14.3 Compound Measures

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: speed, density, pressure, rates and converting compound units. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Understand that a compound measure combines two quantities, such as distance and time
• Use the formulae for speed, density and pressure
• Solve problems with other rates, such as rates of pay and flow rates
• Convert between compound units, such as km/h and m/s

Key words

• Compound measure: A measure made from two other measures, for example speed from distance and time.
• Rate: How much of one quantity per unit of another, for example £12 per hour or 8 litres per minute.
• Density: Mass per unit of volume, in g/cm³ or kg/m³.
• Pressure: Force per unit of area, in newtons per square metre (N/m²) or N/cm².

Rates and their units

The unit of a compound measure tells you how to work it out. “km/h” means kilometres ÷ hours. “g/cm³” means grams ÷ cubic centimetres.

• Speed; Formula speed = distance ÷ time; Common units km/h, m/s, mph
• Density; Formula density = mass ÷ volume; Common units g/cm³, kg/m³
• Pressure; Formula pressure = force ÷ area; Common units N/m², N/cm²
• Rate of pay; Formula pay rate = pay ÷ hours; Common units £ per hour
• Flow rate; Formula flow rate = volume ÷ time; Common units litres per minute

Speed

Worked example: finding speed and time

(a) A coach travels 255 km in 3 hours. Speed = 255 ÷ 3 = 85 km/h.
(b) How long does it take to cycle 42 km at 12 km/h? Time = 42 ÷ 12 = 3.5 hours = 3 hours 30 minutes.

Answer: 85 km/h; 3 h 30 min

There is more on speed, journeys and travel graphs in section 14.4.

Density

Worked example: finding density and mass

(a) A metal block has mass 540 g and volume 60 cm³. Density = 540 ÷ 60 = 9 g/cm³.
(b) Oak has density 0.75 g/cm³. Find the mass of an oak plank with volume 2400 cm³.
Mass = density × volume = 0.75 × 2400 = 1800 g = 1.8 kg.

Answer: 9 g/cm³; 1800 g

Key fact: If the shape is a cuboid or prism, find the volume first, then use the density.

Pressure

The same force spread over a smaller area gives a bigger pressure. That is why a sharp knife cuts well.

Worked example: a crate on the floor

A crate exerts a force of 600 N on an area of 1.5 m². Pressure = 600 ÷ 1.5 = 400 N/m².
Turned onto a smaller face of area 0.5 m²: pressure = 600 ÷ 0.5 = 1200 N/m².

Answer: 400 N/m², then 1200 N/m²

Other rates and converting units

Worked example: a rate of pay

Sam is paid £186 for 15 hours of work. Pay rate = 186 ÷ 15 = £12.40 per hour.
How much is he paid for 22 hours? 22 × 12.40 = £272.80.

Answer: £12.40 per hour; £272.80

Worked example: km/h to m/s

Change 72 km/h to m/s.
72 km = 72 000 m and 1 hour = 3600 seconds. 72 000 ÷ 3600 = 20.

Answer: 20 m/s

Key fact: km/h → m/s: × 1000 then ÷ 3600. m/s → km/h: × 3600 then ÷ 1000 (the same as × 3.6).

Common mistakes

• Dividing the wrong way round. Look at the units: g/cm³ means grams ÷ cm³.
• Writing 3.5 hours as 3 hours 50 minutes. 0.5 hours is 30 minutes.
• Mixing units, for example mass in kg with volume in cm³. Change to matching units first.
• Forgetting the units in the answer.

Revision questions

Try these, then check your answers below each one.

Question 1. A car travels 150 miles in 2.5 hours. Work out its average speed in mph.

Answer: 60 mph

Working: 150 ÷ 2.5 = 60 mph.

Question 2. A block has mass 96 g and volume 12 cm³. Work out its density in g/cm³.

Answer: 8 g/cm³

Working: 96 ÷ 12 = 8 g/cm³.

Question 3. A force of 900 N acts on an area of 3 m². Work out the pressure in N/m².

Answer: 300 N/m²

Working: 900 ÷ 3 = 300 N/m².

Related lessons

• Previous lesson: 14.2 Growth and Decay
• Next lesson: 14.4 Distance, Speed and Time
• In this chapter: 14.1 Percentages
• In this chapter: 14.5 Direct and Inverse Proportion

Continue in the full course

This lesson is section 14.3 (EGCMF1403) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.