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Let the decimal equal x, then multiply by 10 for one repeated digit or 100 for two, so the recurring part lines up. Free IGCSE Maths notes and 9 FAQs.
Recurring Decimals as Fractions is part of Fractions, Decimals and Percentages in IGCSE Edexcel Higher Maths.
1 min
Reading time
30
Lessons in this topic
IGCSE Edexcel Higher
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Reviewed by Revise Right
Content review
Let the decimal equal x, then multiply by 10 for one repeated digit or 100 for two, so the recurring part lines up. Free IGCSE Maths notes and 9 FAQs.
Use this lesson to stay inside Fractions, Decimals and Percentages while connecting the detail back to the wider Topic 1: Numbers and the Number System topic.
Read the core explanations from Recurring Decimals as Fractions before testing yourself from memory.
A recurring decimal is not an approximation. It stands for a precise number, so it can be written as an exact fraction.
The key idea is to line up the repeated block, subtract, and then solve the resulting equation cleanly.
Worked example: Converting 0.3 recurring
Write 0.333... as a fraction.
Answer: 0.333... = 1/3
Worked example: Converting 0.27 recurring
Write 0.272727... as a fraction.
Answer: 0.272727... = 3/11
Worked example: A Mixed Decimal: 0.1666...
Write 0.1666... as a fraction.
Answer: 0.1666... = 1/6
Do not round the decimal and guess the fraction. In an exam, the method earns marks because it proves the exact value.
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Move to the related lessons or back to the topic page when you need the wider sequence before exam-style practice.
Let the decimal equal x, then multiply by 10 for one repeated digit or 100 for two, so the recurring part lines up. Subtract the original equation to remove the recurring tail, solve for x and simplify. For 0.333..., 10x - x = 3, so 9x = 3 and x = 1/3.
A recurring decimal is a decimal in which one digit or a block of digits repeats forever, such as 0.333.... It is not an approximation: it stands for a precise number, so it can be written as an exact fraction.
Let x = 0.272727.... The repeated block is 27, so multiply by 100: 100x = 27.272727.... Subtracting gives 99x = 27, so x = 27/99, which simplifies to 3/11.
Let x = 0.1666.... Only the 6 repeats, so line up the recurring tail with 10x = 1.666... and 100x = 16.666.... Subtracting gives 90x = 15, so x = 15/90 = 1/6.
Multiply by the power of 10 that shifts the decimal by the length of the repeated block: one repeated digit usually means multiply by 10, and two repeated digits usually means multiply by 100. Multiplying by the wrong power is a common pitfall when the repeated block is not identified correctly.
0.3 recurring (0.333...) is 1/3 as a fraction. Let x = 0.333..., so 10x = 3.333...; subtracting gives 9x = 3, so x = 3/9 = 1/3.
A terminating decimal stops after a fixed number of digits, while a recurring decimal has one digit or a block of digits that repeats forever. A recurring decimal still stands for a precise number, so it can be written as an exact fraction.
Subtracting removes the recurring tail: when the multiplied equation is lined up with the original, the repeating parts cancel. For example, 100x - x = 27.272727... - 0.272727... gives 99x = 27.
A recurring decimal is exact, not an approximation, so it can be written as an exact fraction with no rounding. In an exam, rounding the decimal and guessing the fraction does not prove the exact value.