Question
Express the following decimal as a rational number.$0.89$
✓
Answer
Let $x=0 . \overline{89}$
Then, $x=0.898989 \ldots$
Here, the number of digits recurring is $2,$
so we multiply both sides of the equation $(1)$ by $100 .$
$\therefore 100 x=100 \times 0.898989 \ldots $
$=89.8989 \ldots \ldots . \ldots(2)$
On subtracting $(1)$ from $(2),$ we get
$99 x=89 $
$\therefore x=\frac{89}{99} $
$\therefore 0 . \overline{89}$
$=\frac{89}{99}$
Then, $x=0.898989 \ldots$
Here, the number of digits recurring is $2,$
so we multiply both sides of the equation $(1)$ by $100 .$
$\therefore 100 x=100 \times 0.898989 \ldots $
$=89.8989 \ldots \ldots . \ldots(2)$
On subtracting $(1)$ from $(2),$ we get
$99 x=89 $
$\therefore x=\frac{89}{99} $
$\therefore 0 . \overline{89}$
$=\frac{89}{99}$
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