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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

GCSE specification fit

Recurring Decimals and Fractions is part of GCSE Maths Number.

Convert recurring decimals to fractions using simple algebra. Foundation pupils should recognise recurring notation and common fraction-decimal links. Higher pupils should 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

Tier guidanceFoundation recognition; Higher algebraic conversion

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.

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.

Exam 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.

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.

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.

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...?

10x = 2.323232... 20x = 4.646464... 100x = 23.232323...

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

18/100 2/11 18/10

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

Set x equal to the recurring decimal, multiply by the right power of 10 so the repeat lines up, subtract and simplify the final fraction fully.

OCR GCSE Maths

Show the place-value shift clearly, especially when the recurring block has two digits or starts after a non-recurring digit.

Pearson Edexcel GCSE Maths

Use dots or clear recurring notation accurately, then reduce fractions such as 45/99, 27/90 or 306/990 to their simplest form.

Eduqas GCSE Maths

Make it clear whether a decimal is exact, recurring or rounded, and keep exact fraction form unless a decimal approximation is requested.

WJEC Wales

Watch for Numeracy-style contexts where a calculator display may need interpreting as exact, recurring or rounded before a final decision.

CCEA GCSE Maths

Check whether the unit or tier expects calculator interpretation, non-calculator conversion, or a fully simplified exact fraction.

Next lesson

Next, continue with Error Intervals and Limits of Accuracy.

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