A compound measure combines two quantities, such as pounds per hour, litres per minute or people per square kilometre. Each is a rate: one quantity divided by another. Multiply the rate by the second quantity to get the total, or divide the total by the rate to find how much of the second quantity is needed.
Rates and compound measures
- Rate of pay = total pay ÷ hours
- Flow rate = volume ÷ time
- Fuel economy = distance ÷ fuel used
- Population density = population ÷ area
- Total = rate × amount; amount = total ÷ rate
What is a rate?
A rate tells you how much of one quantity there is for each unit of another. A rate of pay of £9 per hour means each hour earns £9. On a graph through the origin, the rate is the gradient of the line. Every compound measure works the same way: divide one quantity by the other.
Rates of pay
Rate of pay = total pay ÷ hours worked. If a worker earns £54 for 6 hours, the rate is £9 per hour. Overtime is often paid at a higher rate, so work out each part separately and add them.
Flow rate and fuel consumption
A flow rate measures volume per unit of time, such as 18 litres per minute; multiply by the time to find the volume. Fuel economy measures distance per unit of fuel, such as 9 km per litre. To find how long something takes, divide the total by the rate: 600 litres at 25 litres per minute takes 24 minutes.
Population density
Population density = population ÷ area, measured in people per km². A town of 84 000 people living in 35 km² has a population density of 2400 people per km². Higher population density means more crowded.
Choosing whether to multiply or divide
If you know the rate and want a total, multiply. If you know the total and the rate and want the time or amount, divide. Checking the units helps: litres per minute × minutes gives litres.
Worked examples
Example 1. A worker earns £54 for 6 hours of work. Work out the rate of pay.
- Rate = £54 ÷ 6.
Answer: £9 per hour.
Example 2. Water flows from a tap at 18 litres per minute. How much flows in 7 minutes?
- Volume = rate × time = 18 × 7.
Answer: 126 litres.
Example 3. A car travels 378 km on 42 litres of fuel. Work out its fuel economy.
- Fuel economy = distance ÷ fuel = 378 ÷ 42.
Answer: 9 km per litre.
Example 4. A town has 84 000 people living in an area of 35 km². Work out the population density.
- 84 000 ÷ 35.
Answer: 2400 people per km².
Example 5. A worker is paid £10.50 per hour for 8 hours, then £15.75 per hour for 2 hours of overtime. Work out the total pay.
- 8 × £10.50 = £84.
- 2 × £15.75 = £31.50.
Answer: £115.50.
Common mistakes
- Dividing the wrong way round. Rate of pay is pay ÷ hours, not hours ÷ pay.
- Mixing time units. Convert minutes and hours to match the rate.
- Using one rate for overtime. Work out each rate of pay separately.
- Forgetting units. State the units of a rate, such as £ per hour.
- Rounding money incorrectly. Give money to 2 decimal places.
How compound measures are examined
Compound measures appear throughout the 1MA1 GCSE and 4MA1 International GCSE, mostly at Foundation tier, often in multi-step problems. Questions ask you to:
- work out a rate of pay, flow rate or fuel economy
- use a rate to find a total, a time or an amount
- work out a population density
- compare two rates to make a decision
There are compound measures questions with answers on the compound measures worksheet; the GCSE proportion engine has plenty more.
Other compound measures include speed and density and pressure.
Frequently asked questions
What is a compound measure?
A measure made from two other quantities, such as speed (distance per time) or density (mass per volume).
How do you work out a rate of pay?
Divide the total pay by the number of hours worked.
What is population density?
The number of people per unit of area, usually per km².