Repeating Decimals and Fractions

Every rational number can be expressed as either a terminating decimal (like $0.5$) or a repeating decimal (like $0.\overline{3}$). A repeating decimal has a digit or a block of digits that repeats infinitely. Because repeating decimals are rational numbers, they can always be written as fractions.

The Algebraic Method

To convert a repeating decimal into a fraction, we use a clever algebraic trick to eliminate the infinite repeating part.

1. Set the repeating decimal equal to a variable, like $x$.
2. Count how many digits make up the repeating block.
3. Multiply both sides of the equation by $10^n$, where $n$ is the number of repeating digits. (Multiply by 10 for 1 digit, 100 for 2 digits, etc.)
4. Subtract the original equation from the new equation to cancel out the repeating decimal.
5. Solve for $x$ and simplify the fraction.

Example 1: Single Repeating Digit

Convert $0.\overline{6}$ to a fraction.

Let $x$ equal the repeating decimal: $x = 0.6666...$

Since only one digit (6) repeats, multiply both sides by 10: $10x = 6.6666...$

Now, subtract the original equation ($x$) from this new equation: $10x - x = 6.6666... - 0.6666...$ $9x = 6$

Divide by 9 to solve for $x$: $x = \frac{6}{9}$

Simplify the fraction by dividing the numerator and denominator by 3: $x = \frac{2}{3}$

Example 2: Multiple Repeating Digits

Convert $0.\overline{27}$ to a fraction.

Let $x$ equal the decimal: $x = 0.272727...$

Because two digits (27) repeat, multiply by 100: $100x = 27.272727...$

Subtract the original equation: $100x - x = 27.272727... - 0.272727...$ $99x = 27$

Solve for $x$ and simplify (divide by 9): $x = \frac{27}{99} = \frac{3}{11}$

Example 3: Fraction to Decimal

Is $\frac{7}{11}$ a terminating or repeating decimal?

To find out, simply divide the numerator by the denominator: $7 \div 11 = 0.636363...$

Because the digits "63" repeat infinitely, it is a repeating decimal, which we write as $0.\overline{63}$.