GB-15-05-001 · Mixed recurring decimals

Use algebra to write $0.5\overline{65}$ as a fraction in its simplest form.

Recurring Decimals (Mixed Recurring) Booklet p. 50

Method

1) Let x equal the recurring decimal

2) Multiply by powers of 10 until the tails match

3) Subtract vertically

4) Divide and simplify

Worked Example (GB-15-05-001)

Let

x=0.5\overline{65}

. Multiply until the recurring tails match.

\begin{array}{rcl}1000x&=&565.\overline{65}\\10x&=&5.\overline{65}\\\hline 990x&=&560\end{array}

Subtract the aligned equations; the infinite tails cancel exactly. Divide and simplify:

x=\frac{560}{990}=\frac{56}{99}

Ans =

\frac{56}{99}