R11: Compound Measures
Compare lengths, areas and volumes using ratio notation and scale factors; make links to similarity
📋 Key Concepts
Definition: Compound measures involve two or more units combined, such as speed (distance/time), density (mass/volume), and pressure (force/area).
Key Formulae
| Measure | Formula | Units |
|---|---|---|
| Speed | Speed = Distance ÷ Time | m/s, km/h, mph |
| Density | Density = Mass ÷ Volume | kg/m³, g/cm³ |
| Pressure | Pressure = Force ÷ Area | N/m² (Pa) |
| Rate of Pay | Rate = Pay ÷ Time | £/hour |
📝 Speed Problems
Distance = Speed × Time
Speed = Distance ÷ Time
Time = Distance ÷ Speed
Speed = Distance ÷ Time
Time = Distance ÷ Speed
Example 1
A car travels 210 miles at an average speed of 60 mph. How long does the journey take?
Solution:
Time = Distance ÷ Speed
Time = 210 ÷ 60 = 3.5 hours = 3 hours 30 minutes
Example 2
Convert 25 m/s to km/h.
Solution:
25 m/s = 25 × 60 × 60 = 90000 m/h
90000 m/h = 90 km/h
Or: 25 × 3.6 = 90 km/h
📝 Density Problems
Mass = Density × Volume
Density = Mass ÷ Volume
Volume = Mass ÷ Density
Density = Mass ÷ Volume
Volume = Mass ÷ Density
Example 3
A block of wood has volume 0.4 m³ and density 650 kg/m³. Find its mass.
Solution:
Mass = Density × Volume
Mass = 650 × 0.4 = 260 kg
Example 4
A 200g block of metal has dimensions 5cm × 4cm × 2cm. Calculate its density in g/cm³.
Solution:
Volume = 5 × 4 × 2 = 40 cm³
Density = Mass ÷ Volume
Density = 200 ÷ 40 = 5 g/cm³
📝 Pressure Problems
Force = Pressure × Area
Pressure = Force ÷ Area
Area = Force ÷ Pressure
Pressure = Force ÷ Area
Area = Force ÷ Pressure
Example 5
A force of 240 N acts on an area of 0.8 m². Calculate the pressure.
Solution:
Pressure = Force ÷ Area
Pressure = 240 ÷ 0.8 = 300 N/m² (or 300 Pa)
📝 Best Value Comparisons
Method: Calculate the unit price or rate to compare value for money.
Example 6
Which is better value: 2.5 kg for £4.00 or 3 kg for £5.10?
Solution:
Option A: £4.00 ÷ 2.5 = £1.60 per kg
Option B: £5.10 ÷ 3 = £1.70 per kg
Option A is better value (lower cost per kg)
🧮 Worked Examples — Exam Style
Speed, density and pressure are all 'compound measures': one unit divided by another. Learn the triangle (or rearrange every time) and convert units BEFORE calculating.
A cyclist rides 42 km in 2 hours 30 minutes. Calculate the average speed in km/h.
Convert time first: 2 h 30 min = 2.5 h (30 ÷ 60).
Speed = distance ÷ time = 42 ÷ 2.5 = 16.8 km/h
M1 for 2.5 (the conversion), A1 for 16.8. Dividing by 2 h 30 min as if it were 2.3 is the classic error.
A metal block has mass 810 g and density 2.7 g/cm³. Find its volume.
Density = mass ÷ volume → volume = mass ÷ density
Volume = 810 ÷ 2.7 = 300 cm³
Check: 2.7 × 300 = 810 ✓
M1 for rearranging, A1 for 300. Rearranging the triangle earns the method mark even with an arithmetic slip.
Exam wording to expect: "Calculate the average speed/density/pressure", "Convert … into …", or a best-value comparison.
❓ Practice Questions
Q1: A train travels 315 km in 3.5 hours. Calculate the average speed.
Q2: A liquid has density 1.2 g/cm³. What is the mass of 150 cm³?
Q3: Convert 72 km/h to m/s.
Q4: A force of 500 N creates a pressure of 2500 Pa. Find the area.
Q5: Which is better value: 400g for £3.20 or 600g for £4.50?
Q6: A crate weighs 300 N and sits on an area of 1.5 m². Calculate the pressure.
Q7: Convert 15 m/s into km/h
✅ Answers
- 90 km/h
- 180 g
- 20 m/s
- 0.2 m²
- 600g for £4.50 (£7.50/kg vs £8/kg)
🧮 Worked Examples — Harder Exam Style
A car journey of 132 km takes 1 hour 30 minutes on the motorway and 30 minutes through town. Calculate the average speed for the WHOLE journey in km/h.
Total distance = 132 km. Total time = 1.5 + 0.5 = 2 hours.
Average speed = 132 ÷ 2 = 66 km/h
M1 for total time 2 h, A1 for 66. Average speed is TOTAL distance ÷ TOTAL time — never the average of the two speeds.
