Question

Express each of the following decimals as a rational number:
(iv) $0.\overline{057}$

Answer 1

Step: Convert decimal into rational number.
$0.\overline{057}$
Let$x=0.\overline{057}$
Then,
$x=0.057057057\dots \left(1\right)$
Here,
The number of digits recurring is $3$,
So, we multiply both sides of equation (1) by $1000$
We get,
$1000x=1000\times 0.057057057\dots$
$=57.057057\dots \left(2\right)$
On subtracting (1) from (2),
We get,
$999x=57$
$x=\frac{57}{999}=\frac{19\times 3}{333\times 3}$
Therefore,
$0.\overline{057}=\frac{19}{333}$.