Step-by-step worked examples and graded practice questions on direct proportion — the unitary method, the constant of proportionality, proportion graphs, and proportion with squares and cubes.
📚 Foundation & Higher✅ 15 Practice Questions🔍 Full Worked Examples⚠️ Common Mistakes
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Two quantities are in direct proportion if, whenever one increases, the other increases by the same scale factor — and whenever one decreases, the other decreases by the same scale factor. Doubling one quantity always doubles the other; trebling one always trebles the other.
Worked Example 1
4 pens cost £3.20. Find the cost of 7 pens, assuming the cost is directly proportional to the number of pens.
1
Find the cost of 1 pen (the unitary method): £3.20 ÷ 4 = £0.80
2
Multiply by the number of pens required: £0.80 × 7 = £5.60
Answer£5.60
The constant of proportionality
Direct proportion can be written as an equation: y = kx, where k is called the constant of proportionality. Once k is known, you can find y for any value of x without repeating the unitary method each time.
Worked Example 2
y is directly proportional to x. When x = 5, y = 15. Find the equation connecting y and x, then find y when x = 12.
1
Write the proportion equation: y = kx
2
Substitute the known values to find k: 15 = k × 5, so k = 3
3
Write the full equation, then substitute x = 12: y = 3x = 3 × 12 = 36
Answery = 3x, and y = 36 when x = 12
Direct proportion graphs
A graph of two directly proportional quantities is always a straight line through the origin (0,0). If the line doesn't pass through the origin, the quantities are not in direct proportion, even if the graph is a straight line.
The line passes through the origin — 0 items cost £0. Doubling the number of items (3 → 6) exactly doubles the cost (£1.50 → £3.00), confirming direct proportion.
Direct proportion with squares and cubes
Some quantities are proportional to a power of another quantity, such as y ∝ x² or y ∝ x³. The method is identical — just substitute the power into the proportion equation.
Worked Example 3
The area A of a circle is directly proportional to the square of its radius r. When r = 2cm, A = 12.56cm². Find A when r = 5cm.
1
Write the proportion equation: A = kr²
2
Substitute the known values to find k: 12.56 = k × 2², so 12.56 = 4k, giving k = 3.14
3
Substitute r = 5 into A = 3.14r²: A = 3.14 × 25 = 78.5
Answer78.5cm²
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q15 chocolate bars cost £4.50. Find the cost of 8 chocolate bars.Foundation
Q2A car travels 150 miles using 6 gallons of petrol. Assuming direct proportion, how far can it travel using 10 gallons?Foundation
Q33kg of apples costs £2.40. Find the cost of 7kg.Foundation
Q4y is directly proportional to x. When x = 4, y = 20. Find y when x = 9.Foundation
Q512 identical books weigh 3.6kg. Find the weight of 20 books.Foundation
Higher (Grade 5–7)
Q6y is directly proportional to x. When x = 6, y = 27. Find the value of x when y = 63.Higher
Q7The cost of a taxi journey is directly proportional to the distance travelled. A 15km journey costs £18. Find the cost of a 40km journey.Higher
Q8p is directly proportional to q. When q = 8, p = 20. Find the constant of proportionality, and write a formula connecting p and q.Higher
Q9y is directly proportional to x². When x = 3, y = 45. Find y when x = 5.Higher
Q10The volume V of a cube is directly proportional to the cube of its side length L. When L = 2cm, V = 16cm³. Find V when L = 5cm.Higher
Higher — Hard (Grade 8–9)
Q11y is directly proportional to x. When x = 12, y = 30. Find x when y = 45.Grade 8–9
Q12F is directly proportional to m. When m = 2.5, F = 20. Find m when F = 52.Grade 8–9
Q13y is directly proportional to x³. When x = 2, y = 40. Find the value of x when y = 135.Grade 8–9
Q14a is directly proportional to b². When b = 3, a = 18. Find b when a = 50.Grade 8–9
Q15Two quantities p and q are in direct proportion. If p increases by 20%, by what percentage does q increase?Grade 8–9
Answers
Foundation (Q1–Q5)
Q1£7.20(£0.90 per bar × 8)
Q2250 miles(25 miles per gallon × 10)
Q3£5.60(£0.80 per kg × 7)
Q445(k = 5, y = 5x, 5 × 9)
Q56kg(0.3kg per book × 20)
Higher (Q6–Q10)
Q6x = 14(k = 4.5, y = 4.5x, 63 ÷ 4.5)
Q7£48(k = 1.2 per km, 40 × 1.2)
Q8k = 2.5, p = 2.5q
Q9125(k = 5, y = 5x², 5 × 25)
Q10250cm³(k = 2, V = 2L³, 2 × 125)
Higher — Hard (Q11–Q15)
Q11x = 18(k = 2.5, y = 2.5x, 45 ÷ 2.5)
Q12m = 6.5(k = 8, F = 8m, 52 ÷ 8)
Q13x = 3(k = 5, y = 5x³, 135 ÷ 5 = 27, cube root of 27 = 3)
Q14b = 5(k = 2, a = 2b², 50 ÷ 2 = 25, square root of 25 = 5)
Q1520% increase(q scales by the same factor as p, since q = p ÷ k)
Common mistakes
Common Mistake 1
Confusing direct and inverse proportion
In direct proportion, both quantities increase together. If a question involves one quantity going up as another goes down (like speed and journey time), that's inverse proportion, not direct — check the relationship carefully.
Common Mistake 2
Forgetting to find k before answering
Jumping straight to an answer without first finding the constant of proportionality often leads to scaling errors — always write y = kx and solve for k as a clear first step.
Common Mistake 3
Forgetting to square or cube x
For y ∝ x², forgetting to square x before multiplying by k is one of the most common slips — always substitute into x² (or x³) first, then multiply by k.
Common Mistake 4
Assuming any straight line graph shows direct proportion
A straight line only represents direct proportion if it passes through the origin. A straight line that crosses the y-axis above zero shows a different kind of relationship.
Exam tips
💡 Exam Tip 1
Always write the proportion equation first
Writing "y = kx" (or the correct power) as your first line of working secures method marks even if a later calculation slip costs the final answer.
💡 Exam Tip 2
Use the unitary method for quick real-life problems
For straightforward word problems, finding the value of "one unit" first is often faster than setting up formal algebra — save the y = kx method for questions that ask for a formula or involve powers.
💡 Exam Tip 3
Check your answer scales sensibly
If x increases, y should increase too (for positive k) — if your final answer decreases when x increases, you've likely mixed up direct and inverse proportion.
💡 Exam Tip 4
Take the correct root for power proportions
When solving for x in y = kx² or y = kx³, remember to take the square root or cube root at the final step, not before dividing by k.
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