Free GCSE Maths lesson: Number
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Lesson 72 · GCSE / Key Stage 4 · Maths · Number

Recurring Decimals and Fractions

Convert recurring decimals to fractions using simple algebra.

Qualification: GCSEKey Stage 4Subject: MathsStrand: Number

Lesson overview

Recurring Decimals and Fractions is part of GCSE Maths Number.

Convert recurring decimals to fractions using simple algebra. Start by learning to recognise recurring notation and common fraction-decimal links. Then learn how to be ready to line up one-digit, two-digit and delayed repeats, subtract, then simplify the exact fraction.

QualificationGCSE Mathematics
Key stageKey Stage 4
StrandNumber

What you will learn

  • Recognise terminating and recurring decimals.
  • Use dot notation for repeated digits.
  • Convert simple recurring decimals to fractions.
  • Convert fractions to recurring decimals.
  • Convert delayed recurring decimals where the repeat starts after a non-recurring digit.
  • Check whether an answer has been rounded or is exact.

Key facts before you start

Recurring decimal A dot or bar shows digits that repeat forever.
Algebra method Use powers of 10 to line up the recurring part.
Subtract equations Subtracting removes the repeating tail.
Simplify fraction Finish by simplifying the fraction fully.

Recurring decimals to fractions infographic

Before viewing

Before viewing, identify the repeating digit or block before starting any algebra.

Infographic explaining how to convert recurring decimals to fractions by spotting the repeat block, multiplying to line it up, subtracting and simplifying.
Use this visual to turn a recurring decimal into an exact simplified fraction by lining up the repeat, subtracting and checking the answer is not rounded.Download visual

After viewing

After viewing, convert a simple recurring decimal to a fraction and explain how subtraction removed the recurring part.

Why this matters

Recurring decimals link number, fractions and algebra. Exact fraction form is often cleaner than a rounded calculator display.

Prior knowledge

You should already be comfortable with:

  • Place value.
  • Fractions.
  • Multiplying by powers of 10.
  • Solving simple equations.

Clear explanation

Main idea

A recurring decimal has a digit or block of digits that repeats forever. For example, 0.333... = 1/3, while 0.3 is a terminating decimal and 0.33 is just two decimal places.

Method

To convert a recurring decimal to a fraction, call it x, multiply by 10, 100 or 1000 so the recurring block lines up, then subtract the original equation. The repeating tails cancel, leaving a simple equation.

Answer tip

For 0.272727..., let x = 0.272727... Then 100x = 27.272727..., so 99x = 27 and x = 27/99 = 3/11. If the repeat is delayed, such as 0.1666..., use 10x and 100x so the recurring 6s line up before subtracting.

Worked examples

Single recurring digit

Convert 0.777... to a fraction.

Reveal answer

Answer: Let x = 0.777... Then 10x = 7.777..., so 9x = 7 and x = 7/9.

Two recurring digits

Convert 0.454545... to a fraction.

Reveal answer

Answer: Let x = 0.454545... Then 100x = 45.454545... Subtracting gives 99x = 45, so x = 45/99 = 5/11.

Delayed recurring digit

Convert 0.1666... to a fraction.

Reveal answer

Answer: Let x = 0.1666... Then 10x = 1.666... and 100x = 16.666... Subtracting gives 90x = 15, so x = 15/90 = 1/6.

Quick checks

Choose an answer, then check your thinking.

1. Which equation lines up the repeat in 0.232323...?

2. What is 0.181818... as a fraction?

Practice questions

Question 1

Convert 0.666... to a fraction by setting x equal to the recurring decimal and subtracting equations.

Reveal answer and marking guidance

Answer: 2/3.

Marking: Let x = 0.666..., so 10x = 6.666... and 10x − x = 6. This gives 9x = 6, so x = 6/9 = 2/3.

Question 2

Convert 0.121212... to a fraction in its simplest form. The repeating block is 12.

Reveal answer and marking guidance

Answer: 4/33.

Marking: Let x = 0.121212... Then 100x = 12.121212..., so 99x = 12 and x = 12/99 = 4/33.

Question 3

Write 1/6 as a decimal and make clear which digit recurs.

