How can a repeating decimal be written as a fraction?

Apr 20, 2015

I will explain it using an example.

Assume you want to write a decimal number 0.(2) as a proper fraction
You can do it so.

1) write an equation $x = 0. \left(2\right)$
2) multiply it by 10 $10 x = 2. \left(2\right)$
3) right-hand side can be written as $2 + x$ so we have $10 x = 2 + x$
So we have $9 x = 2$ and finally $x = \frac{2}{9}$

This algorithm can be used for any fractions which have no digits between a decimal point and the period (repeating digits).

If the period has more digits in point 2 you have to multiply by ${10}^{n}$ where $n$ is the number of digits in period.

The less formal but easier way to write such numbers as fractions could be :
You write $x = \text{Period"/"9...9}$ The number of 9s in the denominator is equal to the length of period.
Finally you must check if the nominator and denominator have any common divisors. If yes you have to divide both parts of the fraction by the divisors. Ex.

$x = 0 , \left(24\right)$
$100 x = 24. \left(24\right)$
$100 x = 24 + x$
$99 x = 24$
$x = \frac{24}{99}$
Both 24 and 99 can be divided by 3 so the answer is $x = \frac{8}{33}$