Converting Repeating Decimals to Fractions

Date: 01/20/98 at 21:00:14
From: Edz LAmy
Subject: Converting repeating decimals to fractions

a. I have to add the decimals. Change the sum of the  decimals to a  
   fraction for 0.77777.......  , 0.4
b. Change each decimal to a fraction. Add the fraction.
c. Compare the answers to parts (a) and (b)

Date: 01/27/98 at 15:45:49
From: Doctor Wolf
Subject: Re: Converting repeating decimals to fractions

Hi Edz,

  Let's take each step in turn.

a) Add the decimals 0.4 and 0.777777... Keeping in mind that 0.4
   is really 0.4000000..... just line up the decimal points and
   add as usual.  You should get 1.1777777...

   Now what is 1.17777.... as a fraction?  This is a good question!
   Here is a technique that will work for any repeating decimal.  

   Let N = 1.17777.....  Since N has one repeating number (7),
   we will multiply N by 10 to shift the decimal point one place
   to the right. So 10N = 11.77777.....  Now for a little elementary
   algebra.
                     If 10N = 11.7777.....
                     and  N =  1.1777.....
                   _________________________
   Subtracting           9N = 10.6000....    or just 10.6

   Finally, since 9N = 10.6, divide both sides by 9 and

                          N = (10.6)/9 = 106/90 = 53/45 (reduced).

   As a check on your work so far, divide 53 by 45.  You should get
   the number 1.177777....

   Up to this point we have added the original decimal numbers, 
   obtained a decimal result, 1.177777..., and have shown this to 
   equal 53/45.

b) Now we'll convert the original decimals to fractions, add them,
   and see if we arrive at the same result; that is, 53/45.

   Here goes ... 0.4 = 4/10, no problem here.  But what is 0.77777...?

   Using the same technique as before, let N = 0.777777....,
   then 10N = 7.77777....  Now to subtract them,

                     If 10N = 7.77777.....
                     and  N =  .77777....
                     ______________________
       subtracting       9N = 7.0000....    or just 7

       Since 9N = 7, divide both sides by 9, and N = 7/9. Okay so far!

   We're just about there. Adding 4/10 and 7/9 (common denominator 
   = 90)
              4/10 + 7/9 = 36/90 + 70/90 = 106/90 = 53/45

c) This is the same result as before, and you've worked your way 
   through a pretty tough but rewarding problem.

   Good luck in your class, and don't hesitate to drop in again.    

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