Year 8 Mathematics · AC9M8N03

Terminating and Recurring Decimals

A rational number’s decimal terminates or eventually repeats. Simplify the fraction first, inspect the denominator’s prime factors, and use digital tools to confirm…

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Key conceptTeach from the board

For a fraction in simplest form, the decimal terminates exactly when the denominator has no prime factors other than 2 and/or 5. This works because powers of 10 contain only factors 2 and 5.

If another prime factor remains, the decimal repeats. For example, 7/25=0.28 terminates, while 7/24=0.291666… recurs because 24=2³×3 contains a factor 3.

Digital tools can expose longer repeating patterns, but the denominator structure explains why the pattern terminates or recurs.

Worked examplesWe do

Worked examples

AC9M8N03 - Terminating and Recurring Decimals
Example 1

Predict whether 3/8 terminates.

Example 2

Simplify 9/30 and predict the decimal type.

Example 3

Classify 5/12 without long division.

Example 4

Write 0.272727… with recurring notation.

Curriculum examplesCopied content

Australian Curriculum v9.0 — AC9M8N03: recognise terminating and recurring decimals, using digital tools as appropriate.

Victorian Curriculum F–10 Version 2.0 — Level 8, VC2M8N03:Partial.

NSW Mathematics K–10 Syllabus (2022) — Stage 4, MA4-FRC-C-01; MAO-WM-01:Partial.

FrameworkMappingRelationship
Australian CurriculumAC9M8N03Canonical
VictoriaVC2M8N03Partial
NSW Stage 4MA4-FRC-C-01; MAO-WM-01Partial
Questions and answersWith answers
  1. When does a rational decimal terminate? When its simplified denominator has only prime factors 2 and/or 5.
  2. Why simplify first? Common factors can remove primes that would otherwise give the wrong prediction.
  3. Are recurring decimals rational? Yes.
  4. What is the role of a digital tool? It can display/verify a pattern; denominator structure supplies the mathematical reason.
Practice and reviewReady for practice
  • Testing an unsimplified denominator. Simplify the fraction first.
  • All non-terminating decimals are irrational. Recurring decimals are rational.
  • Any denominator containing 2 or 5 terminates. It must contain only 2s and 5s after simplification.
  • Writing a few repeated digits without notation proves recurrence. Identify the repeating block clearly.
Curriculum alignmentStart here
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