Question

Express each of the following decimals as a rational number:
(vii) $4.6\overline{724}$

Answer 1

Step: Convert decimal into rational number.
$4.6\overline{724}$
Let $x=4.6\overline{724}$
Then,
$x=4.6724724724\dots \left(1\right)$
Here,
Only numbers $724$ is being repeated, so first, we need to remove $6$ which comes before $724$.
We multiply by $10$ so that only the recurring digits remain after decimal.
Thus,
$10x=10\times 4.6724724\dots$
$10x=46.724724\dots \left(1\right)$
The number of digits recurring in equation (1) is $3$
Hence, we multiply both sides of the equation (1) by $1000$
$10000x=1000\times 46.724724=46724.724\dots \left(2\right)$
On subtracting (1) from (2), we get,
$9990x=46678$
$x=\frac{46678}{9990}=\frac{23339\times 2}{4995\times 2}$
We get,
$4.6\overline{724}=\frac{23339}{4995}$.