1. Introduction
Repeating decimals are decimals in which one digit or a group of digits repeat endlessly. These decimals can always be converted into a rational number (a fraction) using a simple algebraic method. This topic explains the step-by-step process with clear examples.
2. What is a Repeating Decimal?
A repeating decimal is a decimal number where a digit or group of digits repeats forever.
Examples:
- \(0.333\ldots = 0.\overline{3}\)
- \(0.727272\ldots = 0.\overline{72}\)
- \(2.145145145\ldots = 2.\overline{145}\)
3. General Method to Convert a Repeating Decimal to a Fraction
You can convert any repeating decimal into a fraction using algebraic steps. The idea is to create an equation, multiply to shift the decimal, and subtract to eliminate the repeating part.
4. Steps for Conversion
4.1. Step 1: Assign the decimal to a variable
Let \(x\) be the repeating decimal. Example: If \(x = 0.\overline{3}\)
4.2. Step 2: Multiply to shift the repeating digits
If a single digit repeats, multiply by 10; if two digits repeat, multiply by 100; and so on.
Example: Since only '3' repeats, multiply both sides by 10: \(10x = 3.\overline{3}\)
4.3. Step 3: Subtract the two equations
Subtract: \(10x - x = 3.\overline{3} - 0.\overline{3}\)
The repeating part cancels out.
4.4. Step 4: Solve for \(x\)
You get a simple fraction. For example: \(9x = 3\) so \(x = \dfrac{1}{3}\).
5. Examples
5.1. Example 1: Convert \(0.\overline{3}\) to a fraction
Let \(x = 0.\overline{3}\)
Multiply by 10: \(10x = 3.\overline{3}\)
Subtract: \(10x - x = 3.\overline{3} - 0.\overline{3}\)
\(9x = 3\)
\(x = \dfrac{1}{3}\)
5.2. Example 2: Convert \(0.\overline{72}\) to a fraction
Let \(x = 0.\overline{72}\)
Because 2 digits repeat, multiply by 100:
\(100x = 72.\overline{72}\)
Subtract: \(100x - x = 72.\overline{72} - 0.\overline{72}\)
\(99x = 72\)
\(x = \dfrac{72}{99} = \dfrac{8}{11}\)
5.3. Example 3: Convert \(2.\overline{145}\) to a fraction
Let \(x = 2.\overline{145}\)
Multiply by 1000 because 3 digits repeat:
\(1000x = 2145.\overline{145}\)
Subtract: \(1000x - x = 2145.\overline{145} - 2.\overline{145}\)
\(999x = 2143\)
\(x = \dfrac{2143}{999}\)
6. Special Case: Decimals with a Non-Repeating Part
Decimals like \(0.1\overline{6}\) have a non-repeating part followed by a repeating block. These can also be converted using a two-step multiplication method.
6.1. Example: Convert \(0.1\overline{6}\) to a fraction
Let \(x = 0.1\overline{6}\)
Step 1: Multiply by 10 to move non-repeating part:
\(10x = 1.\overline{6}\)
Step 2: Now multiply by 10 again (because 1 digit repeats):
\(100x = 16.\overline{6}\)
Subtract:
\(100x - 10x = 16.\overline{6} - 1.\overline{6}\)
\(90x = 15\)
\(x = \dfrac{15}{90} = \dfrac{1}{6}\)
7. Summary Table
| Repeating Decimal | Method | Fraction |
|---|---|---|
| \(0.\overline{3}\) | Multiply by 10 and subtract | \(\dfrac{1}{3}\) |
| \(0.\overline{72}\) | Multiply by 100 and subtract | \(\dfrac{8}{11}\) |
| \(2.\overline{145}\) | Multiply by 1000 and subtract | \(\dfrac{2143}{999}\) |
| \(0.1\overline{6}\) | Shift non-repeat, then shift repeat | \(\dfrac{1}{6}\) |
8. Key Points to Remember
- All repeating decimals represent rational numbers.
- The number of repeating digits determines the power of 10 used for multiplication.
- Subtracting equations removes the repeating part and allows solving for the fraction.
- Always simplify the resulting fraction.