Improve Tuition · Non-Calculator Maths · Year 8–GCSE
Lesson · Practice · Steps
Recurring Decimals
The dots, the division that never ends, and turning it all back into a fraction.
14 questions · 3 sections · two interactive steppers · no time pressure
What's really happening
A recurring decimal has digits that repeat forever. We don't write them out forever — we mark the repeating part with dots. The straightforward conversions between fractions, decimals and percentages live on the Fractions, Decimals & Percentages page; this page is about the recurring case.
The dot rule: one dot marks a single repeating digit; two dots mark the first and last digit of a repeating block — everything between them repeats too.
0.5 = 0.5555… 0.45 = 0.454545… 0.16 = 0.1666…
One dot · two dots (the 45 block) · a non-repeating 1 then a repeating 6.
1 · Fraction → recurring decimal
Divide the top by the bottom and watch the remainders. When a remainder comes back, the digits start repeating. Tap through 2 ÷ 11:
Fraction → recurring decimal2/11 as a decimal
2 · Recurring decimal → fraction
This is the clever one. Call the decimal x, multiply so one whole block shifts past the point, then subtract — the endless tails cancel. Tap through 0.72:
Recurring decimal → fraction0.72̇ (72 repeats) as a fraction
Watch outOnly the digits under the dots repeat. In 0.16 the 1 does not repeat, so you multiply by 10 and 100 and subtract those — that's why it gives 1/6, not 16/99.
Learning Objective 1 · Year 8–9
Section A · Fraction → recurring decimal
Find the recurring decimal of a fraction by division, and identify the repeating part.
Success criteria — I can:
divide to get the decimal
spot when a remainder repeats
name the digit or block that recurs
Learning Objective 2 · GCSE (Grade 5+)
Section B · Recurring decimal → fraction
Convert a recurring decimal into a fraction using the algebra method, and simplify.
Success criteria — I can:
set x equal to the decimal
multiply to shift one repeating block
subtract, solve and simplify
Learning Objective 3 · GCSE
Section C · Mixed & compare
Work in both directions and compare a recurring decimal with a fraction.
Success criteria — I can:
convert either way confidently
handle a non-repeating digit before the block
compare values across forms
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