Sheet № 127 · Higher only · AQA · Edexcel · OCR
Recurring Decimals to Fractions –
Converting recurring decimals to fractions is a Higher tier GCSE Maths topic that appears regularly on exam papers. The method uses algebra to eliminate the repeating part, producing a clean fraction.
§Key definitions
Question:
This topic is Higher only, but here is a straightforward single-digit recurrence. Convert 0.777... to a fraction.
Answer:
0.2181818... = 12/55 (proven)
Q1 (Higher):
Convert 0.444... to a fraction.
Q2 (Higher):
Convert 0.272727... to a fraction in its simplest form.
Q3 (Higher):
Prove that 0.1666... = 1/6.
§Formulas to memorise
Let x = the recurring decimal, multiply by 10^n where n = number of repeating digits, then subtract x
If one digit repeats: multiply by 10. If two digits repeat: multiply by 100. If three digits repeat: multiply by 1000
Let x equal the recurring decimal.
Worked example
This topic is Higher only, but here is a straightforward single-digit recurrence. Convert 0.777... to a fraction.
Working:
⚠ Common mistakes
- ✗Multiplying by the wrong power of 10. The power must match the number of repeating digits, not the total number of decimal places.
- ✗Forgetting to deal with non-repeating digits before the repeating block. When there are non-repeating digits, you need two multiplications and must subtract carefully.
- ✗Not simplifying the final fraction. Always divide numerator and denominator by their HCF.
✦ Exam tips
- →Set out your working clearly with "Let x = ..." — examiners look for this structure.
- →For "prove" or "show that" questions, you must reach the exact fraction given in the question.
- →Check your answer by dividing the numerator by the denominator on your calculator to verify you get the original recurring decimal.