R1: Compound Units
Convert between standard units; use compound units: speed, rates of pay, prices, density, pressure
📋 Key Concepts
Definition: Compound units are units made from two or more other units, such as speed (distance/time) or density (mass/volume).
Common Compound Units
| Quantity | Formula | Units |
|---|---|---|
| Speed | Speed = Distance ÷ Time | km/h, m/s, mph |
| Density | Density = Mass ÷ Volume | kg/m³, g/cm³ |
| Pressure | Pressure = Force ÷ Area | N/m² (Pascals), N/cm² |
| Rate of Pay | Pay = Rate × Time | £/hour |
| Unit Price | Unit Price = Cost ÷ Quantity | £/kg, £/litre |
📝 Speed Calculations
Speed Triangle: Cover the value you want to find:
• Distance = Speed × Time
• Speed = Distance ÷ Time
• Time = Distance ÷ Speed
• Distance = Speed × Time
• Speed = Distance ÷ Time
• Time = Distance ÷ Speed
Example 1
A car travels 180 miles in 3 hours. Calculate the average speed.
Solution:
Speed = Distance ÷ Time
Speed = 180 ÷ 3 = 60 mph
Example 2
A train travels at 125 km/h for 2.5 hours. How far does it travel?
Solution:
Distance = Speed × Time
Distance = 125 × 2.5 = 312.5 km
Example 3
How long does it take to travel 450 km at an average speed of 90 km/h?
Solution:
Time = Distance ÷ Speed
Time = 450 ÷ 90 = 5 hours
📝 Density Calculations
Density Triangle:
• Mass = Density × Volume
• Density = Mass ÷ Volume
• Volume = Mass ÷ Density
• Mass = Density × Volume
• Density = Mass ÷ Volume
• Volume = Mass ÷ Density
Example 4
A block of metal has mass 2400 g and volume 300 cm³. Calculate its density.
Solution:
Density = Mass ÷ Volume
Density = 2400 ÷ 300 = 8 g/cm³
Example 5
A substance has density 2.7 g/cm³. What is the mass of 500 cm³ of this substance?
Solution:
Mass = Density × Volume
Mass = 2.7 × 500 = 1350 g
📝 Pressure Calculations
Pressure Triangle:
• Force = Pressure × Area
• Pressure = Force ÷ Area
• Area = Force ÷ Pressure
• Force = Pressure × Area
• Pressure = Force ÷ Area
• Area = Force ÷ Pressure
Example 6
A force of 500 N acts on an area of 25 m². Calculate the pressure.
Solution:
Pressure = Force ÷ Area
Pressure = 500 ÷ 25 = 20 N/m² (or 20 Pascals)
📝 Unit Conversions
Converting Speed: To convert km/h to m/s, multiply by 1000 then divide by 3600 (or divide by 3.6)
Example 7
Convert 72 km/h to m/s.
Solution:
72 km/h = 72 × 1000 ÷ 3600 = 20 m/s
Or: 72 ÷ 3.6 = 20 m/s
Example 8
Convert 25 m/s to km/h.
Solution:
25 m/s = 25 × 3.6 = 90 km/h
📝 Best Value Comparisons
Method: Calculate the unit price (cost per unit) for each option to find the best value.
Example 9
Which is better value: 3 kg for £4.50 or 5 kg for £7.20?
Solution:
Option 1: £4.50 ÷ 3 = £1.50 per kg
Option 2: £7.20 ÷ 5 = £1.44 per kg
Best value: 5 kg for £7.20 (£1.44 per kg is cheaper)
🧮 Worked Examples — Exam Style
Compound measures bundle two units: speed (km/h), density (g/cm³), pressure (N/m²). The triangle works; showing the rearrangement earns the method mark.
A stone has mass 72 g and volume 24 cm³. Calculate its density.
Density = mass ÷ volume = 72 ÷ 24 = 3 g/cm³
M1 for mass ÷ volume, A1 for 3. Water is 1 g/cm³ — a useful anchor: 3 means three times denser than water.
Convert 54 km/h into m/s.
÷ 3.6: 54 ÷ 3.6 = 15 m/s
Why 3.6: 1 km/h = 1000 m per 3600 s = 1/3.6 m/s.
M1 for ÷ 3.6, A1 for 15. Deriving 3.6 from 3600/1000 is worth remembering if the number slips your mind.
Exam wording to expect: "Calculate the average speed/density/pressure", "Convert … into …", "Which is better value?".
❓ Practice Questions
Q1: A car travels 240 km in 4 hours. Calculate its average speed.
Q2: A block has mass 450 g and volume 150 cm³. Calculate its density.
Q3: Convert 108 km/h to m/s.
Q4: A force of 800 N acts on an area of 40 m². Calculate the pressure.
Q5: Which is better value: 2 litres for £2.80 or 3 litres for £4.35?
Q6: A car travels 90 km at 60 km/h, then 40 km at 20 km/h. Find the average speed for the whole journey.
Q7: A box weighs 120 N and rests on an area of 0.8 m². Calculate the pressure.
