What is a compound measure?
A compound measure is made by dividing one quantity by another, combining two different units into one — like miles per hour, or grams per cm³. Speed, density, and pressure are the three compound measures you'll meet most often at GCSE.
The formula triangle method
Every compound measure follows the same pattern: Total = Rate × Other quantity. A formula triangle makes it easy to rearrange without memorising three separate equations.
Covering the quantity you want to find shows exactly which operation to use — the triangle never changes, only the labels do.
Applying it to speed
Worked Example 1
A car travels 240 miles in 4 hours. Find its speed.
1
Cover "speed" in the triangle: this leaves distance ÷ time
2
Substitute the values: 240 ÷ 4 = 60
Answer60mph
Applying it to density
Worked Example 2
A block has a mass of 150g and a volume of 25cm³. Find its density.
1
Cover "density" in the triangle: this leaves mass ÷ volume
2
Substitute the values: 150 ÷ 25 = 6
Answer6g/cm³
Applying it to pressure
Worked Example 3
A force of 20N acts over an area of 4m². Find the pressure.
1
Cover "pressure" in the triangle: this leaves force ÷ area
2
Substitute the values: 20 ÷ 4 = 5
Answer5N/m² (5 Pascals)
Rearranging the formula
Worked Example 4
A cyclist travels at a speed of 15mph for 2 hours. Find the distance travelled.
1
This time, distance is uncovered in the triangle: distance = speed × time
2
Substitute the values: 15 × 2 = 30
Answer30 miles
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1A car travels 150 miles in 3 hours. Find its speed, in mph.Foundation
Q2A block has a mass of 40g and a volume of 8cm³. Find its density.Foundation
Q3A force of 12N acts on an area of 4m². Find the pressure.Foundation
Q4A cyclist travels at 15mph for 2 hours. Find the distance travelled.Foundation
Q5A block has a density of 2g/cm³ and a volume of 10cm³. Find its mass.Foundation
Higher (Grade 5–7)
Q6A runner covers 400m in 50 seconds. Find their speed, in m/s.Higher
Q7A metal bar has a mass of 270g and a density of 9g/cm³. Find its volume.Higher
Q8A force of 45N is applied over an area of 0.5m². Find the pressure.Higher
Q9A train travels at 80km/h for 2.5 hours. Find the distance travelled.Higher
Q10A liquid has a density of 1.2g/cm³. Find the mass of 50cm³ of the liquid.Higher
Higher — Hard (Grade 8–9)
Q11A plane travels 1800km in 2 hours 15 minutes. Find its average speed, in km/h.Grade 8–9
Q12A cube of metal has a mass of 216g and a density of 8g/cm³. Find the length of one side of the cube.Grade 8–9
Q13A force of 157N acts over a circular area of radius 5cm (use π = 3.14). Find the pressure, in N/cm².Grade 8–9
Q14200cm³ of liquid A (density 0.8g/cm³) is mixed with 300cm³ of liquid B (density 1.2g/cm³). Find the density of the mixture, assuming the volumes simply add.Grade 8–9
Q15A car travels the first 100km of a journey at 50km/h, then the next 150km at 75km/h. Find the average speed for the whole journey, in km/h.Grade 8–9
Answers
Foundation (Q1–Q5)
Q150mph
Q25g/cm³
Q33N/m²
Q430 miles
Q520g
Higher (Q6–Q10)
Q68m/s
Q730cm³(270 ÷ 9)
Q890N/m²(45 ÷ 0.5)
Q9200km(80 × 2.5)
Q1060g(1.2 × 50)
Higher — Hard (Q11–Q15)
Q11800km/h(2h 15min = 2.25h, 1800 ÷ 2.25)
Q123cm(volume = 216 ÷ 8 = 27cm³, cube root of 27)
Q132N/cm²(area = 3.14 × 25 = 78.5cm², 157 ÷ 78.5)
Q141.04g/cm³(mass = 160g + 360g = 520g, volume = 500cm³, 520 ÷ 500)
Q1562.5km/h(time = 2h + 2h = 4h, total distance = 250km, 250 ÷ 4)
Common mistakes
Common Mistake 1
Placing the wrong quantity at the top of the triangle
The "total" quantity (distance, mass, or force) always goes at the top — the rate (speed, density, or pressure) and the other quantity always go on the bottom.
Common Mistake 2
Forgetting to convert units before calculating
Minutes must be converted to hours (or vice versa) before applying a speed formula in km/h or mph — mixing units gives a meaningless answer.
Common Mistake 3
Averaging speeds directly for multi-stage journeys
Average speed is never found by averaging the individual speeds — always use total distance ÷ total time for the whole journey.
Common Mistake 4
Mixing units within the same calculation
Density in g/cm³ needs mass in grams and volume in cm³ — convert kg to g or m³ to cm³ first if the question mixes units.
Exam tips
💡 Exam Tip 1
Draw the triangle every time
Even for simple questions, quickly sketching the triangle and covering the unknown removes any doubt about whether to multiply or divide.
💡 Exam Tip 2
Write down the units at every step
Carrying units through your working (g, cm³, N, m²) makes it obvious if something doesn't match, and confirms the correct unit for your final answer.
💡 Exam Tip 3
For multi-stage journeys, total everything first
Add up all the distances and all the times separately before dividing — never combine a per-stage calculation partway through.
💡 Exam Tip 4
Check your answer's size makes sense
A car travelling faster than 300mph or a density less than 0 are both signs a calculation has gone wrong — sense-check every compound measure answer.
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