Converting a Repeating Decimal to a Fraction

Date: 9/10/96 at 3:28:58
From: Anonymous
Subject: Converting Repeating Decimal to Fraction

A simple way to convert a recurring decimal to a vulgar fraction is as 
follows: 

Example: 0.942424242...

Write down the whole fraction, 942. Subtract those digits that don't 
recur:s 942 - 9 = 933. Then divide by the following number. For 
every recurring digit write down a 9, and for every non-recurring 
digit write down a nought after your 9's:  933/990. Why does this 
work?

Date: 9/10/96 at 14:34:54
From: Doctor Ana
Subject: Re: Converting Repeating Decimal to Fraction

This actually works for a very good reason, but rather than show you 
why this method works, I'll show you another method for converting 
repeating decimals to fractions and demonstrate why that works, and 
then you can try to work on your problem using similar ideas.  

So here's another way to convert to a fraction. Let's take the decimal

0.759595959...

We can break this up into 2 parts as follows:

0.7 + 0.05959595959... 

and treat each part separately.

       7
.7 =   --
       10
                    1
and .0595959... =  -- x .595959...
                   10

So now we have

              7      1
0.75959... = --  +  -- x .5959...
             10     10

Now we can just worry about the repeating part.

            59       59          59
.5959... =  --  +   ----   +  -------       ...
           100     10,000     1,000,000

This all is actually equal to the following:

            59
.5959...=   --
            99

because:

59     59      59
--  =  --   +  --
99     100    9900

(try it)

Then the 59/9900 term can be split up again to  59/10000 + 59/990000, 
and so on and so on. 

So, let's put all of the pieces together.

              7       1     59
.7595959...=  -  +   --  x  --
              10     10     99

              7       59
           =  -   +   --
              10      990

If you get a common denominator and combine these you should get the 
same answer (after simplifying) that you get with your method. 

Now that we have broken the above method into steps and have a sense 
for why it works, why don't you try to do the same for your method.  
What happens when you break it into steps?  If you need more help, 
please feel free to write back to us.  Good luck!

-Doctor Ana,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/