Question

Express each of the following decimals as a rational number:
(viii) $0.0\overline{17}$

Answer 1

Step: Convert decimal into rational number
$0.0\overline{17}$
Let $x=0.0\overline{17}$
Then,
$x=0.017171717\dots$
Here,
Only numbers $17$ is being repeated, so first, we need
to remove $0$ which comes before $17$.
∴ We multiply by $10$ so that only the recurring digits
remain after decimal
Thus,
$10x=10\times 0.0171717\dots$
$10x=0.171717\dots \left(1\right)$
The number of digits recurring in equation (1) is $2$
Hence, we multiply both sides of the equation (1) by
$100$
$1000x=100\times 0.171717=\mathrm{17.17171717..}\dots \left(2\right)$
On subtracting (1) from (2), we get,
$990x=17$
$x=\frac{17}{990}$
We get,
$0.0\overline{17}=\frac{17}{990}$