Compound Measures (Cambridge O Level Maths)

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
• Density – how heavy something is for each unit of its area or volume
• Pressure – how much force is applied to an object for each unit of its area
• Flow rate – how much the volume changes for each unit of time
• Population density – how many people there are for each unit of area

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
• You write a division as a/b or ab^-1 and pronounce it as “a per b”
• Examples include:
  • 
    • If the distance is measured in km and the time is measured in mins then the speed is measured in km/min
  • 
    • If the force is measured in N and the area is measured in cm² then the pressure is measured in N/cm²

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:
  • kg is a measure of mass and cm³ is a measure of volume
  • Therefore 
• Density can be measured in kg/cm³

Can I use a formula triangle?

• Yes – if you like them!
• STEP 1. Use the units in the question (or other info) to write down a formula

• STEP 2. Create a formula triangle

  • e.g. for Speed, Distance and Time

• STEP 3. For each part of the question write down 
• • what you know
  • what you want to know
• STEP 4. Use the formula triangle to rearrange the formula (if necessary)
• STEP 5. Substitute numbers in and solve

What are speed, density and pressure?

• Speed, density and pressure are compound measures – they are made from other measures
  • Speed is related to the measures distance and time
  • Density is related to mass and volume
  • Pressure is related to force and area
• The relationship between each of these sets of measures follows the same pattern
  • Sometimes referred to as “formula triangles”

• You need to remember the formula triangle for speed, distance and time
• For any other rates, a formula will be given to you in the exam, from which you can write out the formula triangle
• If you do not remember the formula triangles – do not worry, they can often be deduced from information given in the question or from the units

Problems involving speed, distance & time

• Speed is commonly measured in metres per second (m/s) or miles per hour (mph)
  • There are other possibilities such as kilometres per hour (kmph)
  • The units indicate speed is distance per time
    ie speed = distance ÷ time
• “Speed” (in this formula) means “average speed”
• In harder problems there are often two journeys – or two parts to one longer journey

Problems involving density, mass & volume

• Density is usually measured in grams per cubic centimetre (g/cm³) or kilograms per cubic metre (kg/m³)
• The units indicate that density is mass per volume
  • i.e. density = mass ÷ volume

• In harder problems there are often two metals (alloys), liquids or gases that have been combined rather than working with a single substance
  • You may need to use a volume formula to find the volume of an object first

Problems involving pressure, force & 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
  • ie pressure = force ÷ area
• Remember that weight is a force; it is different to mass