Express the following in the form $\frac{p}{q},$ where $p$ and $q$ are integers and $q \neq 0 .$
$0 . \overline{35}$
$0 . \overline{35}$
(A) Let $x = 0 . \overline{35}$.
$\therefore x = 0.353535 \ldots$ $(1)$
Since there are two repeating digits,multiply both sides by $100$:
$100x = 35.353535 \ldots$ $(2)$
Subtract equation $(1)$ from equation $(2)$:
$100x - x = 35.353535 \ldots - 0.353535 \ldots$
$99x = 35$
$\therefore x = \frac{35}{99}$
Thus,$0 . \overline{35} = \frac{35}{99}$.
$\therefore x = 0.353535 \ldots$ $(1)$
Since there are two repeating digits,multiply both sides by $100$:
$100x = 35.353535 \ldots$ $(2)$
Subtract equation $(1)$ from equation $(2)$:
$100x - x = 35.353535 \ldots - 0.353535 \ldots$
$99x = 35$
$\therefore x = \frac{35}{99}$
Thus,$0 . \overline{35} = \frac{35}{99}$.
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