Express the following decimal as a rational number.4.6724 - Vidyadip

Question

Express the following decimal as a rational number.$4.6724$

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Answer

Let $x=4 . \overline{6724}$
$=4.6724724 \ldots$
Here, only numbers $724$ is being repeated,
so first we need to remove $6$ which proceeds $724 .$
We multiply by $10$ so that only the recurring digits remain after decimal.
$\therefore 10 x=46.724724 \ldots (1)$
The number of digits recurring in equation $(1)$ is $3$ ,
so we multiply both sides of the equation $(1)$ by $1000 .$
$\therefore 10000 x =1000 \times 46.724724 \ldots$
$=46724.724 \ldots \ldots(2)$
On subtracting $(1)$ from $(2)$, we get
$9990 x=4678$
$\therefore x=\frac{46678}{9990}$
$=\frac{23339}{4995}$
$\therefore 4 . \overline{6724}=\frac{763}{999}$
$=\frac{23339}{4995}$

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