Writing a Percent as a Fraction and Reducing the Result

Date: 05/03/2004 at 22:38:52
From: Mike
Subject: algebra

I need help.  I just can't seem to figure out how to convert a percent 
into a fraction, such as 20% = 20/100 = 1/5.
 
All I know is that you put it into a fraction like 20/100 but after 
that I get stuck and I just don't know how to get the answer.  I 
thought you did 20 divided by 100 but it does not work.

Date: 05/04/2004 at 17:54:54
From: Doctor Ian
Subject: Re: algebra

Hi Mike,

You're right--you start by putting the percentage over 100, because 
that's what we _mean_ when we write something like 20%--it's just
another name for 20/100. 

So 

         20
  20% = ---
        100

Now what?  Well, the easiest thing to do is break the numerator and
denominator into prime factors:

         20   2 * 2 * 5
  20% = --- = -------------
        100   2 * 2 * 5 * 5

Since we're multiplying by 2 and dividing by 2, we can cancel a pair
of 2's:

              x
         20   2 * 2 * 5
  20% = --- = -------------
        100   2 * 2 * 5 * 5
              x

We can do the same thing for the other pair of 2's, and a pair of 5's:

              x   x   x
         20   2 * 2 * 5
  20% = --- = -------------
        100   2 * 2 * 5 * 5
              x   x   x

Since we're left with nothing in the numerator, we can include a 
factor of 1 up there, as a placeholder:

              x   x   x
         20   2 * 2 * 5 * 1
  20% = --- = -------------
        100   2 * 2 * 5 * 5
              x   x   x

              1
            = - 
              5

Now, we could have approached it slightly differently.  We could have
looked at 

   20
  --- = ?
  100

and asked ourselves:  "What divides both 20 and 100 evenly?"  If we
realize that 5 does, we could divide both the numerator and
denominator by 5 to get

   20    4   5
  --- = -- * - 
  100   20   5

         4
      = --
        20

Now we could do the same thing again:  "What divides both 4 and 20
evenly?"  They're both evenly divided by 2, so 

   4    2   2
  -- = -- * -
  20   10   2

        2
     = --
       10

And again:

   2   1   2
  -- = - * -
  10   5   2

       1
     = -
       5

If you use prime factors instead of going step by step like in the
second method, you're guaranteed never to miss any common factors.  So
sometimes it seems like more work to do the prime factor approach, but
there's something to be said for an approach that you _know_ will
always work, isn't there? 

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