Write \(0.333...\) as a fraction.
This is a well-known recurring decimal.
\(0.333... = \frac{1}{3}\).
Recurring Decimals
Recurring decimals have digits that repeat indefinitely. They are closely linked to decimals and fractions, and in GCSE Maths you need to convert recurring decimals into exact fractions using algebraic methods.
Overview
A recurring decimal is a decimal in which one digit or a group of digits repeats forever.
\( 0.3333\ldots = 0.\dot{3} \) \\ \( 0.272727\ldots = 0.\dot{2}\dot{7} \)
Recurring decimals often come from fractions such as \( \frac{1}{3} \), \( \frac{2}{11} \), or \( \frac{5}{9} \).
What you should understand after this topic
- Understand what a recurring decimal is
- Read and write recurring notation
- Understand how some fractions turn into recurring decimals
- Identify the repeating block
- Convert simple recurring decimals into fractions
Key Definitions
Recurring Decimal
A decimal where a digit or block of digits repeats forever.
Terminating Decimal
A decimal that ends, such as \(0.25\) or \(1.8\).
Recurring Block
The digit or group of digits that repeats.
Dot Notation
A way to show repeating digits with dots above them.
Bar Notation
A way to show repeating digits with a line above them.
Equivalent Fraction
A fraction equal in value to the recurring decimal.
Key Rules
Single recurring digit
\( 0.\dot{3} = 0.3333\ldots \)
Two-digit recurring block
\( 0.\dot{2}\dot{7} = 0.272727\ldots \)
\( \frac{1}{3} \)
\( = 0.\dot{3} \)
\( \frac{1}{9} \)
\( = 0.\dot{1} \)
Quick Recognition
Ends
It is a terminating decimal.
Keeps repeating
It is a recurring decimal.
One digit repeats
Use one recurring marker.
A block repeats
Mark the full repeating block, not just one part.
How to Solve
Step 1: Recognise recurring decimals
A recurring decimal is a decimal that repeats forever in a pattern.
\(0.6666\ldots = 0.\dot{6}\)
\(0.121212\ldots = 0.\dot{1}\dot{2}\)
Exam tip: The dots show the first and last digits of the repeating block.
Step 2: Compare terminating and recurring decimals
Terminating
\(0.4,\ 1.25,\ 3.875\)
Recurring
\(0.\dot{3},\ 2.\dot{7},\ 0.\dot{1}\dot{4}\)
Step 3: Know common recurring fractions
\(\frac{1}{3} = 0.\dot{3}\)
\(\frac{2}{3} = 0.\dot{6}\)
\(\frac{1}{9} = 0.\dot{1}\)
Some fractions become recurring decimals when divided.
Step 4: Convert a recurring decimal to a fraction
Use algebra to cancel the repeating part.
- Let \(x\) equal the recurring decimal.
- Multiply by 10, 100, or 1000 depending on the repeating block.
- Subtract the original equation.
- Solve for \(x\).
- Simplify the fraction.
Step 5: Choose the correct multiplier
Exam thinking: The multiplier moves one full repeating block to the left.
1 repeating digit
Multiply by 10.
2 repeating digits
Multiply by 100.
3 repeating digits
Multiply by 1000.
Step 6: Exam method summary
See fractions for simplifying fractions.
- Identify the repeating block.
- Write \(x =\) the recurring decimal.
- Multiply by the correct power of 10.
- Subtract to remove the recurring part.
- Solve and simplify.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics.
Edexcel
Write \( 0.\overline{3} \) as a fraction in its simplest form.
Edexcel
Write \( 0.\overline{7} \) as a fraction in its simplest form.
Edexcel
Write \( 0.\overline{12} \) as a fraction in its simplest form.
Edexcel
Express \( 0.2\overline{5} \) as a fraction in its simplest form.
Edexcel
Write 0.125 as a fraction and state whether the decimal is terminating or recurring.
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on algebraic methods and reasoning.
AQA
Let \( x = 0.\overline{6} \). Write \( x \) as a fraction in its simplest form.
AQA
Let \( x = 0.\overline{27} \). Express \( x \) as a fraction in its simplest form.
AQA
Convert 0.1\overline{8} into a fraction in its simplest form.
AQA
Show that \( 0.\overline{9} = 1 \).
AQA
Explain why all recurring decimals can be written as fractions.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising reasoning, algebraic manipulation, and classification of decimals.
OCR
Write \( 0.\overline{45} \) as a fraction in its simplest form.
OCR
Convert \( 0.\overline{6} \) into a fraction in its simplest form.
OCR
Express \( 2.\overline{3} \) as a fraction in its simplest form.
