# How can we write 3.11.... (repeated) as fraction?

May 12, 2017

$3.1$ repeated$= \frac{28}{9} = 3 \frac{1}{9}$

#### Explanation:

$3.1$ repeated can be written as $3.11111111111 \ldots \ldots \ldots \ldots$

and when a single digit, say $k$ (where $k$ is a natural number from $1$ and $9$) is repeated after decimal point, the result is $\frac{k}{9}$ plus the number before decimal.

Hence, $3.1$ repeated is $3 \frac{1}{9}$ i.e. $\frac{28}{9}$
There is another way too.

Let $x = 3.1111111 \ldots \ldots .$ and then

$10 x = 31.111111 \ldots \ldots \ldots .$

Subtracting former from latter we get

$9 x = 31 - 3 = 28$

i.e. $x = \frac{28}{9} = 3 \frac{1}{9}$