Question

Write each decimal as a fraction or mixed number on simplest form -0.45 repeating

Answer 1

let's start by moving the repeating part to the left by multiplying by some power of 10, hmmm in this case the repeating part is just "45", so is two digits so let's multiply it by 10 at the 2nd power or 100, so we're moving only two digits to the left

`-0.\overline{454545}\hspace{5em}x=0.\overline{454545}\hspace{5em}-x=-0.\overline{454545} \\\\[-0.35em] ~\dotfill\\\\ \begin{array}{llll} 100x&=&45.\overline{454545}\\\\ &&45+0.\overline{454545}\\\\ &&45+x \end{array}\qquad \implies \qquad 100x=45+x \\\\\\ 99x=45\implies x=\cfrac{45}{99}\implies x=\cfrac{5}{11}\implies -x=-\cfrac{5}{11}`