#### (i) Express the following in the form $\frac{p}{q}$, where p and q are integers and $q\neq 0$. 0.2 (ii) Express the following in the form $\frac{p}{q}$, where p and q are integers and $q\neq 0$.0.888……. (iii)  Express the following in the form $\frac{p}{q}$, where p and q are integers and $q\neq 0$.$5.\bar{2}$    (iv) Express the following in the form $\frac{p}{q}$, where p and q are integers and $q\neq 0$.$0\cdot \overline{001}$ (v) Express the following in the form $\frac{p}{q}$, where p and q are integers and $q\neq 0$.0.2555…… (vii) Express the following in the form $\frac{p}{q}$, where p and q are integers and $q\neq 0$..00323232….. (viii) Express the following in the form $\frac{p}{q}$, where p and q are integers and $q\neq 0$..404040……..

(i) Answer.     $\frac{1}{5}$
Solution.   We know that
0.2 can be written as $\frac{2}{10}$
Now,
$\frac{2}{10}= \frac{1}{5}$
Hence the answer is $\frac{1}{5}$

(ii) Answer.    $\frac{8}{9}$
Solution.    Let x = 0.888…..        .…(i)
Multiply RHS and LHS by 10
10 x = 8.88…….         …(ii)
Subtracting equation (i) from (ii)
We get
10x – x = 8.8 – 0.8
$\Rightarrow 9x= 8$
$\Rightarrow x= \frac{8}{9}$
Hence answer is  $\frac{8}{9}$

(iii) Answer.  $\frac{47}{9}$
Solution.  Let x = $5\cdot \bar{2}$          …eq. (1)
Multiply by 10 on both sides
10x = $52\cdot \bar{2}$                  …eq (2)

Subtracting equation (1) from (2)
We get
10x – x = $52\cdot \bar{2}$ – $5\cdot \bar{2}$
$\Rightarrow$ 9x = 47
$\Rightarrow$ x = $\frac{47}{9}$
Hence the answer is $\frac{47}{9}$

(iv) Answer.   $\frac{1}{999}$
Solution.    Let x = $0\cdot \overline{001}$              …. Eq. (1)
Multiply by 1000 on both sides
1000 x = $1\cdot \overline{001}$            …eq.(2)
Subtracting eq. (1) from (2)
We get
1000 x – x = $1\cdot \overline{001}$ – $1\cdot \overline{001}$

$\Rightarrow$ 999x = 1
$\Rightarrow$ x = $\frac{1}{999}$
Hence the answer is $\frac{1}{999}$

(v) Answer.   $\frac{23}{90}$

Solution. Let x = 0.2555 …..    …eq.(1)
Multiply by 10 on both sides
10x = 2.555…                         …eq.(2)
Multiply by 100 on both sides
100x = 25.55…                       …eq. (3)
Subtracting eq. (2) from (3),
We get
100x – 10x = 25.555… – 2.555…

$\Rightarrow$ 90x = 23
$\Rightarrow$ x = $\frac{23}{90}$
Hence the answer is $\frac{23}{90}$

(vii) Answer.    $\frac{8}{2475}$
Solution.   Let    x = 0.00323232…..      …eq.(1)
Multiply with 100 on both sides.
We get
100x = 0.3232…                …eq(2)
Multiply again with 100 on both sides,
10000x = 32.3232…                           …eq(3)
Equation (3) – (2)
we get,
10000 x – 100x = 32.3232… – 0.3232…
$\Rightarrow$  9900x = 32
$\Rightarrow$ x = $\frac{32}{9900}$
x = $\frac{8}{2475}$
Hence the answer is  $\frac{8}{2475}$

(viii) Answer.     $\frac{40}{99}$
Solution. Let x = 0.404040…….           …(1)
Multiplying by 100 on both sides
we get
100x = 40.40…                       …(2)
Subtracting equation (1) from (2)
we get
100x – x = 40.40 – 0.40
$\Rightarrow$99x = 40
$\Rightarrow$  x = $\frac{40}{99}$
Hence the answer is $\frac{40}{99}$