Terminating and recurring decimals | Year 8

01 / LEARN

The idea, step by step

What happens in division

A decimal terminates when the division reaches remainder zero. It recurs when a nonzero remainder repeats, because the same division steps will then repeat. An overbar marks precisely the repeating digits: $0.1 \bar{6}$ means 0.1666…, not 0.161616…. Every terminating or recurring decimal represents a rational number.

Predict and round

Reduce a fraction before inspecting its denominator. A denominator with only factors 2 and 5 produces a terminating decimal; other prime factors produce recurrence. When a question asks for a rounded decimal, inspect the next digit and use an approximation sign. Do not write a rounded decimal as though it were the exact original value.

A common trap

Every decimal that continues forever is irrational.

Instead: A repeating infinite decimal is rational. For example, $0. \bar{3} = \frac{1}{3}$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

A terminating fraction

Write $\frac{7}{8}$ as a decimal.

$\frac{7}{8} = \frac{875}{1000}$ , because $8 \times 125 = 1000$ .
Therefore the decimal is 0.875, exactly.

0.875

Example 2

Find the repeating digit

Write $\frac{5}{6}$ as a recurring decimal.

Dividing 5 by 6 gives 0.8 with a remainder of 2 after the tenths step.
The next step divides 20 by 6: digit 3 and remainder 2. That same remainder repeats.

0.8 \bar{3}

Check your understanding

Does $\frac{3}{40}$ terminate? Give its exact decimal.

Show answer

Yes; 0.075

$40 = 2^{3} \times 5$ and $\frac{3}{40} = \frac{75}{1000} = 0.075$ .

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Write $\frac{3}{4}$ as a decimal.
2 marks
Write $\frac{7}{20}$ as a decimal.
2 marks
Write $\frac{2}{3}$ using recurring-decimal notation.
2 marks
Write $\frac{11}{6}$ using recurring-decimal notation.
2 marks
Decide whether $\frac{6}{15}$ terminates, and explain.
2 marks
$\frac{2}{7} = 0.285714285714 \dots$ . Round it to three decimal places.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Write $\frac{3}{4}$ as a decimal.
2 marks
$\frac{3}{4} = \frac{75}{100} = 0.75$ .
0.75
Write $\frac{7}{20}$ as a decimal.
2 marks
$\frac{7}{20} = \frac{35}{100} = 0.35$ .
0.35
Write $\frac{2}{3}$ using recurring-decimal notation.
2 marks
Division gives 0.6666…, with the digit 6 repeating.
$0. \bar{6}$
Write $\frac{11}{6}$ using recurring-decimal notation.
2 marks
$\frac{11}{6} = 1 + \frac{5}{6} = 1.8333 \dots$ .
$1.8 \bar{3}$
Decide whether $\frac{6}{15}$ terminates, and explain.
2 marks
$\frac{6}{15} = \frac{2}{5} = 0.4$ ; reduce before checking the denominator.
It terminates at 0.4.
$\frac{2}{7} = 0.285714285714 \dots$ . Round it to three decimal places.
2 marks
Keep 0.285 initially; the next digit is 7.
Round the thousandths digit up, giving 0.286.
0.286

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Four operations with integers; signed fraction/decimal calculations; order of operations; terminating and recurring decimals; rational and irrational numbers; estimating square roots and recognising pi; choosing efficient exact/decimal representations.

Official curriculum source ↗

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