Repeating Decimal → Fraction (Mixed)

Statement

To convert a repeating decimal with a non-repeating and repeating part into a fraction:

\[ x = 0.\text{nonrep rep rep}... \quad \Rightarrow \quad x = \frac{\text{(all digits up to one full repeat)} - \text{(nonrep part)}}{\underbrace{99\ldots900\ldots0}_{\text{rep length} \;|\; \text{nonrep length}}}. \]

Why it’s true

• Let the repeating decimal be written as \(x\).
• Shift the decimal point so that one repeat cycle is to the right.
• Subtract to eliminate the repeating part, leaving a finite integer difference.
• The denominator is built with 9’s for repeating digits and 0’s for non-repeating digits.

Recipe (how to use it)

1. Write down all digits up to the end of the first repeating block.
2. Subtract the non-repeating digits.
3. Denominator: 9’s for each repeating digit, then 0’s for each non-repeating digit.
4. Simplify the fraction.

Spotting it

Use this when decimals have some non-repeating digits before the repeating pattern begins (e.g. 0.167, 0.245).

Common pairings

• Recurring decimal to fraction conversion problems.
• Mixed type decimals in GCSE number questions.
• Proofs or “show that” style questions in exams.

Mini examples

1. \(x=0.1\overline{6}\). All digits=16, nonrep=1, denominator=90 → \(x=(16-1)/90=15/90=1/6\).
2. \(x=0.2\overline{45}\). All digits=245, nonrep=2, denominator=990 → \(x=(245-2)/990=243/990=27/110\).
3. \(x=0.07\overline{1}\). All digits=071=71, nonrep=07=7, denominator=900 → \(x=(71-7)/900=64/900=16/225\).

Pitfalls

• Forgetting to align the denominator (9’s for repeating, 0’s for non-repeating).
• Not subtracting the non-repeating part.
• Leaving the fraction unsimplified.

Exam strategy

• Write clearly “all digits – nonrep” over denominator before simplifying.
• Check with calculator decimal expansion if time permits.
• Always reduce the fraction to simplest form.

Summary

The formula provides a shortcut to convert mixed repeating decimals into fractions. Simply write all digits, subtract the non-repeating part, and divide by the correct combination of 9’s and 0’s.