Converting between decimals and fractions
Every fraction with an integer numerator and denominator is either a terminating decimal or a recurring decimal. Converting in either direction is examined regularly, and the recurring-decimal trick is a popular Higher-tier question.
When is a decimal terminating?
A fraction in lowest terms gives a terminating decimal if and only if the prime factorisation of its denominator contains only 2s and/or 5s.
- ⅛ = 1/2³ → terminates: 0.125.
- 7/20 = 7/(2² × 5) → terminates: 0.35.
- 1/3 = 1/3 → does NOT terminate; recurs as 0.333…
- 1/6 = 1/(2 × 3) → recurs (the 3 in the denominator forces it).
- 1/40 = 1/(2³ × 5) → terminates.
Terminating decimals → fractions
Read the decimal, write it as a fraction over a power of ten, then simplify.
Worked example: 0.45 → 45/100 → divide top and bottom by 5 → 9/20. Worked example: 0.625 → 625/1000 → divide by 125 → 5/8.
For a decimal like 0.1875: it has 4 d.p. so put it over 10⁴ = 10,000. 1875/10000. Divide by 625 (or repeatedly by 5) → 3/16. Final: 3/16.
Recurring decimal notation
A dot (or bar) over a digit means it repeats forever.
- 0.3̇ = 0.333…
- 0.1̇6̇ = 0.161616…
- 0.4ı 5̇3̇ = 0.4535353… (only the 53 recurs).
In typed text we'll write recurring digits in parentheses or use overlines: 0.(3) for 0.333… and 0.4(53) for 0.4535353… The exam paper itself uses dots above the recurring digits.
Recurring decimals → fractions [Higher tier]
The "let x equal" method.
Single-digit recurring
Convert 0.7̇ = 0.777… to a fraction.
- Let x = 0.777…
- 10x = 7.777…
- Subtract: 9x = 7.
- x = 7/9.
Two-digit recurring block
Convert 0.27̇27̇ = 0.272727… to a fraction.
- Let x = 0.272727…
- 100x = 27.272727…
- Subtract: 99x = 27.
- x = 27/99 = 3/11.
Pattern
If a single block of n digits recurs, multiply by 10ⁿ. The denominator before simplifying will be a string of n nines.
- 0.5̇ = 5/9.
- 0.45̇ recurring as a 2-digit block = 45/99 = 5/11.
- 0.123̇ (three-digit block) = 123/999 = 41/333.
Mixed (some digits don't recur)
Convert 0.16̇ = 0.1666… to a fraction.
- Let x = 0.1666…
- 10x = 1.666… (one digit, the 6, is now in front of the recurring tail).
- 100x = 16.666….
- Subtract 100x − 10x: 90x = 15.
- x = 15/90 = 1/6.
The strategy: shift the decimal so the same recurring tail appears in both versions, then subtract.
⚠Common mistakes— Common mistakes (examiner traps)
- Using the wrong number of nines. A 2-digit recurring block needs 99 in the denominator, not 9.
- Forgetting to simplify the final fraction. AQA marks lost almost every time.
- Multiplying by 10 once and stopping when the recurring part is two or more digits.
- Mixing up which subtraction to do when there are leading non-recurring digits — sketch both versions of x and ensure the recurring tails align.
- Calling 0.5 "recurring". It terminates after one digit. Recurring means the digits repeat forever.
➜Try this— Quick check
Convert 0.81̇81̇ to a fraction in lowest terms.
- Let x = 0.818181…; 100x = 81.818181…
- Subtract: 99x = 81 → x = 81/99 = 9/11.
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