Question
Write the following numbers in $\frac{p}{q}$ form. : $357.417417 \ldots .$.
✓
Answer
Let $x=357.417417 \ldots=357 . \overline{417} \ldots$ (i)
Since, three numbers i.e. $4,1$ and $7$ are repeating after the decimal point.
Thus, multiplying both sides by $1000 ,$
$ 1000 x=357417.417417 \ldots$
$\therefore 1000 x=357417.417 \ldots \text { (ii) }$
$\text { Subtracting (i) from (ii), }$
$1000 x-x=357417 . \overline{417}-357 . \overline{417}$
$\therefore 999 x=357060$
$\therefore \quad x=\frac{357060}{999}=\frac{3 \times 119020}{3 \times 333}$
$\therefore \quad 3 5 7 . 4 1 7 4 1 7 \ldots =\frac{119020}{333} $
Since, three numbers i.e. $4,1$ and $7$ are repeating after the decimal point.
Thus, multiplying both sides by $1000 ,$
$ 1000 x=357417.417417 \ldots$
$\therefore 1000 x=357417.417 \ldots \text { (ii) }$
$\text { Subtracting (i) from (ii), }$
$1000 x-x=357417 . \overline{417}-357 . \overline{417}$
$\therefore 999 x=357060$
$\therefore \quad x=\frac{357060}{999}=\frac{3 \times 119020}{3 \times 333}$
$\therefore \quad 3 5 7 . 4 1 7 4 1 7 \ldots =\frac{119020}{333} $
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