Question
Write the following numbers in $\frac{p}{q}$ form. : $9.315315 \ldots .$.
✓
Answer
Let $x=9.315315 \ldots=9 . \overline{315} \ldots$ (i)
Since, three numbers i.e. $3,1$ and $5$ are repeating after the decimal point.
Thus, multiplying both sides by $1000 ,$
$1000 x =9315.315315 \ldots$
$\therefore 1000 x =9315 . \overline{315} \ldots \text { (ii) }$
$\text { Subtracting (i) from (ii), }$
$1000 x - x =9315 . \overline{315}-9 . \overline{315}$
$\therefore 999 x =9306$
$\therefore \quad x=\frac{9306}{999}=\frac{9 \times 1034}{9 \times 111}=\frac{1034}{111}$
$\therefore \quad 9.315315 \ldots=\frac{1034}{111} $
Since, three numbers i.e. $3,1$ and $5$ are repeating after the decimal point.
Thus, multiplying both sides by $1000 ,$
$1000 x =9315.315315 \ldots$
$\therefore 1000 x =9315 . \overline{315} \ldots \text { (ii) }$
$\text { Subtracting (i) from (ii), }$
$1000 x - x =9315 . \overline{315}-9 . \overline{315}$
$\therefore 999 x =9306$
$\therefore \quad x=\frac{9306}{999}=\frac{9 \times 1034}{9 \times 111}=\frac{1034}{111}$
$\therefore \quad 9.315315 \ldots=\frac{1034}{111} $
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