GCSE specification fit
A practical rates skill for journeys, materials and forces.
Compound measures combine two different measurements into one rate or comparison. The main skill is choosing the correct formula, keeping units consistent and rearranging carefully.
What you will learn
Why this matters
Compound measures appear in everyday and scientific situations: average speed for journeys, density for materials, and pressure from forces acting on surfaces.
These questions are usually built from simple division. The challenge is spotting which quantities are being compared and making sure the units match the calculation.
Prior knowledge
You should already be comfortable with:
Clear explanation
What a compound measure is
A compound measure compares two different measurements. The unit often tells you the calculation. For example, km/h means kilometres per hour, so it compares distance with time.
Formula triangles
Formula triangles can be useful if you read them carefully: cover the quantity you want, then use the operation that remains.
For example, speed = distance ÷ time, distance = speed × time, and time = distance ÷ speed.
Unit awareness
Keep the units consistent before calculating. If distance is in km and time is in hours, speed is in km/h. If distance is in metres and time is in seconds, speed is in m/s.
For density, g/cm³ means grams per cubic centimetre, while kg/m³ means kilograms per cubic metre. For pressure, N/m² means newtons per square metre.
Convert both parts of a compound unit when needed. For example, 75 minutes is 1.25 hours, and 600 cm² is 0.06 m² because area units are squared.
Cubed units need the same care. Since 1 cm³ is 0.000001 m³ and 1 g is 0.001 kg, 1 g/cm³ = 1000 kg/m³. This is a common Higher-style density conversion.
Worked examples
Example 1: Find speed
A car travels 150 km in 3 hours. Find its average speed.
speed = distance ÷ time speed = 150 ÷ 3 = 50Example 2: Find distance
A cyclist travels at 6 m/s for 40 seconds. Find the distance travelled.
distance = speed × time distance = 6 × 40 = 240Example 3: Find density
A block has mass 96 g and volume 12 cm³. Find its density.
density = mass ÷ volume density = 96 ÷ 12 = 8Example 4: Find pressure
A force of 300 N acts on an area of 0.5 m². Find the pressure.
pressure = force ÷ area pressure = 300 ÷ 0.5 = 600Example 5: Convert a density
A liquid has density 1.2 g/cm³. Convert this density to kg/m³.
1 g/cm³ = 1000 kg/m³ 1.2 × 1000 = 1200Quick checks
Choose an answer, then check your thinking.
1. A runner travels 400 m in 50 seconds. What is the average speed?
2. A cube has mass 45 g and volume 5 cm³. What is its density?
3. A force of 120 N acts on an area of 6 m². What is the pressure?
Practice questions
Question 1
A train travels 180 km in 2.5 hours. Find its average speed in km/h.
Reveal answer and marking guidance
Answer: 72 km/h.
Marking: speed = distance ÷ time = 180 ÷ 2.5 = 72.
Question 2
A vehicle travels at 15 m/s for 12 seconds. Find the distance travelled.
Reveal answer and marking guidance
Answer: 180 m.
Marking: distance = speed × time = 15 × 12 = 180.
Question 3
A metal object has mass 312 g and volume 40 cm³. Find its density in g/cm³.
Reveal answer and marking guidance
Answer: 7.8 g/cm³.
Marking: density = mass ÷ volume = 312 ÷ 40 = 7.8.
Question 4
A material has density 650 kg/m³ and volume 4 m³. Find its mass.
Reveal answer and marking guidance
Answer: 2600 kg.
Marking: mass = density × volume = 650 × 4 = 2600, with kg as the unit because the density is in kg/m³.
Question 5
A force of 540 N acts on an area of 9 m². Find the pressure in N/m².
Reveal answer and marking guidance
Answer: 60 N/m².
Marking: pressure = force ÷ area = 540 ÷ 9 = 60.
Question 6
A pressure of 35 N/m² acts over an area of 8 m². Find the force.
Reveal answer and marking guidance
Answer: 280 N.
