Terminating and recurring decimals | Year 8

01 / LEARN

Understand the idea.

What happens in division

A terminating decimal ends because long division reaches remainder zero. A recurring decimal has a repeating block: when a nonzero remainder returns, the same next digit and remainder must follow again.

An overbar marks exactly the digits that repeat. Thus $0.1 \bar{6} = 0.1666 \dots$ , whereas $0. \bar{16} = 0.161616 \dots$ . Both recurring and terminating decimals are rational because they can be written as fractions of integers.

Predict and round

Simplify the fraction before inspecting its denominator. A reduced denominator containing only prime factors $2$ and $5$ divides a power of ten, so the decimal terminates. Any other prime factor forces recurrence rather than an exact finite decimal.

To round a recurring decimal, write enough repeated digits to inspect the first discarded place. Use $\approx$ for the rounded result; keep $=$ for the exact fraction or overbar form. A calculator's finite display may hide a continuing pattern.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A terminating fraction

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

Worked solution

1
Choose a power-of-ten denominator.
$8 \times 125 = 1000$
A denominator of one thousand expresses the fraction in thousandths.
2
Scale both parts of the fraction.
$\frac{7}{8} = \frac{7 \times 125}{8 \times 125} = \frac{875}{1000}$
Multiplying numerator and denominator by the same nonzero number preserves value.
3
Read the terminating decimal.
$\frac{875}{1000} = 0.875$
Eight hundred and seventy-five thousandths has exactly three decimal places.

Answer

0.875

EXAMPLE 02

Find the repeating digit

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

Worked solution

1
Find the tenths digit.
$50 = 6 \times 8 + 2$
After writing zero whole units, divide fifty tenths by six; the remainder is two tenths.
2
Find the next digit.
$20 = 6 \times 3 + 2$
The remaining two tenths become twenty hundredths, giving digit three and remainder two.
3
Identify the repeating remainder.
$\frac{5}{6} = 0.8333 \dots = 0.8 \bar{3}$
The same remainder two returns at every later step, so only the digit three repeats.

Answer

$0.8 \bar{3}$

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

Yes; 0.075

The reduced denominator $40 = 2^{3} \times 5$ has no prime factors except two and five. Multiplying by $25$ gives $\frac{3}{40} = \frac{75}{1000} = 0.075$ , exactly.

Keep practising

6 worksheet questions · 12 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

What happens in division

Predict and round

See the method in action.

A terminating fraction

Choose a power-of-ten denominator.

Scale both parts of the fraction.

Read the terminating decimal.

Find the repeating digit

Find the tenths digit.

Find the next digit.

Identify the repeating remainder.

Try it yourself.

Check your understanding

Keep practising