Recurring decimal to fraction
Converting recurring decimals into fractions is a classic application of infinite geometric series, directly targeting the Sequences and Series topic in the HSC Mathematics Extension 1 syllabus. By working through these three decimals, you’ll see how to separate a decimal into a finite place‑value part and an infinite recurring part, then combine them to obtain exact rational forms.
Problem Statement
Convert each to a fraction:
Hints
Use place value for terminating decimals and GP ideas for recurring decimals.
Solutions
We start with the terminating decimal (0.2). Because it stops after one decimal place, place value tells us directly that (0.2 = \frac{2}{10}), which simplifies to (\frac15).
The pure recurring decimal (0.\dot{2} = 0.222\ldots) never terminates, so we express it as an infinite geometric series. The first term is (\frac{2}{10}) and the common ratio is (\frac{1}{10}). Using the sum‑to‑infinity formula (S = \frac{a}{1-r}), we get (S = \frac{2/10}{1-1/10} = \frac{2}{9}).
For the mixed recurring decimal (0.1\dot{2}), the digit 1 is a terminating part and the \dot{2} repeats indefinitely. We split it as (0.1 + 0.0\dot{2}). The recurring part (0.0\dot{2} = 0.0222\ldots) is itself a geometric series with first term (\frac{2}{100}) and ratio (\frac{1}{10}), giving (\frac{2/100}{1-1/10} = \frac{2}{90} = \frac{1}{45}). The terminating part (0.1) is (\frac{1}{10} = \frac{9}{90}). Adding them together, (\frac{9}{90} + \frac{2}{90} = \frac{11}{90}).
Takeaways
- Terminating decimals convert directly via place value; recurring decimals are handled by recognising them as infinite geometric series.
- A pure recurring decimal with a single repeating digit has denominator 9; longer repeating blocks follow the same GP pattern with denominators like 99, 999, etc.
- Mixed recurring decimals are best split into a terminating part and a pure recurring part, solved separately, then combined.
Further Readings
HSC Polynomials, HSC Combinatorics, HSC Functions, HSC Proofs