Step-by-step worked examples and graded practice questions on simplifying ratios — using the highest common factor, and handling decimals, fractions and mixed units.
📚 Foundation & Higher✅ 15 Practice Questions🔍 Full Worked Examples⚠️ Common Mistakes
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Simplifying a ratio means writing it using the smallest possible whole numbers, while keeping the same relationship between the quantities — in the same way a fraction like 6⁄8 simplifies to 3⁄4.
A ratio is fully simplified when the only common factor of all its parts is 1.
Finding the highest common factor
Worked Example 1
Simplify the ratio 18 : 24.
1
List the factors of 18: 1, 2, 3, 6, 9, 18
2
List the factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
3
The highest common factor (HCF) is 6. Divide both sides by 6: 18 ÷ 6 = 3, and 24 ÷ 6 = 4
Answer3 : 4
Both bars are drawn to the same scale, so you can see 24 is genuinely longer than 18. Splitting each into groups of 6 (the HCF) leaves 3 groups and 4 groups — the simplest form of the ratio.
Simplifying ratios with decimals
If a ratio contains a decimal, first multiply every part by a power of 10 to clear the decimal point, then simplify as normal.
Worked Example 2
Simplify the ratio 0.4 : 1.6.
1
Multiply both sides by 10 to remove the decimals: 0.4 × 10 = 4, and 1.6 × 10 = 16
2
Now simplify 4 : 16 using the HCF, which is 4: 4 ÷ 4 = 1, and 16 ÷ 4 = 4
Answer1 : 4
Simplifying ratios with fractions
If a ratio contains a fraction, multiply every part by the denominator (or a common denominator, if there's more than one fraction) to turn it into whole numbers first.
Worked Example 3
Simplify the ratio 2⁄3 : 4.
1
Multiply both sides by 3 to clear the fraction: (2⁄3) × 3 = 2, and 4 × 3 = 12
2
Simplify 2 : 12 using the HCF, which is 2: 2 ÷ 2 = 1, and 12 ÷ 2 = 6
Answer1 : 6
Simplifying ratios in different units
Before simplifying, both quantities must be written in the same unit. Convert one of them first, then simplify as normal.
Worked Example 4
Simplify the ratio 750g : 2kg.
1
Convert 2kg into grams: 2kg = 2000g
2
Simplify 750 : 2000 using the HCF, which is 250: 750 ÷ 250 = 3, and 2000 ÷ 250 = 8
Answer3 : 8
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1Simplify the ratio 12 : 20.Foundation
Q2Simplify the ratio 15 : 35.Foundation
Q3Simplify the ratio 100 : 45.Foundation
Q4Simplify the ratio 500g : 1kg.Foundation
Q5Simplify the ratio 30 minutes : 2 hours.Foundation
Higher (Grade 5–7)
Q6Simplify the ratio 0.8 : 2.Higher
Q7Simplify the ratio 1.2 : 0.3.Higher
Q8Simplify the ratio 3⁄4 : 5.Higher
Q9Simplify the ratio 250ml : 1.5 litres.Higher
Q10Simplify the ratio 40cm : 1.2m.Higher
Higher — Hard (Grade 8–9)
Q11Simplify the ratio 1⁄2 : 2⁄3.Grade 8–9
Q12Simplify the ratio 0.75 : 1⁄4.Grade 8–9
Q13Simplify the ratio 2.5kg : 750g : 1.25kg.Grade 8–9
Q14Simplify the ratio x² : x³ (x ≠ 0), giving your answer in its simplest algebraic form.Grade 8–9
Q15Simplify the ratio 2⁄5 : 3⁄10 : 1⁄2.Grade 8–9
Answers
Foundation (Q1–Q5)
Q13 : 5
Q23 : 7
Q320 : 9
Q41 : 2(500g : 1000g)
Q51 : 4(30 min : 120 min)
Higher (Q6–Q10)
Q62 : 5(8 : 20)
Q74 : 1(12 : 3)
Q83 : 20(3 : 20 after ×4)
Q91 : 6(250ml : 1500ml)
Q101 : 3(40cm : 120cm)
Higher — Hard (Q11–Q15)
Q113 : 4(×6: 3 : 4)
Q123 : 1(0.75 : 0.25)
Q1310 : 3 : 5(2500g : 750g : 1250g, ÷250)
Q141 : x(divide both sides by x²)
Q154 : 3 : 5(×10: 4 : 3 : 5)
Common mistakes
Common Mistake 1
Dividing by a common factor instead of the highest common factor
18:24 can be divided by 2 to give 9:12, but this isn't fully simplified. Always find the highest common factor to reach the simplest form in one step.
Common Mistake 2
Simplifying before converting to the same units
750g : 2kg is not 750:2 — the units must match before simplifying. Convert first, then simplify.
Common Mistake 3
Only clearing the decimal or fraction on one side
When multiplying to remove a decimal or fraction, the same multiplier must be applied to every part of the ratio, not just the one containing the decimal or fraction.
Common Mistake 4
Forgetting a ratio can have three or more parts
The same HCF method applies to ratios with three or more parts — find a factor common to all parts, not just the first two.
Exam tips
💡 Exam Tip 1
Clear decimals and fractions before finding the HCF
It's much easier to spot common factors once every part of the ratio is a whole number — always convert decimals and fractions first.
💡 Exam Tip 2
Check your answer can't be simplified further
After simplifying, quickly check whether the resulting numbers still share a common factor — if they do, the ratio isn't fully simplified yet.
💡 Exam Tip 3
Use prime factors for large or awkward numbers
For ratios with large numbers, breaking each part into prime factors makes the highest common factor much easier to spot than listing every factor.
💡 Exam Tip 4
Convert to the smaller unit when mixing units
Converting to the smaller unit (e.g. grams rather than kilograms) usually avoids decimals appearing in the ratio, keeping the working simpler.
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