Shortcut: Repeating Decimals to Equivalent Fractions

Date: 03/24/2001 at 20:05:01
From: Andrew Wan
Subject: Shortcut to changing repeating decimals into their equivalent 
fractions

My question is, how do you change a repeating decimal into its 
equivalent fraction? Actually, I know how to, but is there some kind 
of shortcut?

Date: 03/24/2001 at 23:05:59
From: Doctor Peterson
Subject: Re: Shortcut to changing repeating decimals into their 
equivalent fractions

Hi, Andrew.

You can read the general method for doing this in our archives:

  Fractions, Decimals, Percentages - Dr. Math FAQ
  http://mathforum.org/dr.math/faq/faq.fractions.html   

It sounds as if you know that, and are looking for something quicker. 
For simple cases, it can be done without going through all the work. 
If the repeat starts right after the decimal point, you can just 
divide the repeat by as many 9's as there are digits in the repeat:

      _   3
    0.3 = - = 1/3
          9

      __   05
    0.05 = -- = 5/99
           99

      ______   142857
    0.142857 = ------ = 1/7
               999999

Compare this with the normal method, and you should quickly see why 
this works:

    100x = 5.0505...
       x = 0.0505...
    ----   ---------
     99x = 5

       x = 5/99

When the repeat doesn't start just after the decimal point, you can 
move it there, do the conversion, and move it back:
        __            __
    0.1234 = 0.12 + 0.34 / 100
           = 12/100 + 34/99 / 100
           = 12/100 + 34/9900
           = 12*99/9900 + 34/9900
           = (1188+34)/9900
           = 1222/9900
           = 611/4950

As you can see, once you have to do this, you might as well use the 
usual method:

    100x = 12.343434...
       x =  0.123434...
    ----   ------------
     99x = 12.22

       x = 12.22/99 = 1222/9900 = 611/4950

Suddenly this doesn't seem so hard, does it?

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