Repeating Decimal → Fraction (Mixed)

\( x=\text{0.nonrep}\,\overline{\text{rep}}\;\Rightarrow\; x=\tfrac{\text{all digits}-\text{nonrep}}{\underbrace{99\dots 9}_{\lvert\text{rep}\rvert}\underbrace{00\dots 0}_{\lvert\text{nonrep}\rvert}} \)
Number GCSE
Question 6 of 20
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Convert 0.12\overline{34 to a fraction.

Tips: use ^ for powers, sqrt() for roots, and type pi for π.
Hint (H)
All digits – nonrep over 9900

Explanation

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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.

Worked examples

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  1. Convert 0.1\overline{6 into a fraction.
    1. \( All digits=16 \)
    2. \( Nonrep=1 \)
    3. \( Denominator=90 \)
    4. \( Fraction=(16-1)/90=15/90=1/6 \)
    Answer: 1/6
  2. Convert 0.2\overline{45 into a fraction.
    1. \( All digits=245 \)
    2. \( Nonrep=2 \)
    3. \( Denominator=990 \)
    4. \( Fraction=(245-2)/990=243/990=27/110 \)
    Answer: 27/110
  3. Convert 0.07\overline{1 into a fraction.
    1. \( All digits=071=71 \)
    2. \( Nonrep=07=7 \)
    3. \( Denominator=900 \)
    4. \( Fraction=(71-7)/900=64/900=16/225 \)
    Answer: 16/225
  4. Convert 0.3\overline{27 into a fraction.
    1. \( All digits=327 \)
    2. \( Nonrep=3 \)
    3. \( Denominator=990 \)
    4. \( Fraction=(327-3)/990=324/990=18/55 \)
    Answer: 18/55
  5. Convert 0.45\overline{8 into a fraction.
    1. \( All digits=458 \)
    2. \( Nonrep=45 \)
    3. \( Denominator=990 \)
    4. \( Fraction=(458-45)/990=413/990 \)
    Answer: 413/990
  6. Convert 0.12\overline{34 into a fraction.
    1. \( All digits=1234 \)
    2. \( Nonrep=12 \)
    3. \( Denominator=9900 \)
    4. \( Fraction=(1234-12)/9900=1222/9900=611/4950 \)
    Answer: 611/4950
  7. Convert 0.0\overline{81 into a fraction.
    1. \( All digits=081=81 \)
    2. \( Nonrep=0 \)
    3. \( Denominator=990 \)
    4. \( Fraction=(81-0)/990=81/990=9/110 \)
    Answer: 9/110
  8. Convert 0.56\overline{7 into a fraction.
    1. \( All digits=567 \)
    2. \( Nonrep=56 \)
    3. \( Denominator=990 \)
    4. \( Fraction=(567-56)/990=511/990 \)
    Answer: 511/990
  9. Convert 0.14\overline{285 into a fraction.
    1. \( All digits=14285 \)
    2. \( Nonrep=14 \)
    3. \( Denominator=99900 \)
    4. \( Fraction=(14285-14)/99900=14271/99900=15857/111000 \)
    Answer: 14271/99900 (simplify if possible)
  10. Convert 0.002\overline{7 into a fraction.
    1. \( All digits=0027=27 \)
    2. \( Nonrep=002=2 \)
    3. \( Denominator=9000 \)
    4. \( Fraction=(27-2)/9000=25/9000=1/360 \)
    Answer: 1/360