Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.
(i) $0.2$
(ii) $0.888 \ldots$
(iii) $5 . \overline{2}$
(iv) $0 . \overline{001}$
(v) $0.2555 \ldots$
(vi) $0.1 \overline{34}$
(vii) $0.00323232 \ldots \quad$
(viii) $0.404040 \ldots$
(i) Let $x=0.2=\frac{2}{10}=\frac{1}{5}$
(ii) Let $x=0.888 \ldots \ldots \ldots$ $\ldots$ (i)
On multiplying both sides of Eq.(i) by 10 , we get
$10 x=8.888$ ... (ii)
On subtracting Eq. (i) from Eq. (ii), we get
$10 x-x=(8.88)-(0.888)$
$\Rightarrow$ $9 x=8$
$\therefore$ $x=\frac{8}{9}$
(iii) Let $x=5 . \overline{2}=5.222 \ldots$ $\ldots$ (i)
On multiplying both sides of Eq. (i) by 10 , we get
$10 x=52222 \ldots \ldots \ldots$$\ldots$ (ii)
On subtracting Eq. (i) from Eq. (ii), we get
$10 x-x=(52.222 \ldots)-(5.222 \ldots)$
$\Rightarrow$ $9 x=47$
$\therefore$ $x=\frac{47}{9}$
(iv) Let $x=0 . \overline{001}$
$\Rightarrow$ $x=0 . \overline{001}=0.001001$ $\ldots$ (i)
On multiplying both sides of Eq. (i) by 1000 , we get
$1000 x=001.001 \ldots \ldots . .$ .......(ii)
On multiplying both sides of Eq. (ii) by 100 , we get
$1000 x=134.3434 \ldots \ldots \ldots$ ...(iii)
On subtracting Eq. (ii) from Eq. (iii), we get
$1000 x-10 x=134.34 \ldots-(1.3434 \ldots)$
$\Rightarrow$ $990 x=133$
$\therefore$ $x=\frac{133}{990}$
(vii) Let $x=0.00323232$ $\ldots($ i)
On multiplying both sides of Eq. (i) by 100 we get
$100 x=0.3232 \ldots \ldots \ldots$ ...(ii)
On multiplying both sides of Eq. (ii) by 100 , we get
$10000 x=32.3232 \ldots \ldots . .$ ...........(iii)
On subtracting Eq. (ii) from Eq. (iii), we get
$10000 x-100 x=32.32 \ldots \ldots-0.3232 \ldots$
$\Rightarrow$ $9900 x=32$
$\therefore$ $x=\frac{32}{9900}=\frac{8}{2475}$ [dividing numerator and denominator by 4 ]
(viii) Let $x=0.404040$ $\ldots$ (i)
On multiplying both sides of Eq. (i) by 100 , we get
$100 x=40.4040$ ...(ii)
On subtracting Eq. (i) from Eq. (ii), we get
$100 x-x=40.4040 \ldots \ldots-0.404040 \ldots .$
$\Rightarrow$ $99 x=40$
$\therefore$ $x=\frac{40}{99}$