Question

What is 2.318 repeating as a mixed number?

Answer 1

the idea being, we first, move the repeating part to the left of the decimal point, by simply multiplying it by a power of 10, in this case, we need to move 318, so we need then 1000, 3 decimals, 3 zeros.

and we'll make the expression a variable, so let's do so first,


`\bf \boxed{x=2.318318\overline{318}}\\\\ -------------------------------\\\\ \begin{array}{llll} 1000\cdot x&=&2318.318\overline{318}\\ &&2316+2.318318\overline{318}\\ &&2316+x \end{array}\implies 1000x=2316+x \\\\\\ 999x=2316\implies x=\cfrac{2316}{999}\implies x=\cfrac{772}{333}\implies x=2(106)/(333)`