IGCSEMaths: CoreCIERevision NotesNumberCompound Measures: Speed, Density, PressureCompound Measures

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First exams 2025

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Compound Measures (CIE IGCSE Maths: Core)

Revision Note

Test Yourself
Naomi C

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Naomi C

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Maths

Compound Measures

What is a compound measure?

  • A compound measure is something that is calculated by using more than one measurement
  • Compound measures can be used to measure rates
    • This measures how much one quantity changes when the other is increased by 1
    • Examples include:
      • Speed – how much the distance changes for each unit of time
      • Flow rate – how much the volume changes for each unit of time
      • Population density – how many people there are for each unit of area
      • Fuel consumption - volume of  fuel used for each unit of distance travelled

How do I find the units for a compound measure?

  • You can use the formula for a compound measure to derive its units
    • Use the units for the quantities in the formula to derive the units of the compound measure
    • Write a division as a/b or ab-1 and pronounce it as “a per b”
  • Examples include:
    • Speed space equals fraction numerator space Distance over denominator Time end fraction
      • If the distance is measured in km and the time is measured in minutes then the speed is measured in km/min or km min-1
    • Flow space rate space equals fraction numerator Volume space over denominator Time end fraction
      • If the volume is measured in m3 and the time is measured in minutes then the flow rate is measured in m3/min or m3min-1

How do I find the formula for a compound measure?

  • You can use the units for a compound measure to help remember its formula
    • You just need to remember what each unit measures
    • If the unit is a/b then the formula will be the quantity that a measures divided by the quantity that b measures
  • Examples include:
    • Density can be measured in kg/cm3
    • kg is a measure of mass and cm3 is a measure of volume
    • Therefore Density space equals space Mass over Volume

What is a formula triangle?

  • A formula triangle shows the relationship between the different measures in a compound formula 
    • E.g. for Speed, Distance and Time

Formula triangle: Speed, Distance, Time

    • If you are calculating a variable on the top of the triangle, multiply the two variables on the bottom
      • For example,  Distance space equals space Speed space cross times space Time 
    • If you are calculating a variable on the bottom of the triangle, divide the top by the other variable on the bottom
      • For example,  Speed space equals space Distance space divided by space Time  and  Time space equals space Distance space divided by space Speed

Exam Tip

  • Make sure all your units are consistent before using a formula.
    • If the answer for speed needs to be in km/h then make sure your distance is in km and your time is in hours.
  • Most compound measure formulas will be given to you in an exam.
    • However, you do need to remember the relationship between speed, distance and time.

Worked example

A high-speed racing car has an average fuel consumption of 3 km per litre during a race.

1 lap of the racing circuit is 5.9 km in length.

(a) Calculate the volume of fuel used, in litres, to complete 15 laps of the circuit.

The units for the fuel consumption are km per litre, which suggests the formula is fuel space consumption space equals fraction numerator distance space over denominator volume end fraction

Calculate the total distance covered for the 15 laps

15 space cross times space 5.9 space km space equals space 88.5 space km

Use the above formula to find the volume of fuel, V litres, used

table row cell fuel space consumption space end cell equals cell fraction numerator distance space over denominator volume end fraction end cell row cell 3 space km space per space litre space end cell equals cell fraction numerator 88.5 space km space over denominator V space litres end fraction end cell end table

Rearrange the equation by multiplying both sides by V, and dividing both sides by 3

table attributes columnalign right center left columnspacing 0px end attributes row cell 3 V end cell equals cell 88.5 end cell row V equals cell fraction numerator 88.5 over denominator 3 end fraction equals 29.5 end cell end table

29.5 litres of fuel

 

The race car then requires a pit-stop to refuel to complete the final laps of the race.

The flow rate of the fuel pump is 720 litres per minute, and fuel is pumped into the car for 3.1 seconds.

(b) Calculate the volume of fuel, in litres, pumped into the car in this time.

