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Decimal Expansions — Terminating and Non-Terminating Repeating

A rational number has either a terminating decimal expansion or a non-terminating repeating decimal expansion.

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Student-friendly explanation

When a rational number is written in lowest form, its decimal terminates if the denominator has only factors 2 and/or 5. If the denominator has any other prime factor, the decimal repeats.

How to write this in exams

  1. 1

    Start with the exact idea

    A rational number has either a terminating decimal expansion or a non-terminating repeating decimal expansion.

  2. 2

    Then show how to use it

    Reduce p/q to lowest form, factorise the denominator, check whether only 2 and 5 appear, then write terminating or non-terminating repeating.

  3. 3

    Add one concrete example

    7/8 = 0.875 terminates because 8 = 2³. But 1/3 = 0.333... repeats because the denominator has factor 3.

  4. 4

    Avoid this incomplete answer

    Saying 17/12 terminates because 12 is even is wrong; factor 3 is also present, so the decimal repeats.

Definition

A rational number has either a terminating decimal expansion or a non-terminating repeating decimal expansion.

Example

7/8 = 0.875 terminates because 8 = 2³. But 1/3 = 0.333... repeats because the denominator has factor 3.

Rule to remember

For p/q in lowest form, the decimal terminates if q = 2^m × 5^n; otherwise it is non-terminating repeating.

Memory hook

Only 2 and 5 make the decimal stop.

Examples and method

Worked example

Check 17/12. It is already in lowest form, and 12 = 2² × 3. Since 3 is present, 17/12 has a non-terminating repeating decimal expansion.

Method to apply

Reduce p/q to lowest form, factorise the denominator, check whether only 2 and 5 appear, then write terminating or non-terminating repeating.

Diagram support

Use a denominator-factor flowchart: reduce fraction, factor denominator, then decide terminating or repeating.

How CBSE asks it

Questions ask whether a decimal will terminate without division, or ask students to classify decimals as rational.

Avoid common mistakes

Common confusion

Students often divide only for a few steps and call a repeating decimal irrational because it does not end.

Common wrong answer

Saying 17/12 terminates because 12 is even is wrong; factor 3 is also present, so the decimal repeats.

Exam tip

Before long division, reduce the fraction to lowest form and factorise the denominator.

Quick check

Will 13/40 have a terminating decimal?

Yes, 13/40 has a terminating decimal because 40 = 2³ × 5, so the denominator has only factors 2 and 5.

Answer writing and exam use

1-mark answer

A rational number has either a terminating decimal expansion or a non-terminating repeating decimal expansion.

2-mark answer

A rational number has either a terminating decimal expansion or a non-terminating repeating decimal expansion. For p/q in lowest form, the decimal terminates if q = 2^m × 5^n; otherwise it is non-terminating repeating. 7/8 = 0.875 terminates because 8 = 2³. But 1/3 = 0.333... repeats because the denominator has factor 3.

3-mark answer

When a rational number is written in lowest form, its decimal terminates if the denominator has only factors 2 and/or 5. If the denominator has any other prime factor, the decimal repeats. For p/q in lowest form, the decimal terminates if q = 2^m × 5^n; otherwise it is non-terminating repeating. Check 17/12. It is already in lowest form, and 12 = 2² × 3. Since 3 is present, 17/12 has a non-terminating repeating decimal expansion. Questions ask whether a decimal will terminate without division, or ask students to classify decimals as rational. Saying 17/12 terminates because 12 is even is wrong; factor 3 is also present, so the decimal repeats.
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