In terminating and repeating decimals you saw that every fraction becomes a decimal that ends or repeats: $\frac{1}{4} = 0.25$ and $\frac{3}{11} = 0.272727\ldots = 0.\overline{27}$. Now go the other way, from a repeating decimal back to a fraction.
Why it works
Let $x = 0.\overline{27}$. Multiplying by 100 moves every digit two places to the left: $100x = 27.\overline{27}$. Both numbers have exactly the same endless tail, so subtracting makes the tail disappear, and what is left is an ordinary equation.
| value | |
|---|---|
| $100x$ | $27.272727\ldots$ |
| $x$ | $0.272727\ldots$ |
| $100x - x = 99x$ | $27$ |
Key idea
To write a repeating decimal as a fraction, call it $x$. Multiply by 10, 100 or 1000 (one zero for each repeating digit) so the repeating parts line up, subtract, and solve for $x$.
Worked examples
Worked example: One repeating digit
Write $0.\overline{4}$ as a fraction.
- Let $x = 0.\overline{4}$. One digit repeats, so $10x = 4.\overline{4}$.
- Subtract: $10x - x = 4$, so $9x = 4$.
Answer: $\frac{4}{9}$
Worked example: Two repeating digits
Write $0.\overline{27}$ as a fraction in lowest terms.
- $100x = 27.\overline{27}$ and $x = 0.\overline{27}$.
- Subtract: $99x = 27$, so $x = \frac{27}{99}$.
- Divide top and bottom by 9: $\frac{3}{11}$.
Answer: $\frac{3}{11}$
Worked example: The repeat starts later
Write $0.1\overline{6} = 0.1666\ldots$ as a fraction.
- Only the 6 repeats. $10x = 1.\overline{6}$ and $100x = 16.\overline{6}$ have the same tail.
- Subtract: $100x - 10x = 16 - 1$, so $90x = 15$.
- $x = \frac{15}{90} = \frac{1}{6}$.
Answer: $\frac{1}{6}$
Common mistake
Thinking $0.\overline{9}$ is a little less than 1. Use the method: $x = 0.\overline{9}$, $10x = 9.\overline{9}$, so $9x = 9$ and $x = 1$. Another check: $\frac{1}{3} = 0.\overline{3}$, and three times each side gives $1 = 0.\overline{9}$. They are two names for one number; there is no number between them.