
    2 1/2  = 5/10
  ---------------
    2       10/10 = 3

Or, you can convert to improper fractions:

  5/2 * 6/5 = (5*6)/(2*5) = 6/2 = 3

Now why is this so much easier?  It's because when you mix 
multiplication and addition, you have to use the distributive 
property, which gives you a lot of parts to add up.  When you mix 
multiplication with pure fractions (no addition), everything works out 
neatly because fractions are the same thing as division, and 
multiplication and division are just different sides of the same 
operation.  So mixed numbers, which are sums, work well when you want 
to add or subtract; while pure fractions, which are divisions, work 
well when you want to multiply or divide.  We convert to whatever form 
works best for what we want to do.

If you have any further questions, feel free to write back.

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