Compound Measures (Edexcel IGCSE Maths A (Modular))

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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:
- $Speed = \frac{Distance}{Time}$
  - 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
- $Flow rate = \frac{Volume}{Time}$
  - 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 $Density = \frac{Mass}{Volume}$

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

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 $Time = Distance \div Speed$

Exam Tip

Check in the exam to see if the answer needs to be in different units
- For example, the question may use metres and seconds but want the answer in km/h
You 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 $fuel consumption = \frac{distance}{volume}$

Calculate the total distance covered for the 15 laps

$15 \times 5.9 km = 88.5 km$

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

$\begin{array}{rcl} fuel consumption & = & \frac{distance}{volume} \\ 3 km per litre & = & \frac{88.5 km}{V litres} \end{array}$

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

$\begin{array}{rcl} 3 V & = & 88.5 \\ V & = & \frac{88.5}{3} = 29.5 \end{array}$

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 $flow rate = \frac{volume}{time}$

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

$720 \div 60 = 12 litres per second$

Substitute these values into the formula

$12 litres per second = \frac{volume in litres}{3.1 seconds}$

Multiply both sides by 3.1

$volume in litres = 12 \times 3.1 = 37.2$

37.2 litres

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