\(\overline{a,b}\times0,9=\overline{a,0b}\)
\(\overline{a,b}\times10\times0,9\times10=\overline{a,0b}\times100\)
\(\overline{ab}\times9=\overline{a0b}\)
\(\left(10a+b\right)\times9=100a+b\)
\(90a+9b=100a+b\)
\(8b=10a\)
\(4b=5a\)
\(\Rightarrow\left\{{}\begin{matrix}a=4\\b=5\end{matrix}\right.\)