Question

How do you convert $0.7777$ as a fraction?

Answer 1

Derivation of Raja's given answer may prove useful!
Method can applied to repeating groups of decimal values
$x=0.7777\overline{7}$ ..... $\left(1\right)$
$10x=0.777\overline{7}$ ..... $\left(2\right)$
$\left(2\right)-\left(1\right)$
$x(10-1)=7$
$x=\frac{7}{9}$