Finding Common Denominators

Date: 06/25/2001 at 11:21:01
From: Mike Aldridge
Subject: Common denominators

Dear Dr. Math, 

Can you please tell me the best way to find a common denominator? I am 
having a very hard time understanding this.

Date: 06/25/2001 at 14:04:30
From: Doctor Ian
Subject: Re: Common denominators

Hi Mike,

When you want to add two fractions with different denominators, the 
trick is to change each fraction into something with the same meaning, 
but a different appearance. The way you do that is by finding a 
creative way to multiply each fraction by some form of 1. 

Suppose you wanted to add 10 francs to 14 marks. One way to do that 
would be to convert them to a common currency... say, the euro. The 
conversion factor for each currency would be derived this way:

  1 franc = f euros        1 mark = m euros

            f euros            m euros
        1 = -------        1 = -------
            1 franc            1 mark

As an American, I have no idea what the values of f and m would be, 
but let's assume that we could look them up.

Your addition would look like this:

                                     f euros              m euros
  10 francs + 14 marks = 10 francs * ------- + 14 marks * -------
                                     1 franc              1 mark

                       = 10f euros + 14m euros

                       = (10f + 14m) euros

This gives you the same result, because multiplying by 1 can't change 
anything.  

With a fraction, the easiest 'conversion factor' to use is the 
denominator of the other fraction. So in a case like

  3   1
  - + -
  8   4

you would multiply the first fraction by 4/4, and the second fraction 
by 8/8, to get:

  3   4     1   8     3 * 4   1 * 8     12 +  8     20
  - * -  +  - * -  =  ----- + -----  =  -- + --  =  --
  8   4     4   8     8 * 4   4 * 8     32   32     32

Now, note that this fraction can be reduced to 10/16, and then to 5/8.  
Why is that? It's because the original denominators, 4 and 8, share 
some common factors.  In this particular case, we _could_ have noticed 
that 4 goes into 8 twice, and simply converted everything into 8ths:

  3     1   2     3    1 * 2     3 + 2     5
  -  +  - * -  =  -  + -----  =  -----  =  --
  8     4   2     8    4 * 2       8       8
  
But it's a trade-off; if you do more work up front to find the 
_lowest_ common denominator, then you won't have to reduce the 
fraction after the addition. On the other hand, you can save time up 
front by finding the _obvious_ common denominator, and then deal with 
the subsequent reduction. 

My own feeling is that it's easier to reduce a fraction than to find 
the lowest common denominator, but that's a personal preference. 

Does this help?  Write back if you'd like to talk about this some 
more, or if you have any other questions. 

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