Pressure and Other Compound Measures

This lesson covers pressure and other compound measures that feature in the Edexcel GCSE Mathematics (1MA1) specification. Compound measures involve a combination of two or more basic measures. As well as speed and density (covered in previous lessons), you need to be able to work with pressure, population density, flow rates and fuel consumption.

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Key Vocabulary

Term	Meaning	Units
Pressure	Force per unit area	Pascals (Pa) or N/m^2
Force	A push or pull on an object	Newtons (N)
Area	The surface area over which the force acts	m^2 or cm^2
Population density	Number of people per unit area	people per km^2
Flow rate	Volume of fluid passing a point per unit time	litres per minute, m^3 per hour
Fuel consumption	Distance travelled per unit of fuel	miles per gallon (mpg) or km per litre

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Pressure

The Pressure Formula

• Pressure = Force / Area ($P = \frac{F}{A}$P=AF​)
• Force = Pressure x Area ($F = P \times A$F=P×A)
• Area = Force / Pressure ($A = \frac{F}{P}$A=PF​)

Worked Example 1

A box exerts a force of 600 N on the ground. The base of the box has an area of 0.15 m^2. Calculate the pressure.

• Pressure = 600 / 0.15 = 4,000 Pa (or 4,000 N/m^2)

Worked Example 2

A woman standing on one foot exerts a pressure of 25,000 Pa. Her weight is 500 N. What is the area of the sole of her shoe?

• Area = Force / Pressure = 500 / 25,000 = 0.02 m^2 = 200 cm^2

Worked Example 3

A hydraulic jack has a piston area of 0.005 m^2. A pressure of 800,000 Pa is applied. What force does the jack exert?

• Force = Pressure x Area = 800,000 x 0.005 = 4,000 N

Edexcel Exam Tip: Pressure questions often involve converting between m^2 and cm^2. Remember: 1 m^2 = 10,000 cm^2. If the area is given in cm^2, either convert to m^2 or ensure your final answer uses the correct units.

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Pressure in Context

Worked Example 4

A rectangular box has a base measuring 40 cm by 25 cm. It exerts a force of 200 N on the table. Find the pressure in N/cm^2 and in Pa.

1. Area = 40 x 25 = 1,000 cm^2
2. Pressure = 200 / 1,000 = 0.2 N/cm^2
3. Convert area to m^2: 1,000 cm^2 = 0.1 m^2
4. Pressure = 200 / 0.1 = 2,000 Pa

Worked Example 5

Two blocks are placed on a surface. Block A has a base area of 50 cm^2 and a weight of 80 N. Block B has a base area of 200 cm^2 and a weight of 160 N. Which exerts the greater pressure?

• Block A: P = 80 / 50 = 1.6 N/cm^2
• Block B: P = 160 / 200 = 0.8 N/cm^2
• Block A exerts greater pressure.

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Population Density

Population density measures how many people live per unit area.

Population density = Population / Area

Worked Example 6

A country has a population of 67 million and an area of 243,610 km^2. Find the population density.

• Population density = 67,000,000 / 243,610 = 275 people per km^2 (to the nearest whole number)

Worked Example 7

A city has a population density of 5,600 people per km^2 and an area of 1,572 km^2. Estimate the population.

• Population = 5,600 x 1,572 = 8,803,200

Worked Example 8

A region has a population of 420,000 and a population density of 350 people per km^2. Find the area.

• Area = 420,000 / 350 = 1,200 km^2

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Flow Rates

A flow rate measures the volume of fluid passing a point per unit time.

Flow rate = Volume / Time

Worked Example 9

A tap fills a 15-litre bucket in 2.5 minutes. What is the flow rate?

• Flow rate = 15 / 2.5 = 6 litres per minute

Worked Example 10

A pipe has a flow rate of 0.4 m^3 per hour. How much water flows through in 3 hours 30 minutes?

• Time = 3.5 hours
• Volume = 0.4 x 3.5 = 1.4 m^3

Worked Example 11

A swimming pool holds 375,000 litres of water. Water is pumped in at a rate of 50 litres per minute. How long will it take to fill the pool?

• Time = 375,000 / 50 = 7,500 minutes
• Convert to hours: 7,500 / 60 = 125 hours (or 5 days 5 hours)

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Fuel Consumption

Fuel consumption can be measured in miles per gallon (mpg), litres per 100 km, or km per litre.

Worked Example 12

A car travels 360 miles on 10 gallons of petrol. What is the fuel consumption in mpg?

• Fuel consumption = 360 / 10 = 36 mpg

Worked Example 13

A car does 45 miles per gallon. How many gallons are needed for a 315-mile journey?

• Gallons needed = 315 / 45 = 7 gallons

Worked Example 14

A car uses 8 litres of petrol per 100 km. How much petrol is needed for a 350 km journey?

• Petrol = (8 / 100) x 350 = 28 litres

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Unit Conversions for Compound Measures

Conversion	Method
N/cm^2 to Pa (N/m^2)	Multiply by 10,000
Pa to N/cm^2	Divide by 10,000
litres per minute to m^3 per hour	Multiply by 0.06
m^3 per hour to litres per minute	Divide by 0.06 (or multiply by 1000/60)

Worked Example 15

Convert 3.5 N/cm^2 to Pascals.

• 3.5 x 10,000 = 35,000 Pa

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Practice Problems