OCR
Arrange the following numbers in order of size:
OCR
\( 0.\overline{4},\; 0.45,\; 0.\overline{43},\; \frac{4}{9} \)
Exam Checklist
Step 1
Check whether the decimal terminates or repeats.
Step 2
Identify the full recurring block.
Step 3
For fractions, divide carefully or use known patterns.
Step 4
When converting to a fraction, choose the correct power of 10 and simplify.
Most common exam mistakes
Pattern mistake
Using the wrong repeating block.
Notation mistake
Marking the recurring digits incorrectly.
Algebra mistake
Multiplying by 10 instead of 100 for a two-digit repeating block.
Simplifying mistake
Leaving the answer as \( \frac{27}{99} \) instead of simplifying it.
Common Mistakes
These are common mistakes students make when working with recurring decimals in GCSE Maths.
Marking the repeating part incorrectly
Incorrect
A student places the recurring dot or bar over only part of the repeating block.
Correct
Identify the full repeating pattern. For example, \(0.3636...\) has a repeating block of \(36\), not just \(6\).
Confusing recurring and rounded decimals
Incorrect
A student treats a recurring decimal as a rounded value.
Correct
Recurring decimals are exact values, not approximations. For example, \(0.333... = \frac{1}{3}\) exactly.
Using the wrong power of 10
Incorrect
A student multiplies by the wrong power of 10 when converting to a fraction.
Correct
Multiply by the correct power of 10 to align the repeating digits. The number of digits in the repeating block determines the power.
Forgetting to simplify the fraction
Incorrect
A student leaves the final answer unsimplified.
Correct
Always simplify the fraction fully to its lowest terms.
Thinking recurring decimals are approximate
Incorrect
A student assumes recurring decimals are not exact.
Correct
Recurring decimals represent exact values and can always be written as fractions.
Try It Yourself
Practise converting recurring decimals into fractions.
Foundation
Foundation Practice
Understand simple recurring decimals and convert them to fractions.
Write \(0.666...\) as a fraction.
Double 1/3.
\(0.666... = \frac{2}{3}\).
Write \(0.111...\) as a fraction.
Think 1 ÷ 9.
\(0.111... = \frac{1}{9}\).
Write \(0.444...\) as a fraction.
Multiply 1/9 by 4.
\(0.444... = \frac{4}{9}\).
Which is equal to \(0.999...\)?
Think carefully about infinite decimals.
\(0.999... = 1\).
Write \(0.222...\) as a fraction.
Use 1/9 pattern.
\(0.222... = \frac{2}{9}\).
A student says \(0.333... = \frac{3}{10}\). What is wrong?
0.3 is not the same as 0.333...
0.333... is repeating, not terminating.
Write \(0.555...\) as a fraction.
Pattern with ninths.
\(\frac{5}{9}\).
Which fraction equals \(0.777...\)?
Use 9 pattern.
\(\frac{7}{9}\).
Write \(0.888...\) as a fraction.
Follow the pattern.
\(\frac{8}{9}\).
Higher
Higher Practice
Convert recurring decimals to fractions using algebra.
Let \(x = 0.3\overline{3}\). What is \(10x\)?
Multiply by 10 shifts decimal.
\(10x = 3.333...\).
Let \(x = 0.3\overline{3}\). Find \(9x\).
Subtract x from 10x.
10x - x = 3 → 9x = 3.
Convert \(0.3\overline{3}\) to a fraction.
Solve 9x = 3.
\(x = \frac{1}{3}\).
Convert \(0.6\overline{6}\) to a fraction.
Use same method.
\(\frac{2}{3}\).
Convert \(0.12\overline{12}\) to a fraction.
Two repeating digits → divide by 99.
\(\frac{12}{99} = \frac{4}{33}\).
Convert \(0.27\overline{27}\) to a fraction.
Two repeating digits.
27/99 simplifies to 3/11.
Convert \(0.1\overline{6}\) to a fraction.
Use algebra method.
0.1666... = 1/6.
Convert \(0.8\overline{3}\) to a fraction.
Split into 0.8 + recurring part.
0.8333... = 5/6.
A student says \(0.27\overline{27} = \frac{27}{100}\). What is wrong?
Recurring decimals use 9s, not 10s.
Should divide by 99, not 100.
Convert \(0.09\overline{09}\) to a fraction.
Two repeating digits.
Simplifies to 1/11.
Games
Practise this topic with interactive games.
Games coming soon.
Recurring Decimals Video Tutorial
Frequently Asked Questions
What is a recurring decimal?
A decimal where one or more digits repeat indefinitely.
How do I convert to a fraction?
Use algebra to eliminate the repeating part.
Why is this important?
It helps express numbers exactly rather than approximately.