the idea being, that we first off move the recurring part to the left-hand-side by simply multiplying it by some power of 10 and then we use the decimal part as variable, let's proceed.
[tex]0.3838\overline{38}~\hspace{10em}x=0.\overline{38} \\\\[-0.35em] ~\dotfill\\\\ \begin{array}{|lll|l}\cline{1-3} &&&\\ 100x&=&38.\overline{38}\\ &&38+0.\overline{38}\\ &&38+x\\[1em]\cline{1-3} \end{array}\qquad \implies \begin{array}{llll} 100x=38+x\implies 99x=38\\\\ x=\cfrac{38}{99} \end{array}[/tex]