✅ Answers
- 60 km/h
- 3 g/cm³
- 30 m/s
- 20 N/m²
- 2 litres for £2.80 (£1.40 per litre vs £1.45 per litre)
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. 240 ÷ 4 = 60 km/h
Q2. 450 ÷ 150 = 3 g/cm³
Q3. 108 ÷ 3.6 = 30 m/s
Q4. 800 ÷ 40 = 20 N/m²
Q5. £2.80 ÷ 2 = £1.40/L vs £4.35 ÷ 3 = £1.45/L → 2 litres for £2.80
Q6. Total 130 km; times: 1.5 h + 2 h = 3.5 h → 130 ÷ 3.5 = 37.1 km/h (3 s.f.)
Q7. 120 ÷ 0.8 = 150 N/m²
📌 Key Facts to Memorise
- Speed = distance ÷ time · Density = mass ÷ volume · Pressure = force ÷ area
- Average speed = TOTAL distance ÷ TOTAL time (never average the speeds)
- km/h ↔ m/s: ÷ or × 3.6
- Best value: price per unit — same unit on both sides before comparing
- Triangles hide the algebra; write the rearranged formula for the method mark
🗺️ In the Exam
Compound measures: 3–5 marks, Papers 2/3. The multi-leg journey (two speeds, one average) is the classic discriminator.
Links to: units (N13/G14), graphs of journeys (A14), proportion (R7).
⚡ Targeted Drills
Cover the answers, work each one, then mark yourself. If any takes over a minute, revisit the matching section above.
- Speed for 210 km in 3 h → 70 km/h · 90 km at 30 km/h → 3 h
- Density: 540 g, 200 cm³ → 2.7 g/cm³
- Pressure: 480 N on 1.2 m² → 400 Pa
- 36 km/h → m/s → 10 m/s · 25 m/s → km/h → 90 km/h
- Best buy: 500 g £2.25 vs 750 g £3.15 → £4.50/kg vs £4.20/kg → 750 g
🧭 Board Focus — OCR
Spec J560. Three papers of 100 marks each: Paper 1 non-calculator, Papers 2 and 3 calculator, at Foundation/Higher tiers. OCR marks problem-solving explicitly, and its papers tend to front-load multi-step worded contexts — practise extracting the mathematics from prose before calculating.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Use the formula triangle: cover what you need to find
- Always check units are consistent before calculating
- For speed conversions: km/h to m/s, divide by 3.6
- When comparing value, calculate cost per single unit
- Watch out for different units in the question (e.g., minutes and hours)
🧠 Problem-Solving Strategies
Problem-Solving
Use formula triangles to rearrange compound unit formulas. Always convert all quantities to consistent base units before calculating. For speed problems, remember: speed = distance ÷ time, so distance = speed × time and time = distance ÷ speed.
Multi-Step Problem
A car travels 45 miles at 30 mph, then 84 km at 60 km/h. Find the average speed for the whole journey in mph. (5 miles ≈ 8 km)
Stage 1: time = 45 ÷ 30 = 1.5 hours. Stage 2: 84 km = 84 × 5/8 = 52.5 miles, time = 84 ÷ 60 = 1.4 hours. Total distance = 45 + 52.5 = 97.5 miles. Total time = 1.5 + 1.4 = 2.9 hours. Average speed = 97.5 ÷ 2.9 = 33.6 mph (1 d.p.)
⚠️ Common Errors
Watch Out!
1. Wrong: Averaging two speeds directly, e.g. (30 + 60) ÷ 2 = 45 mph Correct: Average speed = total distance ÷ total time
2. Wrong: Converting km/h to m/s by dividing by 1000 Correct: Divide by 3.6 (or ×1000÷3600)
3. Wrong: Using density = volume ÷ mass Correct: Density = mass ÷ volume
✍️ 6-Mark Exam Question
Extended Answer
6 marks: A gold bar has mass 12.4 kg and density 19.3 g/cm³. A silver bar has the same volume as the gold bar. Silver has density 10.5 g/cm³. Show that the silver bar costs less than half the price of the gold bar if gold costs £45 per gram and silver costs £0.70 per gram.
Volume of gold = mass ÷ density = 12400 ÷ 19.3 = 642.5 cm³ (1 d.p.). Mass of silver = 642.5 × 10.5 = 6746.1 g. Cost of gold = 12400 × 45 = £558,000. Cost of silver = 6746.1 × 0.70 = £4722.27. Half the gold price = £279,000. Since £4722.27 < £279,000, silver costs less than half the gold price. ✓
Mark scheme: M1 volume formula, M1 correct substitution, A1 volume, M1 silver mass, M1 both costs calculated, A1 comparison and conclusion
📊 AO3: Reason & Interpret
Reasoning and Interpretation
Two cars travel from London to Edinburgh (420 miles). Car A averages 70 mph. Car B averages 56 mph but sets off 30 minutes earlier.
(a) Which car arrives first and by how many minutes?
(b) A speed camera records both cars at the 200-mile point. Does this prove they were both travelling at the same speed at that point? Explain.
(c) Car A's driver says "I averaged 70 mph so I must have been going 70 mph the whole way." Is this correct? Explain.
Answers: (a) Car A: 420÷70=6h, Car B: 420÷56=7.5h, starts 0.5h early so arrives in 7h. Car A arrives 1 hour (60 min) earlier. (b) No — average speed doesn't mean instant speed was equal. (c) No — average speed includes stops and varying speeds; the car likely went faster and slower at different points.
📝 Exam Questions by Topic
🎬 Video Resources
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