Reveal answer and marking guidance

Answer: 0.1666...

Marking: 1 ÷ 6 = 0.1666..., with the 6 recurring after the first decimal place. This is exact recurring notation, not rounding.

Question 4

Convert 0.090909... to a fraction and simplify your answer fully.

Reveal answer and marking guidance

Answer: 1/11.

Marking: The repeating block is 09, so 100x = 9.090909... and 99x = 9. Therefore x = 9/99 = 1/11.

Question 5

Convert 0.272727... to a fraction in its simplest form.

Reveal answer and marking guidance

Answer: 3/11.

Marking: 99x = 27, so x = 27/99 = 3/11.

Question 6

Convert 0.1555... to a fraction in its simplest form.

Reveal answer and marking guidance

Answer: 7/45.

Marking: 10x = 1.555... and 100x = 15.555..., so 90x = 14 and x = 14/90 = 7/45.

Question 7

Convert 0.3060606... to a fraction in its simplest form, where the 06 repeats after the first decimal digit.

Reveal answer and marking guidance

Answer: 101/330.

Marking: 10x = 3.060606... and 1000x = 306.060606..., so 990x = 303 and x = 303/990 = 101/330.

Question 8

Write 5/12 as a decimal and state clearly whether it terminates or recurs.

Reveal answer and marking guidance

Answer: 0.41666..., so it recurs.

Marking: The 6 repeats after the first two decimal places; it is not a rounded answer.

Question 9

Convert 0.0727272... to a fraction in its simplest form, where the 72 repeats after the first decimal digit.

Reveal answer and marking guidance

Answer: 4/55.

Marking: Let x = 0.0727272... Then 10x = 0.727272... and 1000x = 72.727272..., so 990x = 72 and x = 72/990 = 4/55.

Question 10

Convert 0.4181818... to a fraction in its simplest form, where the 18 repeats after the first decimal digit.

Reveal answer and marking guidance

Answer: 23/55.

Marking: Let x = 0.4181818... Then 10x = 4.181818... and 1000x = 418.181818..., so 990x = 414 and x = 414/990 = 23/55.

Answers and marking guidance

The exact practice answers are hidden under each question so you can try first. For recurring decimals, marks usually come from defining x, choosing the correct power of 10, subtracting equations so the repeating part cancels, solving the equation and simplifying the final fraction fully. Use recurring-dot or ellipsis notation clearly so your answer is exact, not rounded.

Common mistakes

  • Treating a recurring decimal as rounded: 0.333... is exactly 1/3, but 0.33 is not.
  • Multiplying by the wrong power of 10: a two-digit repeat such as 45 needs 100x, not 10x.
  • Forgetting delayed repeats: for 0.1666..., subtract 10x from 100x so the recurring 6s line up.
  • Not simplifying the fraction: 27/99 earns method marks, but 3/11 is the fully simplified answer.

Extension challenge

Convert 0.583333... to a fraction. Then explain why multiplying by 10 and 100 is more useful than multiplying by 1000.

Reveal answer

Example answer: Let x = 0.583333... Then 10x = 5.8333... and 100x = 58.3333..., so 90x = 52.5 and x = 52.5/90 = 525/900 = 7/12. The 3s line up after multiplying by 10 and 100, so 1000 is unnecessary.

Exam-board guidance

Recurring Decimals and Fractions appears within GCSE Maths number content. Exact wording, tiering and calculator expectations can vary, but the core skill is the same: decide whether a decimal is terminating, recurring or rounded, then use exact notation and simplified fraction form.

AQA GCSE Maths

Focus on the shared method, notation and checking habit; wording and context can vary by route.

OCR GCSE Maths

Show clear method steps and use precise notation; question wording may vary by route.

Pearson Edexcel GCSE Maths

Keep working visible and check units or notation; contexts may vary by route.

Eduqas GCSE Maths

Use the same core skill in practical contexts, with clear reasoning and final checks.

WJEC Wales

Connect the method to the context and state what the result means.

CCEA GCSE Maths

Make each method step visible and keep notation, units and final checks clear.

Next lesson

Next, continue with Error Intervals and Limits of Accuracy.

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