Procedure

Convert a recurring decimal to a fraction

Arithmetic • Number System • Rational Numbers

Statement

If a decimal has

p

digits after the point before the bar and

q

digits under it, then multiplying it by

1 0^{p + q}

and by

1 0^{p}

produces two numbers with identical tails; subtracting leaves

(1 0^{p + q} - 1 0^{p}) N

equal to a whole number, so

N = \frac{( digits up to the end of one period ) - ( digits before the bar )}{q 9 \dots 9 p 0 \dots 0}

How to use it

1Count $q$ , the digits under the bar, and $p$ , the digits after the point that are not under it; the denominator is then $q$ nines followed by $p$ zeros.
2For a purely recurring decimal ( $p = 0$ ) the denominator is all nines: $0. ab = \frac{ab}{99}$ .
3Multiply by the two powers of ten and subtract if the rule is not remembered; the tails must match exactly for the subtraction to clear them.

Express

0.1 6

as a fraction in lowest terms.

Step-by-step

1
$N = 0.1 6$ , so $100 N = 16. 6$ and $10 N = 1. 6$
Why: One digit sits before the bar and one under it, so the two multipliers are $1 0^{2}$ and $1 0^{1}$ .
2
$100 N - 10 N = 16. 6 - 1. 6 = 15$
Why: The identical recurring tails cancel in the subtraction.
3
$90 N = 15 \Rightarrow N = \frac{15}{90} = \frac{1}{6}$

Answer

\frac{1}{6}

Watch out

The two multipliers must leave the SAME tail after the point: using

10

and

100

0. 123

does not clear the recurrence, because its period is three digits and needs

1 0^{3}

Also known as

recurring decimal to fraction, repeating decimal as a fraction, mixed recurring decimal, decimal to fraction, bar notation decimal