Express each of the following as a fraction in simplest form:

(i) Let $x=0 . \overline{8}$

∴ x = 0.888                                             ...(1)
10x = 8.888                                            ...(2)
On subtracting equation (1) from (2), we get

$9 x=8 \Rightarrow x=\frac{8}{9}$

$\therefore 0 . \overline{8}=\frac{8}{9}$

(ii) Let $x=2 . \overline{4}$

∴ x = 2.444                                         ...(1)
10x = 24.444                                      ...(2)
On subtracting equation (1) from (2), we get

$9 x=22 \Rightarrow x=\frac{22}{9}$

$\therefore 2 \cdot \overline{4}=\frac{22}{9}$

(iii) Let $x=0 . \overline{24}$

∴​ x = 0.2424                                   ...(1)
100x = 24.2424                              ...(2)
 On subtracting equation (1) from (2), we get

$99 x=24 \Rightarrow x=\frac{8}{33}$

$\therefore 0 . \overline{24}=\frac{8}{33}$

(iv) Let $x=0.1 \overline{2}$

$10 x=1.22222 \ldots \quad \ldots(1)$

$100 x=12.22222 \ldots \quad \ldots(2)$

On subtracting equation (1) from (2), we get

$100 x-10 x=(12.22222 \ldots)-(1.22222 \ldots)$

$\Rightarrow 90 x=11$

$\Rightarrow x=\frac{11}{90}$

(v) Let $x=2.2 \overline{4}$

∴  x = 2.2444                               ...(1)
10x = 22.444                              ...(2)
100x = 224.444                          ...(3)       
On subtracting equation (2) from (3), we get

$90 x=202 \Rightarrow x=\frac{202}{90}=\frac{101}{45}$

Hence, $2.2 \overline{4}=\frac{101}{45}$

(vi) Let $x=0 . \overline{365}$

∴ x = 0.3656565                          ...(1)
10x = 3.656565                       ...(2)
1000x = 365.656565                 ...(3)
On subtracting (2) from (3), we get

$990 x=362 \Rightarrow x=\frac{362}{990}=\frac{181}{495}$

Hence, $0 . \overline{365}=\frac{181}{495}$