Repeating Decimal → Fraction (Pure Repeat)

Statement

A pure repeating decimal can be converted directly into a fraction:

\[ x = 0.\overline{a_1a_2\ldots a_k} \quad \Rightarrow \quad x = \frac{a_1a_2\ldots a_k}{99\ldots9} \]

The denominator has as many 9’s as there are repeating digits.

Why it’s true

• Let \(x = 0.\overline{a_1a_2\ldots a_k}\).
• Multiply by \(10^k\) to shift one full repeat cycle.
• Subtract to cancel the repeating part, leaving an integer difference.
• The result is \((10^k-1)x = a_1a_2\ldots a_k\).
• Since \(10^k-1\) is all 9’s, \(x = \frac{a_1a_2\ldots a_k}{99\ldots9}\).

Recipe (how to use it)

1. Write down the repeating block as the numerator.
2. Denominator is made of 9’s, same length as block.
3. Simplify if possible.

Spotting it

Use this formula when the decimal repeats immediately from the decimal point, like 0.\overline{3}, 0.\overline{142}, etc.

Common pairings

• Recurring decimals to fractions.
• Proof-style questions in GCSE maths.
• Checking equivalences (e.g. 0.\overline{9} = 1).

Mini examples

1. \(x=0.\overline{3}\). Numerator=3, Denominator=9 → \(3/9=1/3\).
2. \(x=0.\overline{27}\). Numerator=27, Denominator=99 → \(27/99=3/11\).
3. \(x=0.\overline{142}\). Numerator=142, Denominator=999 → \(142/999\).

Pitfalls

• Forgetting to use the same number of 9’s as digits in the repeating cycle.
• Leaving fractions unsimplified.
• Confusing with mixed repeating decimals (needs different formula).

Exam strategy

• Always check whether the decimal is pure repeat or mixed repeat.
• Simplify the final fraction.
• If needed, use calculator to check decimal expansion.

Summary

Pure repeating decimals convert neatly to fractions: numerator is the repeating block, denominator is all 9’s. A very efficient method for quick conversions.