GB-15-04-001 · Pure recurring decimals

Use algebra to write $0.\overline{71}$ as a fraction in its simplest form.

Recurring Decimals (Pure 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-15-04-001)

Let

x=0.\overline{71}

. Multiply until the recurring tails match.

\begin{array}{rcl}100x&=&71.\overline{71}\\x&=&0.\overline{71}\\\hline 99x&=&71\end{array}

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

x=\frac{71}{99}=\frac{71}{99}

Ans =

\frac{71}{99}