Compound Measures (AQA GCSE Maths)

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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 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:
    • 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 mins then the speed is measured in km/min
    • Pressure space equals space Force over Area
      • 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 space equals space Mass over Volume
    • 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 TimeSpeed distance equation triangle, IGCSE & GCSE revision notes

  • 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

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

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, so 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.

MASS DENSITY VOLUME equation triangle

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.)

Speed, Density & Pressure

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”

SDPHard Notes fig1, downloadable IGCSE & GCSE Maths revision notes

  • 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
  • It is important you understand and can do the basic questions with speed, density and pressure

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

SDPHard Notes fig2, downloadable IGCSE & GCSE Maths revision notes

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

SDPHard Notes fig3, downloadable IGCSE & GCSE Maths revision notes

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

Exam Tip

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

Worked example

SDPHard Example fig1 sol, downloadable IGCSE & GCSE Maths revision notes

SDPHard Example fig2 alt sol, downloadable IGCSE & GCSE Maths revision notes

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Amber

Author: Amber

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.