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Compound measures combine two different units to describe a rate or relationship. The three compound measures you need for AQA GCSE Mathematics are speed, density, and pressure. This lesson covers the formulae, unit conversions, and worked examples for each.
A compound measure is a measure that involves two or more different units. For example, speed is measured in miles per hour (mph) — it combines distance (miles) with time (hours).
| Compound Measure | Formula | Common Units |
|---|---|---|
| Speed | Speed = Distance / Time | m/s, km/h, mph |
| Density | Density = Mass / Volume | g/cm cubed, kg/m cubed |
| Pressure | Pressure = Force / Area | N/m squared (Pascals), N/cm squared |
Exam Tip: You can rearrange each formula using the formula triangle method. Cover the quantity you want to find, and the remaining two quantities show you the calculation.
graph TD
A[Distance] --> B[Speed x Time]
C["Speed = Distance / Time"]
D["Time = Distance / Speed"]
E["Distance = Speed x Time"]
Worked Example 1: A car travels 180 miles in 3 hours. What is its average speed?
Worked Example 2: A cyclist rides at 12 km/h for 2.5 hours. How far do they travel?
Worked Example 3: A train travels 225 km at an average speed of 90 km/h. How long does the journey take?
Exam Tip: If your answer for time is a decimal, convert it properly. 2.5 hours = 2 hours 30 minutes (NOT 2 hours 50 minutes). Multiply the decimal part by 60 to get minutes.
| Conversion | Method |
|---|---|
| m/s to km/h | Multiply by 3.6 |
| km/h to m/s | Divide by 3.6 |
| mph to km/h | Multiply by 1.6 (approx.) |
| km/h to mph | Divide by 1.6 (approx.) |
Worked Example 4: Convert 15 m/s to km/h.
Worked Example 5: A block of metal has a mass of 540 g and a volume of 200 cm cubed. What is its density?
Worked Example 6: A liquid has a density of 1.2 g/cm cubed. What is the mass of 350 cm cubed of the liquid?
Worked Example 7: A substance has a density of 8 g/cm cubed and a mass of 2,400 g. What is its volume?
| Material | Typical Density (g/cm cubed) |
|---|---|
| Gold | 19.3 |
| Iron | 7.9 |
| Aluminium | 2.7 |
| Water | 1.0 |
| Air | 0.0012 |
| Wood (oak) | 0.6-0.9 |
Exam Tip: If an object's density is less than 1 g/cm cubed, it floats in water. If its density is greater than 1 g/cm cubed, it sinks. This fact can appear in exam questions.
Worked Example 8: A force of 600 N acts on an area of 0.5 m squared. What is the pressure?
Worked Example 9: A box exerts a pressure of 200 N/m squared on the ground. The base of the box has an area of 1.5 m squared. What is the weight of the box?
Worked Example 10: A woman weighing 600 N stands on one heel with an area of 2 cm squared. What pressure does she exert?
Worked Example 11: A journey of 120 km takes 1 hour 20 minutes. Calculate the average speed in km/h.
Worked Example 12: Two objects are made of different materials. Object A has a mass of 480 g and a volume of 60 cm cubed. Object B has a mass of 720 g and a volume of 80 cm cubed. Which is denser?
graph LR
A[Compound Measures] --> B[Speed = Distance / Time]
A --> C[Density = Mass / Volume]
A --> D[Pressure = Force / Area]
B --> E[Units: m/s, km/h, mph]
C --> F[Units: g/cm cubed, kg/m cubed]
D --> G[Units: N/m squared, N/cm squared]
| Mistake | Correct Approach |
|---|---|
| Not converting minutes to hours | 45 minutes = 0.75 hours, not 0.45 |
| Mixing units (g with kg, cm with m) | Check all values use consistent units |
| Using the wrong formula | Identify which three quantities are involved |
| Writing 2.5 hours as "2 hours 50 minutes" | Multiply the decimal by 60: 0.5 x 60 = 30 minutes |
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