Question

Convert 6.1212 into a rarional number

Answer 1

in short, we will start off by making the left-side of the dot and the recurring numbers a variable, say "x", then multiplying it by some power of 10 that moves the recurring numbers over to the left, let's do so

$\bf x = 6.\overline{12}~\hspace{10em} \begin{array}{llll} 100\cdot x&=&612.\overline{12}\\\\ &&606+6.\overline{12}\\\\ &&606+x \end{array} \\\\\\ 100x=606+x\implies 99x=606\implies x = \cfrac{606}{99}\implies x = \cfrac{202}{33}\implies x = 6\frac{4}{33}$