Marking: force = pressure × area = 35 × 8 = 280.
Question 7
A coach travels 90 km in 75 minutes. Find its average speed in km/h.
Reveal answer and marking guidance
Answer: 72 km/h.
Marking: 75 minutes = 1.25 hours, so speed = 90 ÷ 1.25 = 72 km/h.
Question 8
A force of 240 N acts on an area of 600 cm². Find the pressure in N/m².
Reveal answer and marking guidance
Answer: 4000 N/m².
Marking: 600 cm² = 0.06 m². Pressure = force ÷ area = 240 ÷ 0.06 = 4000 N/m². Do not divide by 600 because the required area unit is m².
Question 9
A liquid has density 0.85 g/cm³. Convert this density to kg/m³.
Reveal answer and marking guidance
Answer: 850 kg/m³.
Marking: Use 1 g/cm³ = 1000 kg/m³, so 0.85 × 1000 = 850. The factor is 1000 because both the mass unit and the cubed length unit change.
Question 10
Two delivery vans travel the same route. Van A travels 96 km in 1 hour 20 minutes. Van B travels 54 km in 45 minutes. Which van has the greater average speed?
Reveal answer and marking guidance
Answer: The vans have the same average speed, 72 km/h.
Marking: Convert the times to hours: 1 hour 20 minutes = 1⅓ hours and 45 minutes = 0.75 hours. Van A: 96 ÷ 1⅓ = 72 km/h. Van B: 54 ÷ 0.75 = 72 km/h.
Answers and marking guidance
The exact practice answers are hidden under each question so you can try first. For compound measures, marks usually come from selecting the right formula triangle or rearrangement, substituting the correct quantities, converting units before calculating when needed, and writing the compound unit such as km/h, g/cm³, kg/m³ or N/m². In multi-step questions, show the unit conversion separately so the examiner can follow the method.
Common mistakes
- Mixing units: do not combine metres with hours unless that is the unit you want.
- Using the wrong operation: speed, density and pressure are all found by division.
- Dropping the compound unit: 12 on its own is not as clear as 12 m/s or 12 g/cm³.
- Confusing mass and volume: density compares how much mass is packed into each unit of volume.
- Ignoring area in pressure: the same force gives a larger pressure when the area is smaller.
Extension challenge
A block has density 2.7 g/cm³ and mass 189 g. Find its volume.
Reveal answer
Answer: 70 cm³.
volume = mass ÷ density = 189 ÷ 2.7 = 70.
Exam-board guidance
Compound measures are common across GCSE Maths. Expect questions involving formulae, rearranging, rates, measures, unit conversion, practical contexts and checks that the answer is realistic.
AQA GCSE Maths
Write the formula first, convert units before substituting if needed, then rearrange only the part you need. In speed questions, convert minutes to hours or seconds before calculating if the final unit demands it; in density or pressure questions, check whether squared or cubed units need converting.
OCR GCSE Maths
Check the compound unit tells you the calculation, such as km/h meaning kilometres divided by hours. When finding distance, mass, force, time, volume or area, reverse the division with multiplication or division carefully and keep the converted unit attached to the answer.
Pearson Edexcel GCSE Maths
Expect questions that mix formula substitution, rearranging and a unit conversion before or after the calculation. Keep the compound unit attached to the final answer, for example g/cm³ or N/m², and convert cm² to m² where required.
Eduqas GCSE Maths
Show the formula, substituted values and units clearly so your method is easy to follow. A short sentence explaining whether the result is realistic can help in practical contexts.
WJEC Wales
Compound measures are practical numeracy skills, so explain what each unit means and check whether the answer is realistic for the journey, material or force described. Watch for mixed units in everyday contexts.
CCEA GCSE Maths
Keep the numerator and denominator units clear, especially when rearranging a formula or converting units. Use the unit structure to decide whether to divide or multiply, and keep calculator working organised.
Next lesson
Next, build on compound measures with Rates of Change and Units.