Recurring decimals · Grade 10
In Grade 10 you learn: recurring decimals as fractions.
Some fractions become decimals that never end but repeat a pattern for ever, like 1/3 = 0.333…. Every recurring decimal is a fraction in disguise, and a neat bit of algebra (multiplying by 10, 100 or 1,000 and subtracting) turns it back into one.
Key facts
- Dots mark the repeating digits: 0.3̇ = 0.333…, 0.1̇2̇ = 0.121212…, 0.1̇23̇ = 0.123123….
- A fraction (in simplest form) terminates only if its denominator has no prime factors other than 2 and 5.
- One repeating digit: multiply by 10. Two: multiply by 100. Three: by 1,000.
- Then subtract to cancel the repeating part: 100x − x = 99x.
- Shortcuts: 0.ȧ = a/9, 0.ȧḃ = ab/99.
- 0.9̇ = 1 exactly.
Worked examples
Example 1
Write 0.7̇ as a fraction.
Answer: 7/9
Example 2
Write 0.3̇6̇ as a fraction in its simplest form.
Answer: 4/11
Example 3
Write 0.25̇ (0.2555…) as a fraction.
Answer: 23/90