Question
Express the following decimal as a rational number.$2.67$
✓
Answer
Let $x=2 . \overline{67}$
Then, $x =2.676767 \ldots (1)$
Here, the number of digits recurring is $2$ ,
so we multiply both sides of the equation $(1)$ by $100 .$
$\therefore 100 x =100 \times 2.676767 \ldots $
$=267.8989 \ldots \ldots(2)$
On subtracting $(1)$ from $(2),$ we get
$99 x=265 $
$\therefore x=\frac{265}{99} $
$\therefore 2 . \overline{67}=\frac{265}{99}$
Then, $x =2.676767 \ldots (1)$
Here, the number of digits recurring is $2$ ,
so we multiply both sides of the equation $(1)$ by $100 .$
$\therefore 100 x =100 \times 2.676767 \ldots $
$=267.8989 \ldots \ldots(2)$
On subtracting $(1)$ from $(2),$ we get
$99 x=265 $
$\therefore x=\frac{265}{99} $
$\therefore 2 . \overline{67}=\frac{265}{99}$
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