Year 8 - Fractions and decimals - Free practice

Year 8 Fractions, Terminating and Recurring Decimals Worksheets & Practice

Convert between fractions and decimals, from neat terminating ones to the tricky repeating kind.

Quick answer

How do you convert fractions to terminating or recurring decimals?

Divide the numerator by the denominator. If the denominator's only prime factors are 2s and 5s, the decimal terminates (e.g. 3/8 = 0.375). Otherwise it recurs. Write recurring decimals with a dot above the first and last digit of the repeating block (e.g. 1/3 = 0.3̇, 1/7 = 0.1̇4285̇7).

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Six questions across Foundation, Standard and Extension, with full worked solutions, written to the Australian Curriculum.

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Worked examples

Year 8 fractions and decimals: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

Example 1 · Identifying and converting a fraction to a terminating decimal
Convert $\dfrac{9}{16}$ to a decimal and state whether it is terminating or recurring.
Show worked solution
Key idea: If the denominator's only prime factors are 2s and 5s, the decimal terminates. Divide to find the decimal.
1. Check the denominator: $16 = 2^4$ . Only 2s, so the decimal terminates.
2. Divide: $9 \div 16$ .
  $\begin{aligned} 9 \div 16 &= 0.5625 \end{aligned}$
3. Verify: $0.5625 \times 16 = 9$ ✓
Answer: $\dfrac{9}{16} = 0.5625$ (terminating decimal)
Example 2 · Converting a fraction to a recurring decimal using dot notation
Convert $\dfrac{5}{12}$ to a recurring decimal. Write your answer using dot notation.
Show worked solution
Key idea: Divide the numerator by the denominator. When a remainder repeats, the decimal recurs. Use a dot above the first and last repeating digit.
1. Check the denominator: $12 = 2^2 \times 3$ . It has a factor of 3 (not just 2s and 5s), so the decimal recurs.
2. Perform the division $5 \div 12$ :
  $\begin{aligned} 5 \div 12 &= 0.41666\ldots \end{aligned}$
3. The digit 6 repeats forever, so the recurring part is just 6.
4. In dot notation, place one dot above the 6: $0.41\dot{6}$
Answer: $\dfrac{5}{12} = 0.41\dot{6}$
Example 3 · Choosing container sizes using decimal conversions
A recipe needs $\dfrac{7}{9}$ of a litre of olive oil. A shop sells olive oil in bottles labelled $0.7\overline{7}$ litres and $0.\dot{7}\dot{8}$ litres. Which bottle size exactly matches the amount the recipe needs? Show all working.
Show worked solution
Key idea: Convert $\dfrac{7}{9}$ to a recurring decimal and compare with the two bottle labels.
1. Check the denominator: $9 = 3^2$ . It has no factors of 2 or 5, so the decimal recurs.
2. Divide: $7 \div 9 = 0.7777\ldots$
3. Write in dot notation: $0.\dot{7}$
4. Compare with the bottle labels:
- $0.7\overline{7} = 0.7777\ldots = 0.\dot{7}$ ✓
- $0.\dot{7}\dot{8} = 0.787878\ldots$ ✗
1. The first bottle, $0.7\overline{7}$ litres, matches $\dfrac{7}{9}$ exactly.
Answer: The $0.7\overline{7}$ litre bottle exactly matches the recipe.

Common mistakes

Where students usually lose marks

Putting dots over the wrong digits

$\dfrac{5}{12} = 0.41666\ldots$ written as $0.\dot{4}1\dot{6}$ instead of $0.41\dot{6}$

Dots go only above the digits that actually repeat. In $0.41666\ldots$ only the 6 repeats, so only the 6 gets a dot: $0.41\dot{6}$ . The digits 4 and 1 appear only once and are not part of the repeating block.

Thinking a large denominator always means a recurring decimal

$\dfrac{3}{40}$ labelled as recurring because 40 is large

The size of the denominator does not decide the decimal type. Write the denominator in prime factor form. If it contains only 2s and 5s (like $40 = 2^3 \times 5$ ), the decimal terminates, no matter how large the denominator looks.

Forgetting to simplify when converting a terminating decimal to a fraction

$0.64 = \dfrac{64}{100}$ left unsimplified

Always divide the numerator and denominator by their highest common factor. Here $\text{HCF}(64, 100) = 4$ , giving $\dfrac{64}{100} = \dfrac{16}{25}$ . An unsimplified fraction is not fully correct.

Stopping the division too early and missing the recurring pattern

$\dfrac{2}{3} = 0.6$ (division stopped after one step)

Keep dividing until you see a remainder that has appeared before. That is the signal that the digits are repeating. For $\dfrac{2}{3}$ , the remainder 2 recurs immediately, giving $0.\dot{6}$ , not $0.6$ .

Try one yourself

Your turn

Convert $\dfrac{9}{16}$ to a decimal.

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About this worksheet

Why we built it

In Year 8, students move beyond simple fractions to understand why some decimals terminate (like 0.25) and others repeat forever (like 0.333...). These worksheets build the skills to convert confidently in both directions, with worked solutions for every question so students can pinpoint exactly where their reasoning went off track.

What's covered

Sub-skills your student will practise

✓Identifying whether a fraction gives a terminating or recurring decimal
✓Converting fractions to terminating decimals by division
✓Converting fractions to recurring decimals and using dot or bar notation
✓Converting terminating decimals back to fractions in simplest form
✓Converting recurring decimals back to fractions
✓Using the denominator's prime factors (2s and 5s) to predict decimal type

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Frequently asked

Questions parents ask about fractions and decimals

What is the difference between a terminating and a recurring decimal?

A terminating decimal has a finite number of digits after the decimal point, like 0.75. A recurring decimal has one or more digits that repeat forever, like 0.333... Written with a dot or bar above the repeating part.

How do I know if a fraction will terminate or recur before I divide?

Write the fraction in simplest form and look at the denominator. If its only prime factors are 2s and 5s, the decimal terminates. Any other prime factor, like 3 or 7, means the decimal will recur.

How do you convert a recurring decimal back to a fraction?

Let x equal the recurring decimal, then multiply both sides by a power of 10 to shift one full repeating block. Subtracting the original equation cancels the repeating part, leaving a whole number you can divide to find the fraction.

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