📋 Model Answers — Full Working
Every mark in GCSE maths comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. Speed = 315 ÷ 3.5 = 90 km/h
Q2. Mass = density × volume = 1.2 × 150 = 180 g
Q3. km/h → m/s: divide by 3.6. 72 ÷ 3.6 = 20 m/s
Q4. Pressure = force ÷ area → area = 500 ÷ 2500 = 0.2 m²
Q5. Price per kg: £3.20 ÷ 0.4 = £8.00/kg; £4.50 ÷ 0.6 = £7.50/kg → 600 g for £4.50 is better value
Q6. Pressure = 300 ÷ 1.5 = 200 Pa
Q7. 15 × 3.6 = 54 km/h
📌 Key Facts to Memorise
- Speed = distance ÷ time (units: km/h, m/s)
- Density = mass ÷ volume (g/cm³)
- Pressure = force ÷ area (N/m² = Pa)
- Triangle method: cover the one you want; rearrange algebraically for the method mark
- Conversions: km/h ÷ 3.6 = m/s; m/s × 3.6 = km/h; 30 min = 0.5 h
🗺️ In the Exam
Compound measures: 3–4 marks on Papers 2/3 (calculator). Unit conversion inside the question (km/h → m/s, grams → kg) is where the marks are protected or lost.
Links to: units (N13/G14), ratio and proportion (R7), graphs of journeys (A14).
🧭 Board Focus — AQA
Spec 8300. Three papers, each 80 marks and 1h30: Paper 1 non-calculator, Papers 2 and 3 calculator. Foundation/Higher tiers. Roughly a third of the marks sit under AO2 (reasoning) and AO3 (problem-solving), so show every line of working — method marks survive arithmetic slips.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Use triangles to remember formulae (cover what you need)
- Check units are consistent before calculating
- km/h to m/s: divide by 3.6
- m/s to km/h: multiply by 3.6
- For best value, always find the cost per single unit
🧠 Problem-Solving Strategies
Problem-Solving
Compound measures combine two or more quantities. Speed = distance/time, density = mass/volume, pressure = force/area. Use formula triangles to help rearrange. Always check units are consistent before substituting. Convert all units to the same system before calculating.
Multi-Step Problem
A metal block has mass 2.7 kg and dimensions 5 cm × 3 cm × 2 cm. Find its density in g/cm³. Would it float in water (density 1.0 g/cm³)?
Solution: Volume = 5 × 3 × 2 = 30 cm³. Mass = 2.7 kg = 2700 g. Density = 2700/30 = 90 g/cm³. Since 90 > 1.0, it would NOT float.
⚠️ Common Errors
Watch Out!
1. Wrong: Calculating density as volume ÷ mass = 30/2700 = 0.011 g/cm³ Correct: Density = mass ÷ volume = 2700/30 = 90 g/cm³
2. Wrong: Forgetting to convert units: using mass in kg with volume in cm³ Correct: Convert mass to grams (or volume to m³) before dividing. 2.7 kg = 2700 g.
3. Wrong: For pressure, using P = area/force instead of P = force/area Correct: Pressure = force ÷ area. A force of 50N over 2m² gives P = 50/2 = 25 Pa.
✍️ 6-Mark Exam Question
Extended Answer
6 marks: A cyclist travels 18 km at an average speed of 15 km/h, then 12 km at 20 km/h. (a) Calculate the total time taken. (b) Calculate the average speed for the whole journey. (c) Explain why the average speed for the whole journey is NOT simply (15 + 20) ÷ 2.
(a) Time for first part = 18/15 = 1.2 hours. Time for second part = 12/20 = 0.6 hours. Total time = 1.2 + 0.6 = 1.8 hours.
(b) Total distance = 18 + 12 = 30 km. Average speed = 30/1.8 = 16.67 km/h.
(c) Averaging the speeds ignores the different times spent at each speed. More time was spent at 15 km/h, so the average is closer to 15 than to 20. Average speed = total distance ÷ total time, not the mean of the speeds.
Mark scheme: M1 for time formula, A1 both times, A1 total 1.8h, M1 for average speed, A1 16.67 km/h, A1 explanation
📊 AO3: Reason & Interpret
Reasoning and Interpretation
An ice cube has sides of 2 cm and mass 7.2 g. A stone has mass 45 g and volume 18 cm³.
(a) Which has the higher density?
(b) A student says "The stone is heavier so it must be denser." Is this reasoning correct?
(c) The ice melts. Does the density of the resulting water increase, decrease, or stay the same compared to the ice?
Answers: (a) Ice: density = 7.2/8 = 0.9 g/cm³. Stone: density = 45/18 = 2.5 g/cm³. Stone is denser. (b) No — density depends on mass AND volume, not mass alone. A large fluffy pillow is heavier than a small diamond but far less dense. (c) Water is denser than ice (ice floats because its density is lower). When ice melts, the water produced has density ≈1.0 g/cm³, so density increases from 0.9 to 1.0.
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🎬 Video Resources
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