[tex]\bf 3.8888888\overline{8}\\\\
-------------------------------\\\\
x=3.8888888\overline{8}\qquad thus\qquad
\begin{array}{llcll}
10x&=&38.888888\overline{8}\\
&&\downarrow \\
&&35+3.8888888\overline{8}\\
&&\downarrow \\
&&35+x
\end{array}
\\\\\\
10x=35+x\implies 9x=35\implies x=\cfrac{35}{9}[/tex]
the idea being, you multiply the "x" by some power of 10 that moves the repeating part to the left of the decimal point and the split it like above.
