A car travels 120 km in 2 hours. What is its speed?
Speed = distance ÷ time.
120 ÷ 2 = 60 km/h.
Compound Measures
Compound measures involve quantities that combine two different units, such as speed or density. Understanding how to use formulas, convert units using units and conversions, and rearrange equations is essential in GCSE Maths.
Overview
A compound measure combines two or more different measurements into one rate.
It helps describe how one quantity changes compared with another.
Compound measure = one measurement divided by or linked to another
What you should understand after this topic
- Understand what a compound measure is
- Recognise common compound measures
- Use the correct formula
- Interpret compound units correctly
- Avoid unit mistakes in exam questions
Speed
\( \dfrac{\text{distance}}{\text{time}} \)
Density
\( \dfrac{\text{mass}}{\text{volume}} \)
Pressure
\( \dfrac{\text{force}}{\text{area}} \)
Flow Rate
\( \dfrac{\text{volume}}{\text{time}} \)
Key Definitions
Compound Measure
A measure made by combining two different quantities.
Rate
A quantity compared with another quantity, often using division.
Speed
Distance travelled per unit of time.
Density
Mass per unit of volume.
Pressure
Force per unit area.
Flow Rate
Volume of liquid or gas per unit of time.
Key Rules
Use the right formula
Different compound measures use different linked quantities.
Check units carefully
Units must match before calculating.
Rearrange when needed
Sometimes you must find a missing value, not the compound measure itself.
Write units clearly
Compound answers need compound units, such as km/h or g/cm³.
Main GCSE Compound Measures
Speed
\( \text{Speed} = \dfrac{\text{Distance}}{\text{Time}} \)
Common units: km/h, m/s
Density
\( \text{Density} = \dfrac{\text{Mass}}{\text{Volume}} \)
Common units: g/cm³, kg/m³
Pressure
\( \text{Pressure} = \dfrac{\text{Force}}{\text{Area}} \)
Common units: N/cm², N/m²
Flow Rate
\( \text{Flow rate} = \dfrac{\text{Volume}}{\text{Time}} \)
Common units: litres/min, cm³/s
How to Solve
What is a compound measure?
A compound measure combines two different quantities into one useful value. Instead of measuring one thing, it compares one measurement to another.
Exam tip: Always identify what is being compared (e.g. distance per time, mass per volume).
Key idea
Compound measures answer questions like: how much per one unit of something else?
Recognising compound units
Compound units often use words like per, or symbols such as \( / \).
km/h
Kilometres per hour
g/cm³
Grams per cubic centimetre
N/m²
Newtons per square metre
litres/min
Litres per minute
Speed
Speed tells you how far something travels in a given time. For more detail, see speed, distance and time.
\( \text{Speed} = \dfrac{\text{Distance}}{\text{Time}} \)
Example: A car travels \(120\) km in \(2\) hours.
\( \text{Speed} = \frac{120}{2} = 60 \)
Answer: \(60\) km/h
Speed questions often involve units and conversions.
Density
Density compares mass to volume. For more detail, see density, mass and volume.
\( \text{Density} = \dfrac{\text{Mass}}{\text{Volume}} \)
Example: Mass \(=40\) g, volume \(=10\text{ cm}^3\).
\( \text{Density} = \frac{40}{10} = 4 \)
Answer: \(4\text{ g/cm}^3\)
Pressure
Pressure compares force to area.
\( \text{Pressure} = \dfrac{\text{Force}}{\text{Area}} \)
Example: Force \(=50\) N, area \(=10\text{ cm}^2\).
\( \text{Pressure} = \frac{50}{10} = 5 \)
Answer: \(5\text{ N/cm}^2\)
Flow rate
Flow rate compares volume to time.
\( \text{Flow rate} = \dfrac{\text{Volume}}{\text{Time}} \)
Example: \(18\) litres in \(6\) minutes.
\( \text{Flow rate} = \frac{18}{6} = 3 \)
Answer: \(3\) litres/min
Rearranging formulas
You may need to rearrange formulas to find a different variable.
Rearranging formulas uses the same skills as solving equations.
Speed
\( d = s \times t \)
\( t = \frac{d}{s} \)
Density
\( m = d \times v \)
\( v = \frac{m}{d} \)
Pressure
\( F = p \times a \)
\( a = \frac{F}{p} \)
Flow rate
\( v = f \times t \)
\( t = \frac{v}{f} \)
Units matter
Many mistakes come from using the wrong units. Always check them before calculating.
Examples of unit checks
- Convert minutes to hours for km/h
- Convert kg to g for density
- Match area units for pressure
- Match volume units for flow rate
Step-by-step exam method
Exam tip: Always include the correct compound unit in your final answer.
- Identify the type of compound measure.
- Choose the correct formula.
- Decide what you need to find.
- Convert units first if needed.
- Substitute values carefully.
- Write the final answer with correct units.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics, focusing on calculating and interpreting compound measures.
Edexcel
A car travels 180 km in 3 hours. Calculate its average speed.
Edexcel
A container holds 400 kg of sand in a volume of 0.5 m³. Calculate the density of the sand.
Edexcel
A machine produces 2,400 bottles in 8 hours. Calculate the production rate in bottles per hour.
Edexcel
A cyclist travels 15 km in 45 minutes. Calculate the average speed in km/h.
Edexcel
A water tank contains 900 litres of water. It is emptied in 30 minutes. Calculate the flow rate in litres per minute.
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, emphasising unit conversions and real-life applications.
