Question

What is 0.612 ( 12 repeating) as a fraction

Answer 1

the idea behind the recurring decimal as a fraction, is to first off, multiply or divide by some power of 10, in order that we leave the recurring decimal to the right of the decimal point.

then we multiply by a power of 10, in order to move the repeating digits to the left of the decimal point, anyhow, let's proceed.

`\bf 0.6\overline{1212}\implies \cfrac{06.\overline{1212}}{10}\implies \cfrac{6+0.\overline{1212}}{10}\qquad \qquad \stackrel{\textit{now let's make}}{x=0.\overline{1212}} \\\\[-0.35em] ~\dotfill\\\\ \begin{array}{llll} 100\cdot x &=& 12.\overline{1212}\\\\ &&12+0.\overline{1212}\\\\ &&12+x\\\\ 100x&=&12+x\\\\ 99x&=&12\\\\ x&=&\cfrac{12}{99}\implies x = \cfrac{4}{33} \end{array} \\\\[-0.35em] ~\dotfill`

`\bf \cfrac{06.\overline{1212}}{10}\implies \cfrac{6+x}{10}\implies \cfrac{6+(4)/(33)}{10}\implies \cfrac{~~(202)/(33)~~}{10}\implies \cfrac{~~(202)/(33)~~}{(10)/(1)}\implies \cfrac{202}{330} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill \cfrac{101}{165}~\hfill`

notice, we first divided by 10, to move the decimal point over to the right by 1 slot, then we multiplied by 100, to move it two digits over the decimal point, namely the repeating "12", thus we use 100.