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### Deriving Fractions from Decimals

```
Date: 07/19/2001 at 00:03:33
From: Amanda
Subject: Percents and Fractions

Dear Dr. Math,

I am trying to re-learn math. I am currently reviewing percents and
fractions. Here is my question: how do I get the fraction in problems
like:

12.5% = 0.125--????????? = 1/8

66.7% = 0.667--?????????? = 2/3

I don't understand how they come up with this.

Your help would be greatly appreciated. Thanks!
Amanda
```

```
Date: 07/19/2001 at 16:18:38
From: Doctor Ian
Subject: Re: Percents and Fractions

Hi Amanda,

It's easy to forget that a decimal is just a shorthand way of writing
a fraction. For example,

0.125 = 125 / 1000

The two expressions mean exactly the same thing.  Now, if you start
with the fraction 125/1000 and try to simplify it,

125               5 * 5 * 5
---- = ---------------------
1000   2 * 2 * 2 * 5 * 5 * 5

the 5's in the numerator and denominator cancel out, leaving you with
125/1000 = 1/8.

In the same way, you can convert something like

0.5625 = 5625/10000

3 * 3 * 5 * 5 * 5 * 5
= -----------------------------
2 * 2 * 2 * 2 * 5 * 5 * 5 * 5

3 * 3
= -------------
2 * 2 * 2 * 2

= 9/16

In the case of a repeating decimal, you use (10...0 - 1) in the
denominator, rather than 10...0.  Here's why:

x = 0.666...

10x = 6.666...

10x - x = 6

9x = 6

x = 6/9

So the terminating decimal 0.667 is the same as 667/1000, but the
repeating decimal 0.666... is the same as 6/9, or 66/99, or 666/999.
And all of those are equal to 2/3.

Here's another example:

x = 0.521521...

1000x = 521.521521...

1000x - x = 521

999x = 521

x = 521/999

Note that

66.7% = 0.667 = 2/3                     (False!)

is NOT a true equation!  It's only true if the percentage is a
repeating decimal, i.e.,

66.6...% = 0.666... = 2/3               (True!)

Having said all of this, a handful of fractions appear so often that
it will simplify your life considerably to memorize their decimal
equivalents:

1/2 = 0.5
1/3 = 0.333...
1/4 = 0.25
1/5 = 0.2
1/8 = 0.125

One you know these fractions, you can compute others (like 3/8) by
multiplying by the appropriate numerator, e.g.,

3/8 = 3 * (1/8)

= 3 * (0.125)

= 0.375

more, or if you have any other questions.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Elementary Fractions
Middle School Fractions

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