Deriving Fractions from Decimals

Date: 07/19/2001 at 00:03:33
From: Amanda
Subject: Percents and Fractions

Dear Dr. Math, 

I am trying to re-learn math. I am currently reviewing percents and 
fractions. Here is my question: how do I get the fraction in problems 
like:

12.5% = 0.125--????????? = 1/8

66.7% = 0.667--?????????? = 2/3

I don't understand how they come up with this.

Your help would be greatly appreciated. Thanks!
Amanda

Date: 07/19/2001 at 16:18:38
From: Doctor Ian
Subject: Re: Percents and Fractions

Hi Amanda,

It's easy to forget that a decimal is just a shorthand way of writing 
a fraction. For example,

  0.125 = 125 / 1000

The two expressions mean exactly the same thing.  Now, if you start 
with the fraction 125/1000 and try to simplify it, 

   125               5 * 5 * 5
  ---- = ---------------------
  1000   2 * 2 * 2 * 5 * 5 * 5

the 5's in the numerator and denominator cancel out, leaving you with
125/1000 = 1/8.

In the same way, you can convert something like 

  0.5625 = 5625/10000 


                   3 * 3 * 5 * 5 * 5 * 5
         = -----------------------------
           2 * 2 * 2 * 2 * 5 * 5 * 5 * 5

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

         = 9/16

In the case of a repeating decimal, you use (10...0 - 1) in the 
denominator, rather than 10...0.  Here's why:

       x = 0.666...  

     10x = 6.666...

 10x - x = 6

      9x = 6

       x = 6/9

So the terminating decimal 0.667 is the same as 667/1000, but the 
repeating decimal 0.666... is the same as 6/9, or 66/99, or 666/999.  
And all of those are equal to 2/3. 

Here's another example:
  
         x = 0.521521...

     1000x = 521.521521...

 1000x - x = 521

      999x = 521

       x = 521/999
  
Note that 

  66.7% = 0.667 = 2/3                     (False!)

is NOT a true equation!  It's only true if the percentage is a 
repeating decimal, i.e., 

  66.6...% = 0.666... = 2/3               (True!)

Having said all of this, a handful of fractions appear so often that 
it will simplify your life considerably to memorize their decimal 
equivalents:


  1/2 = 0.5
  1/3 = 0.333...  
  1/4 = 0.25      
  1/5 = 0.2       
  1/8 = 0.125     

One you know these fractions, you can compute others (like 3/8) by
multiplying by the appropriate numerator, e.g., 

   3/8 = 3 * (1/8)

       = 3 * (0.125)

       = 0.375

I hope this helps.  Let me know 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/