Repeating Decimals and Fractions

Date: 11/27/2001 at 20:59:25
From: Lisa Paez
Subject: Changing repeating decimals into fractions

I have a chart of repeating decimals and their fraction equivalents. 
The decimal is .71 and they give a fraction of 5/7. I cannot figure 
out how (step by step) they came up with this number. I came up with 
71/100 and not think it would reduce any further than that. 

Another is .4545; the chart says that should be 5/11, and I can't 
explain that either. 

Thank you, 
Lisa Paez

Date: 11/28/2001 at 00:15:56
From: Doctor Tom
Subject: Re: Changing repeating decimals into fractions

Hi Lisa,

The chart is wrong. Here's how to do it, by means of examples:

  .71717171... (continuing the alternating 71 forever) = 71/99

  .123123123... = 123/999

  .71428571428571428575... = 714285/999999 = 5/7

Trickier are these types:

  .1112323232323... (just the 23 repeats) = 111/1000 + (1/1000)(23/99)

The easy way to see this is that:

  .11123232323... = .111 + .00023232323...

   = 111/1000 + (1/1000)(.232323...)

All others can be handled similarly.

Just for fun, why don't you try the reverse for the fraction 1/81?
Find the decimal expansion that repeats.

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