Step-by-step worked examples and graded practice questions on decimals — converting between fractions and decimals, and the four operations with decimal numbers.
📚 Foundation & Higher✅ 15 Practice Questions🔍 Full Worked Examples⚠️ Common Mistakes
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A decimal is another way of writing a fraction, using place value columns to the right of a decimal point — tenths, hundredths, thousandths and so on (see the Place Value page for the full column chart). Decimals and fractions represent exactly the same values, just written differently.
Converting between fractions and decimals
Worked Example 1 — Fraction to decimal
Convert 5/8 to a decimal.
1
A fraction means "numerator ÷ denominator" — divide 5 by 8
2
5 ÷ 8 = 0.625
Answer0.625
Worked Example 2 — Decimal to fraction
Convert 0.35 to a fraction in its simplest form.
1
Write the decimal over the place value of its last digit: 0.35 has 2 decimal places, so write it as 35/100
2
Simplify using the HCF of 35 and 100 (which is 5): 35 ÷ 5 = 7, 100 ÷ 5 = 20
Answer7/20
Adding and subtracting decimals
Always line up the decimal points before adding or subtracting — this keeps each digit in its correct place value column.
Worked Example 3
Work out (a) 3.45 + 2.7 and (b) 8.2 − 3.65
1
(a) Line up the decimal points, using a placeholder zero: 3.45 + 2.70 = 6.15
2
(b) Line up the decimal points: 8.20 − 3.65 = 4.55
Answer(a) 6.15 (b) 4.55
A placeholder zero was added to 2.7 so both numbers have 2 decimal places — this makes column addition straightforward.
Multiplying and dividing decimals
To multiply decimals, ignore the decimal points and multiply as whole numbers, then place the decimal point in the answer so it has the same total number of decimal places as both numbers combined. To divide, it's often easiest to first multiply both numbers by a power of 10 to remove the decimal point from the divisor.
Worked Example 4
Work out (a) 3.4 × 0.6 and (b) 7.2 ÷ 0.4
1
(a) Multiply as whole numbers: 34 × 6 = 204. Both numbers had 1 decimal place each (2 total), so place the point 2 digits from the right: 2.04
2
(b) Multiply both numbers by 10 to clear the decimal in the divisor: 7.2 ÷ 0.4 becomes 72 ÷ 4 = 18
Answer(a) 2.04 (b) 18
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1Convert 3/4 to a decimal.Foundation
Q2Convert 0.6 to a fraction in its simplest form.Foundation
Q3Work out 4.6 + 3.85Foundation
Q4Work out 7.2 − 4.55Foundation
Q5Work out 2.5 × 4Foundation
Higher (Grade 5–7)
Q6Convert 7/8 to a decimal.Higher
Q7Convert 0.24 to a fraction in its simplest form.Higher
Q8Work out 5.6 × 0.3Higher
Q9Work out 8.4 ÷ 0.7Higher
Q10Work out 12.6 − 3.482, giving your answer to 2 decimal places.Higher
Higher — Hard (Grade 8–9)
Q11Convert 5/6 to a decimal, giving your answer correct to 3 decimal places.Grade 8–9
Q12Convert the recurring decimal 0.7̇ (0.7777…) to a fraction.Grade 8–9
Q13Work out 0.35 × 0.04, giving your answer in standard form.Grade 8–9
Q14A number rounded to 1 decimal place is 6.4. Find the smallest possible value the number could have been.Grade 8–9
Q15Prove algebraically that the recurring decimal 0.9̇ (0.9999…) is exactly equal to 1.Grade 8–9
Answers
Foundation (Q1–Q5)
Q10.75
Q23/5
Q38.45
Q42.65
Q510
Higher (Q6–Q10)
Q60.875
Q76/25
Q81.68
Q912
Q109.12(12.600 − 3.482 = 9.118, rounded to 9.12)
Higher — Hard (Q11–Q15)
Q110.833(5 ÷ 6 = 0.8333…)
Q127/9
Q131.4 × 10⁻²(0.35 × 0.04 = 0.014)
Q146.35(the lower bound — any value from 6.35 up to but not including 6.45 rounds to 6.4)
Q15Let x = 0.9999…, so 10x = 9.9999…. Subtracting, 10x − x = 9x = 9.9999… − 0.9999… = 9, so x = 1.
Common mistakes
Common Mistake 1
Not lining up decimal points before adding or subtracting
3.45 + 2.7 is not 3.72. Line up the decimal points (adding a placeholder zero to 2.7 to make 2.70) before adding column by column.
Common Mistake 2
Miscounting decimal places when multiplying
3.4 × 0.6 has 1 + 1 = 2 decimal places in total, so the answer needs 2 decimal places: 2.04, not 20.4 or 0.204.
Common Mistake 3
Writing the wrong denominator when converting to a fraction
The denominator matches the number of decimal places: 1 decimal place → tenths (÷10), 2 decimal places → hundredths (÷100), and so on. 0.35 becomes 35/100, not 35/10.
Common Mistake 4
Rounding too early in a multi-step calculation
Keep full accuracy through every step of a calculation and round only the final answer — rounding partway through can shift your final answer outside the range examiners accept.
Exam tips
💡 Exam Tip 1
Add placeholder zeros to match decimal places
Before adding, subtracting, or comparing decimals, give every number the same number of decimal places using placeholder zeros — it removes almost all careless errors.
💡 Exam Tip 2
Estimate first to check your answer's size
Before multiplying or dividing decimals, do a rough estimate (e.g. 3.4 × 0.6 ≈ 3 × 0.5 = 1.5) — this catches decimal-point placement errors instantly.
💡 Exam Tip 3
Clear the divisor's decimal point when dividing
Multiply both numbers in a division by the same power of 10 to make the divisor a whole number first — dividing by a whole number is much less error-prone.
💡 Exam Tip 4
Know your common fraction-decimal equivalents
Memorising conversions like 1/4 = 0.25, 1/3 = 0.333…, and 1/8 = 0.125 saves valuable time on non-calculator questions.
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