Question

Express $0.6+0.\overline{7}+0.4\overline{7}$ In the form $\frac{p}{q}$, where p and q are integers and $q\ne 0$.

Answer 1

Let $x=0.\overline{7}=0.777$ ....(i)
On multiplying both sides of eq.(i) by 10 , we get,
10x = 7.77 ---(ii)
On subtracting eq.(i) from eq(ii), we get
10x – x = (7.77…) – ( 0.77…..)
$\Rightarrow 9x=7$
$\therefore x=\frac{7}{9}$
Now,
Let $y=0.4\overline{7}=0.4777$ .....(iii)
On multiplying eq.(iii) by 10, we get 10y = 4.7777... (iv)
On multiplying both sides of eq.(iv) by 10, we get 100y=47.777….. …(v)
On Subtracting eq.(iv) from eq (v), we get,
(100y – 10y) = ( 47.777…) – ( 4.777….)
$\Rightarrow 90y=43=\frac{43}{90}\therefore 0.6+0.\overline{7}+0.4\overline{7}=\frac{6}{10}+\frac{7}{9}+\frac{43}{90}=\frac{54+70+43}{90}=\frac{167}{90}$