we'll start off by making x = 25.41414141.....
then we'll move one of the repeating part to the left with *100
then we'll move 2 of the repeating part to the left with *10000
anyhow, let's proceed
[tex]\bf x = 25.\overline{41}\qquad \quad \begin{array}{llll} 100\cdot x&=&2541.\overline{41}\\\\ 10000\cdot x&=&254141.\overline{41} \end{array}\qquad \qquad \begin{array}{rllll} 10000x\\ -100x\\\cline{1-1} 9900x \end{array}= \begin{array}{rllll} 254141.\overline{41}\\ -2541.\overline{41}\\\cline{1-1} 251600~~~~ \end{array} \\\\\\ 9900x = 251600\implies x = \cfrac{251600}{9900}\implies x = \cfrac{2516}{99}\implies x = 25 \frac{41}{99}[/tex]