AQA
A car travels at 20 m/s. Convert this speed to km/h.
AQA
A factory produces 5,400 units in 6 hours. Calculate the production rate in units per minute.
AQA
Water flows through a pipe at 3.5 litres per second. How much water flows in 4 minutes?
AQA
A metal block has a mass of 2.4 kg and a volume of 300 cm³. Calculate its density in g/cm³.
AQA
Explain why consistent units are necessary when calculating compound measures.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, focusing on reasoning, efficiency, and interpreting compound measures in practical contexts.
OCR
A pump fills 1.8 m³ of water in 12 minutes. Calculate the flow rate in m³ per minute.
OCR
A car uses 45 litres of fuel to travel 675 km. Calculate the fuel consumption in km per litre.
OCR
A printer produces 360 pages in 9 minutes. Calculate the printing rate in pages per minute.
OCR
A delivery driver travels 240 km in 4 hours using 20 litres of fuel. Calculate the average fuel consumption in km per litre.
OCR
Describe the steps required to calculate a compound measure from given information.
Exam Checklist
Step 1
Identify which compound measure is being used.
Step 2
Write down the correct formula.
Step 3
Check and convert units if needed.
Step 4
Give the answer with the correct compound unit.
Most common exam mistakes
Wrong formula
Using a speed formula in a density or pressure question.
Wrong rearrangement
Not solving properly for the missing quantity.
Unit mistake
Not converting minutes to hours, or grams to kilograms, when needed.
Missing units
Final answers need full compound units such as km/h or g/cm³.
Common Mistakes
These are common mistakes students make when working with compound measures in GCSE Maths.
Using the wrong formula
Incorrect
A student uses the speed formula when solving a density question.
Correct
Make sure you use the correct formula for the situation, such as speed = distance ÷ time, density = mass ÷ volume, or pressure = force ÷ area.
Rearranging formulas incorrectly
Incorrect
A student rearranges a formula but changes the operations incorrectly.
Correct
When rearranging, use inverse operations carefully and keep the equation balanced. Work step by step to avoid mistakes.
Ignoring unit conversions
Incorrect
A student uses metres with kilometres or seconds with hours without converting.
Correct
All units must be consistent before using a formula. Convert values first, for example km to m or hours to seconds.
Missing units in the answer
Incorrect
A student gives a numerical answer but does not include units.
Correct
Always include the correct compound unit, such as km/h, m/s or g/cm³, as marks are often awarded for units.
Confusing different compound measures
Incorrect
A student mixes up formulas for density, pressure and speed.
Correct
Each compound measure has its own formula and unit. Learn them clearly and check which one the question is asking for.
Try It Yourself
Practise calculating speed, density and pressure using compound units.
Foundation
Foundation Practice
Use formulas for speed, density and pressure.
A block has mass 40 g and volume 10 cm³. Find its density.
Density = mass ÷ volume.
40 ÷ 10 = 4 g/cm³.
Force = 20 N, Area = 5 m². What is the pressure?
Pressure = force ÷ area.
20 ÷ 5 = 4 N/m².
A cyclist travels 30 km in 2 hours. Find the speed in km/h.
Divide distance by time.
30 ÷ 2 = 15 km/h.
A liquid has density 3 g/cm³ and volume 6 cm³. What is its mass?
Mass = density × volume.
3 × 6 = 18 g.
Force = 50 N, Area = 10 m². Find the pressure.
Divide force by area.
50 ÷ 10 = 5 N/m².
Which formula is used to calculate density?
Density measures how packed mass is.
Density = mass ÷ volume.
A car travels at 80 km/h for 3 hours. Find the distance travelled.
Distance = speed × time.
80 × 3 = 240 km.
A student uses pressure = area ÷ force. What is wrong?
Pressure uses force first.
Correct formula is pressure = force ÷ area.
A material has density 2 g/cm³ and mass 10 g. Find the volume.
Volume = mass ÷ density.
10 ÷ 2 = 5 cm³.
Higher
Higher Practice
Solve multi-step problems involving compound measures and unit conversions.
A car travels 150 km in 2.5 hours. What is its average speed?
Convert time to hours if needed.
150 ÷ 2.5 = 60 km/h.
A block has mass 3 kg and volume 0.5 m³. Find its density in kg/m³.
Density = mass ÷ volume.
3 ÷ 0.5 = 6 kg/m³.
Force = 120 N, Area = 4 m². What is the pressure?
Divide force by area.
120 ÷ 4 = 30 N/m².
A cyclist travels 18 km/h for 40 minutes. Find the distance in km.
Convert minutes to hours.
40 minutes = 2/3 hour → 18 × 2/3 = 12 km.
A material has density 0.8 g/cm³. What is the mass of 50 cm³?
Mass = density × volume.
0.8 × 50 = 40 g.
Force = 200 N, pressure = 50 N/m². Find the area.
Rearrange pressure formula.
Area = force ÷ pressure = 200 ÷ 50 = 4 m².
A student converts 30 minutes to 0.3 hours. What is wrong?
30 minutes is half an hour.
30 ÷ 60 = 0.5 hours.
A car travels 180 km at 90 km/h. How many minutes does it take?
Find time in hours first.
180 ÷ 90 = 2 hours = 120 minutes.
Which formula calculates pressure?
Pressure spreads force over area.
Pressure = force ÷ area.
A material has density 4 g/cm³ and mass 80 g. Find the volume.
Volume = mass ÷ density.
80 ÷ 4 = 20 cm³.
Games
Practise this topic with interactive games.
Games coming soon.