Question
Write the following rational numbers in $\frac{p}{q}$ form. : $2 . \overline{514}$
✓
Answer
$\text { Let } x =2 . \overline{514}$
$ x = 2.514514…$
Since, three numbers i.e. $5, 1$ and $4$ are repeating after the decimal point.
Thus, multiplying both sides by $1000,$
$1000x = 2514.514514…$
$1000 x =2514 . \overline{514} \text {...(ii) }$
$\text { Subtracting (i) from (ii), }$
$1000 x - x =2514 . \overline{514}-2 . \overline{514}$
$\therefore 999 x =2512$
$\therefore \quad x=\frac{2512}{999}$
$\therefore 2 . \overline{514}=\frac{2512}{999}$
$ x = 2.514514…$
Since, three numbers i.e. $5, 1$ and $4$ are repeating after the decimal point.
Thus, multiplying both sides by $1000,$
$1000x = 2514.514514…$
$1000 x =2514 . \overline{514} \text {...(ii) }$
$\text { Subtracting (i) from (ii), }$
$1000 x - x =2514 . \overline{514}-2 . \overline{514}$
$\therefore 999 x =2512$
$\therefore \quad x=\frac{2512}{999}$
$\therefore 2 . \overline{514}=\frac{2512}{999}$
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