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

(i) $0.\overline{5}$

(ii) $0.\overline{43}$

(iii) $0.\overline{158}$

(iv) $1.\overline{3}$

(v) $4.\overline{17}$

(vi) $0.\overline{12}$

(vii) $0.1\overline{36}$

(viii) $1.5\overline{7}$

(i) $0.\overline{5}$

Let x = $0.\overline{5}$.

⇒ x = 0.555..     ………..(1)

⇒ 10x = 5.555..     ……….(2)

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

⇒ 10x - x = 5.555… - 0.555…

⇒ 9x = 5

⇒ x = $\dfrac{5}{9}$

Hence, $0.\overline{5} = \dfrac{5}{9}$.

(ii) $0.\overline{43}$

Let x = $0.\overline{43}$.

⇒ x = 0.434343..     ………(1)

⇒ 100x = 43.4343..     ……..(2)

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

⇒ 100x - x = 43.4343…. - 0.4343……

⇒ 99x = 43

⇒ x = $\dfrac{43}{99}$

Hence, $0.\overline{43} = \dfrac{43}{99}$.

(iii) $0.\overline{158}$

Let x = $0.\overline{158}$

⇒ x = 0.158158..     ……(1)

⇒ 1000x = 158.158158..     ……(2)

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

⇒ 1000x - x = 158.158158…. - 0.158158…..

⇒ 999x = 158

⇒ x = $\dfrac{158}{999}$

Hence, $0.\overline{158} = \dfrac{158}{999}$.

(iv) $1.\overline{3}$

Let x = $1.\overline{3}$

⇒ x = 1.3333..     ……(1)

⇒ 10x = 13.333..     ……(2)

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

⇒ 10x - x = 13.333.. - 1.333..

⇒ 9x = 12

⇒ x = $\dfrac{12}{9}$

⇒ x = $\dfrac{4}{3}$

Hence, $1.\overline{3} = \dfrac{4}{3}$.

(v) $4.\overline{17}$

Let x = $4.\overline{17}$

⇒ x = 4.1717..     ……..(1)

⇒ 100x = 417.1717..     …….(2)

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

⇒ 100x - x = 417.1717….. - 4.1717…..

⇒ 99x = 413

⇒ x = $\dfrac{413}{99}$

Hence, $4.\overline{17} = \dfrac{413}{99}$.

(vi) $0.\overline{12}$

Let x = $0.\overline{12}$

⇒ x = 0.1222..     ………..(1)

⇒ 100x = 12.222…(ii)

On subtracting (i) from (ii), we get

⇒ 99x = 12

⇒ x = $\dfrac{12}{99}$

⇒ x = $\dfrac{12÷3}{99÷3}$

⇒ x = $\dfrac{4}{33}$

Hence, 0.12… = $\dfrac{4}{33}$.

(vii) $0.1\overline{36}$

Let x = $0.1\overline{36}$

⇒ x = 0.13636..     ………..(1)

⇒ 10x = 1.3636..     ………..(2)

⇒ 1000x = 136.3636..     ………..(3)

On subtracting (2) from (3), we get

⇒ 1000x - 10x = 136.3636.. - 1.3636..

⇒ 990x = 135

⇒ x = $\dfrac{135}{990}$

⇒ x = $\dfrac{135 ÷ 45}{990 ÷ 45}$

⇒ x = $\dfrac{3}{22}$

Hence, $0.1\overline{36}\dfrac{3}{22}$.

(viii) $1.5\overline{7}$

Let x = $1.5\overline{7}$.

⇒ x = 1.5777..     ………..(1)

⇒ 10x = 15.777..     ………..(2)

⇒ 100x = 157.7777..     ………..(3)

On subtracting (i) from (iii), we get

⇒ 100x - 10x = 157.7777.. - 15.777……

⇒ 90x = 142

⇒ x = $\dfrac{142}{90}$

⇒ x = $\dfrac{142 ÷ 2}{90 ÷ 2}$

⇒ x = $\dfrac{71}{45}$

Hence, $1.5\overline{7} = \dfrac{71}{45}$.