Recurring Decimals to Fractions
Master recurring decimals to fractions for GCSE Maths with structured, exam-style practice. This Higher resource covers converting recurring decimals to fractions and includes a complete mark scheme showing the steps examiners reward. Suitable for AQA, Edexcel and OCR. Free to download as a PDF. Multiply by a power of 10, then subtract to eliminate the recurring part.
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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.
This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.
Topic overview
A recurring decimal has a digit or block of digits repeating forever, such as \(0.\dot{3}\) or \(0.\dot{1}\dot{8}\). Every recurring decimal can be written as an exact fraction.
The algebraic method works by multiplying the decimal by a power of ten so the repeating parts line up, then subtracting. The infinite tail cancels, leaving an ordinary equation to solve.
Which power of ten you use depends on how many digits repeat. One repeating digit needs \(10\), two need \(100\), three need \(1000\). Getting this right is the whole method, because if the tails do not align the subtraction leaves a recurring decimal behind.
Revision notes
Setting up the algebra
Let \(x\) equal the recurring decimal. Multiply by \(10\) for one repeating digit, \(100\) for two, and so on.
For \(0.\dot{4}\): let \(x = 0.444\ldots\), then \(10x = 4.444\ldots\). The decimal parts now match exactly.
Subtracting to remove the tail
Subtract the smaller equation from the larger. The infinite decimal parts cancel, leaving whole numbers.
\(10x - x = 4.444\ldots - 0.444\ldots\) gives \(9x = 4\), so \(x = \frac{4}{9}\).
Two or more repeating digits
With a two-digit block, multiply by \(100\) so the blocks align.
For \(0.\dot{2}\dot{7}\): \(100x = 27.2727\ldots\) and \(x = 0.2727\ldots\), so \(99x = 27\) and \(x = \frac{27}{99} = \frac{3}{11}\).
Key points
- Every recurring decimal equals an exact fraction.
- Let \(x\) be the decimal, then multiply by a power of ten.
- One repeating digit uses 10, two use 100.
- Subtract to cancel the infinite tail.
- Solve the resulting equation for \(x\).
- Simplify the final fraction.
Worked examples
Example 1
Write \(0.\dot{7}\) as a fraction.
Working
Example 2
Write \(0.\dot{3}\dot{6}\) as a fraction in its simplest form.
Working
Example 3
Write \(0.1\dot{2}\) as a fraction.
Working
Common mistakes
Using the wrong power of ten.
Two repeating digits need 100, not 10, or the tails will not cancel.
Forgetting to simplify.
36/99 must be given as 4/11 when the simplest form is asked for.
Mishandling a non-recurring digit at the start.
For 0.1̇2 you must shift twice, subtracting 10x from 100x rather than x from 100x.
Subtracting the larger from the smaller.
Always subtract the smaller equation from the larger so the coefficient of x is positive.
Exam tips
- Write out several digits of the decimal so you can see the repeating block.
- Count the repeating digits to choose the power of ten.
- Set the working out as two labelled equations before subtracting.
- Simplify the fraction and check by dividing it back.
Key terms
- Recurring decimal
- A decimal with a digit or block repeating forever.
- Dot notation
- Dots marking the repeating digits, such as \(0.\dot{3}\).
- Terminating decimal
- A decimal that ends after finitely many digits.
- Simplest form
- A fraction reduced to its smallest whole numbers.
Related topics
Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.