1. \({\textbf{Let}\space} x=0.3777...\)
2. \({\textbf{Multiply both sides of the equation by 10}}\)
\({\space\space\space\space\space\space\space\space\space\space\space}10\cdot x=10\cdot0.3777...\)
$${\space\space\space\space\space\space\space\space\space\space\space\space\space\space}10x=3.777...$$
3. $$(10x-x)=3.777…−0.377...$$
$${\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space}9x=3.4$$
$${\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space}x=\frac{3.4}{9}=\frac{1.7}{4}$$
Therefore, 0.3777...= 1.7/4