Recurring decimals as fractions
Convert a recurring decimal to a fraction by setting up an equation that eliminates the recurring part.
Higher tier method: let x = the recurring decimal, multiply by an appropriate power of 10 to align the recurring blocks, subtract.
Worked examples
0.˙3 = 1/3. 10x = 3.˙3, x = 0.˙3, 9x = 3, x = 1/3.
0.˙1˙2 = 12/99 = 4/33. 100x − x = 12.
0.5˙7 = 0.5777... = 26/45. Mixed: handle the non-recurring part first.
Frequently asked questions
Notation?
Dots over recurring digits: 0.˙3 means 0.333... 0.˙1˙7 means 0.171717...
Always converts to a fraction?
Yes — every recurring decimal is rational. Non-recurring (e.g. π, √2) are irrational.