Simplifying Fractions

Date: 12/08/2002 at 21:01:13
From: Lillian
Subject: Simplifying Fractions

How do I simplify a fraction?

Date: 12/08/2002 at 22:05:45
From: Doctor Ian
Subject: Re: Simplifying Fractions

Hi Lillian,

It depends on the fraction, but the general idea is this. Suppose you
have a fraction,

   bibbity
   -------
   bobbity

and it turns out that both the numerator and denominator share a
common factor, e.g., 

   bibbity   bibble * itty
   ------- = -------------
   bobbity   bobble * itty

then you can eliminate the common factor:

   bibbity   bibble * itty
   ------- = -------------
   bobbity   bobble * itty


             bibble   itty
           = ------ * ----
             bobble   itty

             bibble   
           = ------ * 1
             bobble   

             bibble   
           = ------ 
             bobble 

In practice, it might look like this:

   12   6 * 2   6
   -- = ----- = - 
   18   9 * 2   9

Of course, there can be more than one common factor!

   12   6 * 2   6   2 * 3   2
   -- = ----- = - = ----- = -
   18   9 * 2   9   3 * 3   3

So the easiest way to deal with numbers is to find all the prime
factors right away, and kill them off in pairs:

            /   /
   12   2 * 2 * 3       2
   -- = ------------- = - 
   18       2 * 3 * 3   3
            /   /

It might be too soon for you to be thinking about 'variables', but
when you get to algebra you'll want to use the same rule:

                   /   /           /   /
  21 x^3 y^2   3 * 7 * x * x * x * y * y
  ---------- = -------------------------------------
  35 x   y^5   5 * 7 * x         * y * y * y * y * y
                   /   /           /   /


               3 x^2
             = -----
                y^3

I'm just telling you this now because it's not uncommon for students
who have no trouble reducing fractions with numbers to freeze up when
they have to do the same thing to a fraction with variables, because
they don't realize that it _is_ the same thing. 

The important point is not to get confused or intimidated when the
numbers get big, or you have lots of variables, or whatever. The main
idea is _always_ the same: If you have 

   a bunch of stuff multiplied together
   ------------------------------------------
   another bunch of stuff multiplied together

then if you can identify common factors, you can kill them off (cancel 
them) in pairs. What is tricky is that sometimes it's hard to remember 
that something like 12 actually _is_ a bunch of stuff multiplied 
together.
                  
   12   a bunch of stuff multiplied together
   -- = ------------------------------------------
   18   another bunch of stuff multiplied together

        2 * 2 * 3
      = ---------
        2 * 3 * 3

And one of the secrets to happiness in math is to get into the habit
of looking at every integer you see this way. For example, as soon as
you _see_ a number like 72, you should already be thinking 

  72 = 8 * 9

     = (2 * 2 * 2) * (3 * 3)

It's kind of like what Superman does when he uses his x-ray vision to
look through people's clothes so he can see if they're hiding things. 

It takes a little work to get into this habit (learning the most
common divisibility rules,

   Divisibility Rules - Dr. Math FAQ
   http://mathforum.org/dr.math/faq/faq.divisibility.html 

can help a lot - especially the rules for 2, 3, 5, 9, and 10), but you
wouldn't believe how much work it can save you down the road. 

I hope this helps. Write back if you'd like to talk more about this,
or anything else. 

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