let's first off make the recurring value to say "x", thus x = 0.12222.....
then let's multiply "x" by some power of 10, that will move the recurring decimal to the left of the decimal point, then we'll do the same to move another digit over, let's do so
[tex]\bf x = 0.1\overline{2}\qquad \begin{array}{llll} 100x&=&12.\overline{2}\\\\ 1000x&=&122.\overline{2} \end{array} \\\\[-0.35em] ~\dotfill\\\\ 1000x - 100x = 122.\overline{2} - 12.\overline{2}\implies 900x=110\implies x=\cfrac{110}{900}\implies x = \cfrac{11}{90}[/tex]
