Converting Repeating Decimals to Fractions

Date: 12/17/2006 at 19:06:40
From: John
Subject: Repeating decimals to fractions

How do you change a repeating decimal, where the first few numbers are
not repeating, into a fraction?  For example, what does -2.41666666...
equal to as a fraction?  (in this example, the 6 is the only number
repeating)

I read another of your articles where someone asked how to turn a
repeating decimal (for example 0.4444444...) into a fraction, but it
didn't work in my case.

Date: 12/17/2006 at 23:14:54
From: Doctor Peterson
Subject: Re: Repeating decimals to fractions

Hi, John.

We don't have a lot of examples of this sort of problem; I found one here:

  Fractions for Repeating Decimals  [3.04050505...]
    http://mathforum.org/library/drmath/view/58106.html 

There's another in section IIB of our FAQ:

  Fractions, Decimals, Percentages  [1.3481481481481...]
    http://mathforum.org/dr.math/faq/faq.fractions.html 

Both of these just follow the same rule as for the simpler case:
multiply by 10^n, where n is the number of digits in the repetend, and
subtract.  Then you have to eliminate a decimal in the fraction you
get, by multiplying the numerator and denominator by a power of ten.

I'll show a slightly different method that I saw in a text the other
day, which avoids that last step by putting in an extra first step.
Let's try 1.2343434... .

First, we get the repeating part to start just after the decimal 
point, as in most examples you see; we want to move the decimal point
one place to the right, so we multiply by 10:

      x = 1.2343434...
    10x = 12.343434...

Now we want to move the decimal point two more places, so that the
repetend will match up.  So we multiply by 100 more:

  1000x = 1234.343434...

Now we subtract these:

  1000x = 1234.343434...
    10x =   12.343434...
  -----   --------------
   990x = 1222.000000...

Solving for x,

      x = 1222/990 = 611/495

I don't know that I like this method any more than the other, since
you have to pay more attention at the front rather than just being
reminded of the need to get rid of the extra decimals after you do the
routine first step. But it's there if you're interested.

There are two other things to point out.  First, if you want a mixed
number rather than an improper fraction, you can work ONLY with the
decimal part of your number, and then add on the whole part at the end:

      x = 1.2343434...

      y = x - 1 = 0.2343434...   [remove whole part]
    10y = 2.343434...
  1000y = 234.343434...

  1000y = 234.343434...
    10y =   2.343434...
  -----   -------------
   990y = 232

      y = 232/990 = 116/495

      x = 1 + y = 1 116/495      [add whole part back on]

Finally, your question involves a negative number.  You would just
convert the absolute value, 2.41666666..., to a fraction, and then
take the negative of that.

If you have any further questions, feel free to write back.


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