Compound Measures: Speed, Density and Rates
A compound measure combines two different quantities into a single rate: speed combines distance and time, speed = distance / time, density combines mass and volume, density = mass / volume, and other rates, such as pay per hour, follow the same pattern. Working with compound measures means rearranging the formula to find any one of the three quantities from the other two, while keeping every unit, such as mph, m/s or g/cm^3, consistent.
Before you start
Make sure you're comfortable with these topics first:
Method
- Learn the three compound measure formulas: speed = distance / time, density = mass / volume, and, where relevant, pressure = force / area.
- Use a formula triangle, or simple rearranging, to find whichever quantity is missing: covering the quantity being found shows whether to multiply or divide the other two.
- To find distance from speed and time, multiply: distance = speed x time. To find time from distance and speed, divide: time = distance / speed.
- Keep units consistent before calculating - convert time into hours (or minutes) to match the units of the speed, and convert mass and volume into compatible units (for example, g and cm^3, or kg and litres) before dividing for density.
- To convert a speed from km/h to m/s, multiply by 1000 and divide by 3600 (or simply divide by 3.6); to convert from m/s to km/h, multiply by 3.6.
- For average speed over a whole journey with different stages, always use total distance divided by total time - never average the individual stage speeds directly.
- Give the answer with the correct compound unit, such as m/s, mph, g/cm^3 or kg/m^3, since the unit is part of a correct answer for a rate.
Worked example
A stone has a mass of 172.5 g and a volume of 30 cm^3. (a) Calculate its density. (b) A different stone made of the same material has a volume of 52 cm^3. Work out its mass.
- Use the formula density = mass / volume: 172.5 / 30.
- Calculate: 172.5 / 30 = 5.75 g/cm^3.
- Since the second stone is the same material, it has the same density, 5.75 g/cm^3.
- Rearrange the formula to find mass: mass = density x volume.
- Calculate: 5.75 x 52 = 299 g.
Practice questions
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Q1A cyclist travels 52.5 km in 3 hours at a constant speed. Work out her average speed in km/h.Show answer
Answer: 17.5 km/h (52.5 / 3)
Q2A block of oak has a mass of 2.4 kg and a volume of 3200 cm^3. Work out its density in g/cm^3.Show answer
Answer: 0.75 g/cm^3 (2400 g / 3200 cm^3)
Q3Convert a speed of 108 km/h into m/s.Show answer
Answer: 30 m/s (108 / 3.6)
Q4A runner completes a 15 km race in 50 minutes at a constant pace. Work out how far she would run in 80 minutes at the same pace.Show answer
Answer: 24 km (15 / 50 = 0.3 km per minute, then 0.3 x 80)
Q5A hose pours water into a paddling pool at a constant rate, filling 250 litres in 20 minutes. Work out the rate of flow in litres per minute, then use it to find how long it would take to fill a larger pool holding 700 litres.Show answer
Answer: 12.5 litres per minute, then 56 minutes (700 / 12.5)
Q6A metal bar has a volume of 45 cm^3 and a density of 8.9 g/cm^3, the density of copper. Work out the mass of the bar, in grams.Show answer
Answer: 400.5 g (45 x 8.9)
Q7A delivery van covers 60 miles in the first hour of a journey in heavy traffic, then a further 140 miles over the next 3 hours on the motorway. Work out the van's average speed for the whole 4-hour journey.Show answer
Answer: 50 mph (total distance 200 miles / total time 4 hours)
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
A capsule of medicine is a cylinder-shaped tablet with a mass of 1.35 g and a volume of 0.9 cm^3. (a) Calculate the density of the tablet, in g/cm^3. (b) A pharmaceutical company designs a larger tablet from the same material with a volume of 2.4 cm^3. Work out the mass of the larger tablet.
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A ferry sails at an average speed of 18 knots (nautical miles per hour) for 2 hours 30 minutes, then slows to an average speed of 12 knots for a further 45 minutes to enter harbour. (a) Work out the total distance sailed, in nautical miles. (b) Work out the ferry's average speed for the whole journey, in knots, giving your answer to 1 decimal place.
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Free printable worksheet
Want more practice on paper? Download the compound measures: speed, density and rates worksheet pack - 6 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 42 of KS3 Maths Workbook 1, the whole course as one free printable PDF.
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