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Speed = distance ÷ time, density = mass ÷ volume, and pressure is measured in pascals (newtons per square metre). Free notes and 13 questions answered.
Standard Units and Compound Measures is part of Measures and accuracy in GCSE AQA Higher Maths.
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Speed = distance ÷ time, density = mass ÷ volume, and pressure is measured in pascals (newtons per square metre). Free notes and 13 questions answered.
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Use this lesson to stay inside Measures and accuracy while connecting the detail back to the wider Number topic.
Read the core explanations from Standard Units and Compound Measures before testing yourself from memory.
Every time you calculate a speed, read a medicine label, or check a fuel price, you are using units. Without agreed standard units, a measurement is meaningless — 'the journey took 3' tells you nothing without knowing whether that is 3 seconds, 3 hours, or 3 days.
GCSE Higher maths expects you to work fluently with standard units of mass, length, time, and money, and to calculate compound measures such as speed, density, and pressure. These topics appear on both calculator-allowed questions and questions without a calculator, so you need the methods locked in.
The good news is that every compound measure follows the same logical pattern: one quantity divided by another. Once you see that pattern, the whole topic clicks into place.
Standard units are the agreed measurements that everyone uses so that results can be compared and communicated. In the UK and in GCSE Maths, you work within the metric system for most quantities.
The table below shows the key standard units you are expected to know and use. Notice that each quantity has a base unit and then larger or smaller versions created by multiplying or dividing by powers of ten.
| Quantity | Base unit | Common alternatives | Key conversions |
|---|---|---|---|
| Length | metre (m) | km, cm, mm | 1 km = 1000 m; 1 m = 100 cm; 1 cm = 10 mm |
| Mass | kilogram (kg) | g, mg, tonnes | 1 kg = 1000 g; 1 tonne = 1000 kg |
| Time | second (s) | min, h | 1 min = 60 s; 1 h = 3600 s |
| Money | pound (£) | pence (p) | £1 = 100p |
| Volume | cubic metre (m³) | cm³, litres | 1 litre = 1000 cm³; 1 m³ = 1 000 000 cm³ |
When a question mixes units — for example, giving a distance in kilometres and a time in minutes — you must convert to a consistent unit system before calculating. Failing to do this is the single most common source of errors on compound measure questions.
A reliable way to check a conversion is to ask: am I moving to a smaller unit or a larger unit? Moving to a smaller unit (e.g. km to m) means the number gets bigger. Moving to a larger unit (e.g. cm to m) means the number gets smaller. If your answer goes the wrong way, you have divided when you should have multiplied, or vice versa.
Before moving on to compound measures, check your unit instincts with this question.
A compound measure combines two different quantities. Speed, for instance, combines distance and time. The word 'per' is your signal: metres per second, grams per cubic centimetre, newtons per square metre. 'Per' always means divided by.
There are three compound measures you need to know for GCSE Higher. Each one has a formula triangle that helps you rearrange for any of the three variables. We will look at each in turn, starting with speed.
Speed tells you how much distance is covered in a given time. A car travelling at 60 km/h covers 60 kilometres for every hour it travels. The formula connects speed, distance, and time in a straightforward relationship.
The formula triangle for speed, distance, and time is a useful memory aid. Cover the quantity you want to find: if you cover Speed, you see Distance over Time (divide). If you cover Distance, you see Speed × Time (multiply). If you cover Time, you see Distance over Speed (divide).
Worked example: Calculating Speed from Distance and Time
Mohammed cycles 24 km in 1.5 hours. Calculate his average speed in km/h. [2 marks]
Answer: 16 km/h
Notice that step 2 is often skipped by students in a hurry — and that is exactly where marks are lost. Always check your units before substituting. Now try a question where the units need converting first.
Density describes how much mass is packed into a given volume. A small block of lead feels much heavier than a same-sized block of foam because lead has a much higher density. This is why density is mass divided by volume, not the other way round.
Worked example: Finding the Volume of an Object Using Density
A block of aluminium has a mass of 540 g and a density of 2.7 g/cm³. Find its volume. [3 marks]
Answer: 200 cm³
The unit check in step 5 is a powerful self-checking tool. If your units do not simplify to the correct unit for the quantity you are finding, something has gone wrong in your working.
Pressure measures how much force is applied over a given area. Think of walking on snow: wide snowshoes spread your weight over a large area, reducing the pressure and stopping you from sinking. High heels concentrate force onto a tiny area, creating very high pressure.
Worked example: Calculating Pressure
A box exerts a force of 300 N on a floor. The base of the box has an area of 0.5 m². Calculate the pressure the box exerts on the floor. [2 marks]
Answer: 600 Pa
Step 3 highlights something that trips up many students: dividing by a number less than 1 makes the result larger, not smaller. 300 ÷ 0.5 = 600 because dividing by 0.5 is the same as multiplying by 2. If your pressure answer came out smaller than the force, check your division.
