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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.

In one line: Only 2 and 5 make the decimal stop.

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Understand it

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.

Example

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

Formula and rule

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

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Check 17/12.

  1. 1It is already in lowest form, and 12 = 2² × 3.
  2. 2Since 3 is present, 17/12 has a non-terminating repeating decimal expansion.

Method you can reuse

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

Sketch it in your answer

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Will 13/40 have a terminating decimal?

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

Common wrong answer

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

Don't mix this up

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

Exam tip

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

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Questions ask whether a decimal will terminate without division, or ask students to classify decimals as rational.
  1. 1

    Enough for 1 mark

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

  2. 2

    Add for 2 marks

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

  3. 3

    Add for 3 marks

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

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