Express $0.6+0 . \overline{7}+0.4 \overline{7}$ in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0 .$
Let $x=0 . \overline{7}=0.777$ $\ldots(\mathrm{i})$
On multiplying both sides of Eq. (i) by 10, we get
$10 x=7.77 \ldots$ ...(ii)
On subtracting Eq. (i) from Eq. (ii), we get
$10 x-x=(7.77 \ldots)-(0.77 \ldots)$
$\Rightarrow$ $9 x=7$
$\therefore$ $x=\frac{7}{9}$
Now, let $\quad y=0.47=0.4777 \ldots . \quad \ldots$ (iii)
On multiplying both sides of Eq. (iii) by 10 , we get
$10 y=4.777 \ldots$ $\ldots$ (iv)
On multiplying both sides, Eq. (iv) by 10, we get
$100 y=47.777 \ldots$ $\ldots(\mathrm{v})$
On subtracting Eq. (iv) from Eq. (v), we get
$(100 y-10 y)=(47.777 \ldots)-(4.777 \ldots)$
$\Rightarrow$ $90 y=43=\frac{43}{90}$
$\therefore$ $0.6+0 . \overline{7}+0.4 \overline{7}=\frac{6}{10}+\frac{7}{9}+\frac{43}{90}$
$=\frac{54+70+43}{90}=\frac{167}{90}$
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