I need an explanation of how to do this question
0.1(57) Where the brackets mark recurring signs.
I know you start it by saying let x = 0.157575757... just not quite sure where to go from here.
Well, if we let $\displaystyle x=0.157575757\ldots$ then $\displaystyle 100x=15.757575757\ldots.$ Now subtract.
$\displaystyle \begin{tabular}{crcr@{.}l}
&$100x$&$=$&15&757575757\ldots\\
$-$&$x$&$=$&0&157575757\ldots\\\hline
&$99x$&=&15&6
\end{tabular}$
So $\displaystyle x=\frac{15.6}{99}=\frac{156}{990}=\frac{26}{165}.$