The flow rate is 720 litres per minute which suggests the formula is flow space rate space equals fraction numerator volume space over denominator time end fraction

Before we can use the formula, we need to change the units of time to both be the same

Change 720 litres per minute, into litres per second (to match the time fuel is pumped for, which is in seconds)

If 720 litres are pumped in 1 minute, 60 times less will be pumped in 1 second

720 space divided by space 60 space equals space 12 space litres space per space second

Substitute these values into the formula

12 space litres space per space second space equals fraction numerator volume space in space litres over denominator 3.1 space seconds end fraction

Multiply both sides by 3.1

volume space in space litres space equals space 12 space cross times space 3.1 space equals space 37.2

37.2 litres

Speed, Density & Pressure

What are speed, density and pressure?

  • Speed, density and pressure are frequently used compound measures
    • Speed is equal to distance divided by time
    • Density is equal to mass divided by volume
    • Pressure is equal to force divided by area

Formula Triangles for Speed, Density and Pressure

What should I know about speed, distance and time?

  • Speed is commonly measured in metres per second (m/s) or kilometres per hour (kmph)
    • The units indicate speed is distance per time
      Speed equals Distance over Time
  • 'Speed' (in this formula) means 'average speed'
  • In harder problems there are often two journeys or two parts to one longer journey

What should I know about density, mass and volume?

  • Density is usually measured in grams per centimetre cubed (g/cm3) or kilograms per metre cubed (kg/m3)
    • The units indicate that density is mass per volume
      Density equals Mass over Volume
  • You may need to use a volume formula to find the volume of an object first

What should I know about pressure, force and area?

  • Pressure is usually measured in Newtons per square metre (N/m2)
    • The units of pressure are often called Pascals (Pa) rather than N/m2
    • The units indicate that pressure is force per area
      Pressure equals Force over Area
  • Remember that weight is a force
    • It is different to mass

Exam Tip

  • Look out for a mixture of units
    • Time can be given as minutes but common phrases like 'half an hour' (30 minutes) could also be used in the same question
    • Any mixed units should be those in common use and easy to convert, e.g., g to kg or m to km etc

Worked example

A box exerts a force of 140 newtons on a table.
The pressure on the table is 35 newtons/m2.

p equals F over A        p space equals space pressure F space equals space force A space equals space area

Calculate the area of the box that is in contact with the table.

 

Method 1

Substitute the numbers you know into the formula

35 space equals space 140 over A      

Solve the equation for A
First multiply both sides by A to get A out of the denominator

35 A equals 140

Then divide by 35 to find the value of A

table row cell A space end cell equals cell space 140 over 35 space end cell end table

The units will be m2, matching the units seen in newtons/m2

4 m2
 

Method 2

Use the given formula to create a formula triangle for pressure, force and area

space space F p space space space space space space A      

A is on the bottom of the triangle, so this tells us to divide F by P

A space equals space F over p space equals space 140 over 35 space equals space 4      

4 m2 

Worked example

The density of pure gold is 19.3 g/cm3.

What is the volume of a gold bar that has a mass of 0.454 kg?

 

Begin by checking that all of the units are consistent.
Density is given in g/cm3
Convert the volume of the gold bar into grams to match the units

0.454 kg = (0.454 × 1000) g = 454 g

Use the units given for density to help you decide what calculation needs to be carried out
The units of density are g/cm3, so divide the mass (g) by the volume (cm3)

Or, write out the formula triangle

Formula triangle: Mass, Density, Volume

Write out the formula that you will need

Volume space equals fraction numerator space mass over denominator density end fraction

Substitute the given values for the mass and the density

Volume space space equals space fraction numerator space 454 over denominator 19.3 end fraction space equals space 23.5233...

Make sure you give the correct units with your final answer
The density is given in g/cm3, so the volume should be cm3

Volume = 23.5 cm3 (1 d.p.)

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    Naomi C

    Author: Naomi C

    Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.