📘 GCSE Maths · Notes & worksheet
Proportion, Compound Measures and Growth.
Notes
Direct and inverse proportion
| Statement | Equation |
|---|---|
| y is proportional to x | y = kx |
| y is proportional to x² | y = kx² |
| y is inversely proportional to x | y = k/x |
| y is inversely proportional to √x | y = k/√x |
| y is inversely proportional to x² | y = k/x² |
Method: (1) write the equation with k, (2) substitute the given pair to find k, (3) rewrite the equation with the value of k, (4) use it to answer the question.
Graph shapes
- y = kx: straight line through the origin.
- y = kx²: parabola through the origin.
- y = k/x: reciprocal curve that never touches the axes.
Compound measures
- Speed = distance ÷ time
- Density = mass ÷ volume
- Pressure = force ÷ area
- Average speed = total distance ÷ total time. It is not the mean of the speeds.
Percentage change
- Percentage change = (change ÷ original) × 100
- Reverse percentages: after a 15% decrease the price is 85% of the original, so original = new ÷ 0.85.
Compound growth and decay
Final amount = Original × (multiplier)ⁿ
- 3.5% growth: multiplier 1.035
- 18% decrease: multiplier 0.82
Worked example
y is directly proportional to x². y = 48 when x = 4.
- k = 48 ÷ 16 = 3, so y = 3x².
- When x = 7, y = 3 × 49 = 147.
Common mistakes
- Using y = kx when the question says "proportional to x²".
- Calculating average speed as the average of two speeds.
- Applying 15% to the new price instead of dividing by 0.85 in reverse percentage questions.
Exam tips
- For "show that" questions, state the equation form before substituting.
- Keep full calculator values through compound-interest calculations and round only at the end.
Worksheet
Prints the questions only, without answers. Choose “Save as PDF” in the print window to download.
Total: 24 marks. Calculator allowed.
- y is directly proportional to x². When x = 4, y = 48. Find y when x = 7. (3)
- y is inversely proportional to √x. When x = 9, y = 6. Find x when y = 4. (4)
- The time t hours taken to fill a pool is inversely proportional to the number of pumps n. Four pumps take 9 hours. How long do 6 pumps take? (2)
- The force F between two magnets is inversely proportional to d², where d is the distance between them. F = 20 when d = 3. Find F when d = 6. (3)
- After a 15% decrease in a sale, a jacket costs £42.50. Find its original price. (2)
- £8000 is invested at 3.5% compound interest per year. Find the value after 6 years, to the nearest penny. (3)
- A car worth £14,000 loses 18% of its value each year. Find its value after 3 years, to the nearest penny. (3)
- A cyclist travels 24 km at 18 km/h, then 36 km at 27 km/h. Find the average speed for the whole journey. (4)
Show mark scheme and answers (tutor copy)
Try the worksheet first, then check your work here.
- y = kx² (M1). k = 48/16 = 3 (M1). y = 3 × 49 = 147. (A1)
- y = k/√x (M1). 6 = k/3, so k = 18 (M1). 4 = 18/√x, so √x = 4.5 (M1). x = 20.25. (A1)
- t = k/n. k = 4 × 9 = 36 (M1). t = 36/6 = 6 hours. (A1)
- F = k/d² (M1). k = 20 × 9 = 180 (M1). F = 180/36 = 5. (A1)
- 85% of the original = £42.50 (M1). Original = 42.50 ÷ 0.85 = £50. (A1)
- 8000 × 1.035⁶ (M1, M1) = £9,834.04. (A1)
- 14000 × 0.82³ (M1, M1) = 14000 × 0.551368 = £7,719.15. (A1)
- Time 1 = 24/18 = 4/3 h and time 2 = 36/27 = 4/3 h (M1, M1). Total distance 60 km, total time 8/3 h (M1). Average speed = 60 ÷ 8/3 = 22.5 km/h. (A1) Here the two times happen to be equal, so the average speed equals the mean of 18 and 27. That is a coincidence, not a rule.