$0.6 + 0.\overline{7} + 0.4\overline{7}$ is equal to:

1. $\dfrac{155}{90}$

2. $\dfrac{147}{90}$

3. $\dfrac{167}{90}$

4. none of these

Convert repeating decimals to fraction,

Let, x = $0.\overline{7}$

⇒ x = 0.7777….     …….(1)

⇒ 10x = 7.777…     …….(2)

Subtracting equation (1) from (2), we get :

⇒ 10x - x = 7.777….. - 0.777…..

⇒ 9x = 7

⇒ x = $\dfrac{7}{9}$

Let, y = $0.4\overline{7}$

⇒ 10y = 4.7777….     …….(3)

⇒ 100y = 47.777…     …….(4)

Subtracting equation (3) from (4), we get :

⇒ 100y - 10y = 47.777….. - 4.777…..

⇒ 90y = 43

⇒ y = $\dfrac{43}{90}$

$\Rightarrow 0.6 + 0.\overline{7} + 0.4\overline{7} = 0.6 + \dfrac{7}{9} + \dfrac{43}{90} \\[1em] = \dfrac{6}{10} + \dfrac{7}{9} + \dfrac{43}{90} \\[1em] = \dfrac{54 + 70 + 43}{90} \\[1em] = \dfrac{167}{90}.$

Hence, Option 3 is the correct option.