\
"One neat way to do this is the following:"
"Let:"
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad x = .3777777777... \qquad \qquad \qquad \qquad \qquad \ "(Equation 1)"
"Multipy both sides by" \ 10, "and then move the decimal point one place to the right:"
\qquad \qquad \qquad \qquad \qquad \qquad \quad \ 10 x = 10 \cdot (.3777777777... )
\qquad \qquad \qquad \qquad \qquad \qquad \quad \ 10 x = 3.777777777... \qquad \qquad \qquad \qquad \qquad \ \ "(Equation 2)"
"Subtract (Equation 1) from (Equation 2):"
\ 10 x - x = 3.777777777... - .3777777777... \quad \ "(Eqn 2) - (Eqn 1)"
\qquad \qquad \ \ \ 9 x = 3.777777777... - .3777777777...
\qquad \qquad \qquad \quad \ = \qquad \quad " " \ 3.777777777...
\qquad \qquad \qquad \qquad \qquad \quad \ \ \ - \ \ \ \ .3777777777...
\qquad \qquad \qquad \quad \ = \qquad " " \ \ \ 3.400000000...
\qquad \qquad \qquad \quad \ = \qquad \quad " " \ 3.4
"So we conclude:"
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad\qquad \qquad 9 x = 3.4
"Multiply both sides by 10:"
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad 10 (9 x) = 10 \cdot (3.4)
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ 90 x = 34
"This is easy to solve (!!):"
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad x = 34/90.
"Simplify to lowest terms:"
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad x = 17/45.
"Remembering what" \ \ x \ \ "is, by (Equation 1), we have:"
\qquad \qquad \qquad \qquad \qquad \qquad \quad \quad 3.777777777... = 17/45.
"This is our answer."
\
"Summarizing:"
\qquad \qquad \qquad \qquad \qquad \qquad \quad \quad 3.777777777... = 17/45 \quad.