Multiplying Mixed Numbers

Date: 06/02/2003 at 16:07:16
From: Natasha
Subject: Mixed numbers

Why can you not multiply mixed numbers without changing them into 
improper fractions? 

I understand that it is easier and I understand the process, but is 
there a theory?

Date: 06/04/2003 at 14:32:21
From: Doctor Douglas
Subject: Re: Mixed numbers

Hi Natasha,

Thanks for writing to the Math Forum.

You can indeed multiply mixed numbers without converting them to
improper fractions.  

  3 1/2 x 4 1/5 = (3 + 1/2) x (4 + 1/5) 
                = (3 x 4) + (1/2 x 4) + (3 x 1/5) + (1/2 x 1/5)
                = 12 + 2 + 3/5 + 1/10
                = 14 + 6/10 + 1/10
                = 14 7/10

Doing it this way (multiplying all the terms out using the 
distributive property of multiplication over addition) was probably 
more work than converting:

  3 1/2 x 4 1/5 = 7/2 x (21/5)
                = (7 x 21)/(2 x 5)
                = 147/10
                = 14 7/10

But there might be situations where doing the conversion is more work.  
For example, we could have the following situation: let there be two 
years where some monetary quantity increases by 2% the first year and 
decreases by 2% the following year. Then the total overall percentage 
change is

  (1 + 2%)(1 - 2%) = (1 + 2/100)(1 - 2/100)
                   = 102/100)(98/100)
                   = (102 x 98)/(100 x 100)
                   = 9996/10000
                   = 0.9996

or a 0.04 % drop overall. Notice how we had to execute the 
multiplication 102 x 98.

We could also have done the following:

  (1 + A)(1 - A) = 1x1 + A - A - AxA
                 = 1 - AxA
                 = 1 - (0.02)(0.02)
                 = 1 - 0.0004
                 = 0.9996

which is a lot easier in terms of the number crunching, especially 
if we don't have access to a calculator. So there may be times where 
the conversion of a mixed number to a fraction may not be the most 
efficient way to proceed.  

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

Date: 06/04/2003 at 15:34:36
From: Natasha
Subject: Mixed numbers

Thank you for the response. 

I tried to do the same with division (using the distributive property) 
but it doesn't seem to be working. Is it not possible to divide mixed 
numbers?

Date: 06/04/2003 at 18:58:16
From: Doctor Douglas
Subject: Re: Mixed numbers

Hi again, Natasha,

There is no distributive property of division over addition, so
the most you can do for a ratio of mixed numbers is

  A+B    A     B
  --- = --- + ---          here, B and D are fractions
  C+D   C+D   C+D          while A and C are whole numbers.

I confess that it is more difficult to contrive an example where one 
would divide the mixed numbers without first converting to an improper 
fraction. But here's one:

  12 1/2    24/2 + 1/2   25/2
  ------- = ---------- = ---- = 5
   2 1/2     4/2 + 1/2    5/2

  12 1/2   10 + 2 1/2
  ------ = ---------- = 4 + 1 = 5
   2 1/2     2 1/2

Here the second method is easy because the numerator splits 
conveniently into two pieces, each of which is relatively easy to 
divide (the second term, in fact, is trivial). But you do have to 
recognize that 10/(2+1/2) is an easy computation: 10/2.5 = 100/25 = 4.

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