Converting recurring decimals to fractions

What is a recurring decimal?

A recurring decimal has a digit or group of digits that repeats infinitely. Notation:

• 0.333… = 0.3̄ (dot above the repeating digit)
• 0.142857142857… = 0.̇142857̇ (dots above first and last of repeating group)
• 0.16666… = 0.16̄

The algebraic method

This is the CCEA Higher method. The idea is to multiply by a power of 10 to shift the decimal so that subtracting eliminates the repeating part.

Step-by-step:

1. Let x = the recurring decimal.
2. Multiply by 10ⁿ where n = number of recurring digits.
3. Subtract to eliminate the recurring part.
4. Solve for x and simplify.

Example 1: Convert 0.7̄ (= 0.777…) to a fraction. Let x = 0.777… 10x = 7.777… Subtract: 9x = 7 → x = 7/9.

Example 2: Convert 0.36̄ (= 0.3666…) to a fraction. Let x = 0.3666… 10x = 3.666… 100x = 36.666… Subtract: 100x − 10x = 36.666… − 3.666… → 90x = 33 → x = 33/90 = 11/30.

Example 3: Convert 0.ṁ18ẋ (= 0.181818…) to a fraction. Let x = 0.181818… 100x = 18.181818… Subtract: 99x = 18 → x = 18/99 = 2/11.

Quick shortcuts

For simple cases:

• One recurring digit: numerator = digit; denominator = 9. e.g. 0.4̄ = 4/9.
• Two recurring digits: numerator = number formed by digits; denominator = 99. e.g. 0.27̄ = 27/99 = 3/11.
• Three recurring digits: denominator = 999.

CCEA context

This is exclusively Higher tier. CCEA Paper 1 tests this without a calculator. You must show the algebraic method — just writing the fraction without working will not earn full marks. Questions may also ask you to prove that a given recurring decimal equals a specific fraction.