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Year 12 Maths Advanced (2027) Sequences and series

Recurring Decimals as Infinite GPs

20 practice questions 2 video lessons Theory + worked examples

Learn to write recurring decimals as infinite GPs for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-03). A decimal such as \(0.272727\ldots\) is really a sum of ever-smaller pieces that form a geometric series, so its exact value can be found with a limiting sum.

You will learn to split a recurring decimal into its repeating blocks, identify the first term and common ratio, apply the limiting sum and simplify the answer to an exact fraction — a clear example of an infinite process giving a finite, exact result.

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  • Repeating decimal as infinite geometric series | Precalculus | Khan Academy Watch
  • Repeating Decimal to Fraction Using Geometric Series(Challenging) Watch
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Theory

Every recurring decimal is an infinite geometric series, so the limiting sum turns it into an exact fraction. This Year 12 Mathematics Advanced topic converts recurring decimals to fractions.

A recurring decimal repeats a block of digits forever, which is a geometric series with \(|r|<1\). Summing it gives an exact fraction.

Shortcut: one repeating digit is over \(9\) (\(0.\dot{d}=\dfrac{d}{9}\)); a two-digit block is over \(99\); a three-digit block over \(999\). A useful check is \(0.\dot{9}=\dfrac{9}{9}=1\).

A recurring decimal as a shrinking sumThe parts of 0.6 recurring form a geometric series 0.6, 0.06, 0.006 that shrink toward zero. 0.6 0.06 0.006 ...
\(0.\dot{6}=0.6+0.06+0.006+\cdots\), a geometric series.
\[0.\dot{d}=\dfrac{d}{9},\qquad 0.\dot{a}\dot{b}=\dfrac{ab}{99}\]
0 point d recurring equals d over 9; 0 point a b recurring equals a b over 99

Method

  1. Identify the repeating block and its length.
  2. Write it over \(9\), \(99\), \(999\), \(\ldots\) (matching the block length), or sum it as a GP.
  3. Simplify the fraction.
Example 1 — One digit
Write \(0.\dot{4}\) as a fraction.
Solution
\(0.\dot{4}\)\(=\)\(\dfrac{4}{9}\)
Example 2 — Two digits
Write \(0.\dot{2}\dot{7}\) in simplest form.
Solution
\(0.\dot{2}\dot{7}\)\(=\)\(\dfrac{27}{99}=\dfrac{3}{11}\)
Example 3 — Via the GP sum
Use \(S_\infty\) to write \(0.\dot{6}\) as a fraction.
Solution
\(0.\dot{6}\)\(=\)\(\dfrac{0.6}{1-0.1}\)
\(=\)\(\dfrac{0.6}{0.9}=\dfrac{2}{3}\)
Example 4 — Two conversions
Write \(0.\dot{1}\dot{2}\) and \(0.\dot{9}\) as fractions.
Solution
\(0.\dot{1}\dot{2}\)\(=\)\(\dfrac{12}{99}=\dfrac{4}{33}\)
\(0.\dot{9}\)\(=\)\(\dfrac{9}{9}=1\)

Common pitfalls

Match block length to the denominator. One digit over \(9\); two digits over \(99\).
Only the repeating part is the GP. Handle any non-repeating start separately.
Always simplify (e.g. \(\dfrac{27}{99}=\dfrac{3}{11}\)).

Frequently asked questions

How do you turn a recurring decimal into a fraction?

A recurring decimal is an infinite geometric series. Put the repeating block over as many nines as its length: one digit over 9, two digits over 99, then simplify.

What is 0.6 recurring as a fraction?

It is a geometric series with first term 0.6 and ratio 0.1, so the sum is 0.6 divided by 0.9, which simplifies to two thirds.

Is 0.9 recurring equal to 1?

Yes. As a fraction 0.9 recurring is 9 over 9, which equals exactly 1.

What is 0.27 recurring as a fraction?

The two-digit block 27 goes over 99, giving 27 over 99, which simplifies to 3 over 11.

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