Compound Measures

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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
- Write a division as a/b or ab^-1 and pronounce it as “a per b”
Examples include:
- $Speed = \frac{Distance}{Time}$
  - If the distance is measured in miles and the time is measured in hours then the speed is measured in miles per hour or mph
- $Pressure = \frac{Force}{Area}$
  - If the force is measured in N and the area is measured in cm² then the pressure is measured in N/cm² or N cm^-2

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 $Density = \frac{Mass}{Volume}$

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, $Distance = Speed \times Time$
- If you are calculating a variable on the bottom of the triangle, divide the top by the other variable on the bottom
  - For example, $Speed = Distance \div Time$ and $Time = Distance \div Speed$

Examiner Tip

Make sure all your units are consistent before using a formula.
- If the answer for speed needs to be in mph then make sure your distance is in miles 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

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/cm³Convert 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

$Volume = \frac{mass}{density}$

Substitute the given values for the mass and the density

$Volume = \frac{454}{19.3} = 23.5233 . . .$

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 frequently used compound measures
- Speed is related to distance and time
- Density is related to mass and volume
- Pressure is related to force and area

Formula Triangles for Speed, Density and Pressure

What are important things to know about problems involving 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
  $Speed = \frac{Distance}{Time}$
'Speed' (in this formula) means 'average speed'
In harder problems there are often two journeys or two parts to one longer journey

What are important things to know about problems involving density, mass and 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
  $Density = \frac{Mass}{Volume}$

You may need to use a volume formula to find the volume of an object first

What are important things to know about problems involving 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
  $Pressure = \frac{Force}{Area}$
Remember that weight is a force
- It is different to mass

Examiner 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.

$p = \frac{F}{A}$ $p = pressure F = force A = area$

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

Method 1

Substitute the numbers you know into the formula

$35 = \frac{140}{A}$

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

$35 A = 140$

Then divide by 35 to find the value of A

$\begin{array}{rcl} A & = & \frac{140}{35} \end{array}$

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

$F p A$

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

$A = \frac{F}{p} = \frac{140}{35} = 4$

4 m²

Compound Measures (Edexcel GCSE Maths: Foundation)

Revision Note