Question
Express the following decimal as a rational number.$0.763$
✓
Answer
Let $x=0 . \overline{763}$
Then, $x=0.763763 \ldots (1)$
Here, the number of digits recurring is $3,$
so we multiply both sides of the equation $(1)$ by $1000 .$
$\therefore 1000 x =1000 \times 0.763763 \ldots $
$=763.763 \ldots . . \ldots(2)$
On subtracting $(1)$ from $(2),$ we get
$999 x=763 $
$\therefore x=\frac{763}{999} $
$\therefore 0 . \overline{763}=\frac{763}{999}$
Then, $x=0.763763 \ldots (1)$
Here, the number of digits recurring is $3,$
so we multiply both sides of the equation $(1)$ by $1000 .$
$\therefore 1000 x =1000 \times 0.763763 \ldots $
$=763.763 \ldots . . \ldots(2)$
On subtracting $(1)$ from $(2),$ we get
$999 x=763 $
$\therefore x=\frac{763}{999} $
$\therefore 0 . \overline{763}=\frac{763}{999}$
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