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### Repeating Decimals and Fractions

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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/
```
Associated Topics:
Elementary Fractions
Middle School Fractions

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