Converting a simple unit like km to m is straightforward. Converting a compound unit — such as km/h to m/s — requires you to convert both parts of the unit separately. This is where many students lose marks, because they only convert one part.
Here is the key insight: km/h stands for kilometres per hour. To convert this to m/s, first convert kilometres to metres (since 1 km = 1000 m) and hours to seconds (since 1 h = 3600 s). This means that speed in m/s is calculated as (speed in km/h × 1000) ÷ 3600, which simplifies to dividing by 3.6. Ensure that all parts of the unit conversion use consistent units.
Worked example: Converting 54 km/h into m/s
54 km/h: convert km to m (54 × 1000 = 54000 m) and hours to seconds (1 h = 3600 s). Speed = 54000 ÷ 3600 = 15 m/s.
Answer: 15 m/s
Earlier we saw that unit conversions for simple measures involve multiplying or dividing by a single factor. Now you can see the same logic applied to compound units — you handle each part of the unit in turn. The structure is identical; there is just one extra step.
Before tackling exam-style questions, it helps to have all three compound measure formulae side by side. This reference block is the kind of summary you might write at the top of your working in an exam to keep yourself on track.
Now put the formulae to work. The practice set below covers all three compound measures. Try each question before checking the solution — the act of attempting it first is what builds exam confidence.
Speed questions sometimes appear in a graphical form on GCSE Higher exam papers. A distance–time graph shows distance on the vertical axis and time on the horizontal axis. The gradient of the line at any point gives the speed at that moment.
A steeper line means a higher speed. A horizontal line means the object is stationary — distance is not changing, so speed is zero. This connects directly back to the formula: if distance is constant, speed = 0 ÷ time = 0.
In the graph above, the steeper blue line ($y = 2x$) represents a faster speed — for every 1 unit of time, distance increases by 2 units. The shallower red line ($y = 0.5x + 6$) represents a slower speed. The gradient of each line equals the speed: rise ÷ run = distance ÷ time.
Compound measure questions are reliable mark-earners if you follow a consistent method. The marks are structured so that showing the correct formula and substitution earns method marks even if your arithmetic slips. That means a clear, structured layout is worth marks in itself.
examiners report that the most common errors are: using the wrong formula (confusing density and pressure), forgetting to convert units before substituting, and omitting units from the final answer. Each of these is avoidable with a brief check before and after your calculation.
In questions without a calculator, you must show all arithmetic working in full. On calculator-allowed exam questions , you still need to show the formula and substitution — the calculator does the arithmetic, but the method marks come from your written working.
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Move to the related lessons or back to the topic page when you need the wider sequence before exam-style practice.
Speed = distance ÷ time, so distance = speed × time and time = distance ÷ speed. For example, cycling 24 km in 1.5 hours gives an average speed of 24 ÷ 1.5 = 16 km/h.
The unit of pressure is the pascal (Pa), which is N ÷ m², newtons per square metre. For example, a force of 300 N on an area of 0.5 m² gives a pressure of 300 ÷ 0.5 = 600 Pa.
Time is found by dividing distance by speed: Time = Distance ÷ Speed. In the formula triangle, covering Time leaves Distance over Speed, and units must match before substituting.
Volume is found from mass and density using Volume = Mass ÷ Density. For a block of aluminium with mass 540 g and density 2.7 g/cm³, Volume = 540 ÷ 2.7 = 200 cm³.
The main metric unit conversions are 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm, 1 kg = 1000 g and 1 tonne = 1000 kg. For time, 1 min = 60 s and 1 h = 3600 s, and for money £1 = 100p.
54 km/h is 15 m/s. Converting both parts, 54 km = 54 × 1000 = 54000 m and 1 h = 3600 s, so 54000 ÷ 3600 = 15 m/s; the shortcut is dividing by 3.6.
To convert km/h to m/s, multiply by 1000 to change kilometres to metres and divide by 3600 to change hours to seconds, which is the same as dividing by 3.6. For example, 90 km/h = 90 000 ÷ 3600 = 25 m/s.
A compound measure combines two or more standard measures using multiplication or division, such as speed, density and pressure. The word per signals division, so a speed in m/s means metres divided by seconds.
There are 1000 cm³ in 1 litre, and 1 m³ = 1 000 000 cm³. The base unit of volume is the cubic metre (m³), with cm³ and litres as common alternatives.
90 km/h is 25 m/s. 90 km = 90 000 m and 1 hour = 3600 seconds, so 90 000 ÷ 3600 = 25 m/s, which matches the shortcut 90 ÷ 3.6 = 25.
Per means divided by. It signals a compound measure, such as metres per second, grams per cubic centimetre or newtons per square metre, where one quantity is divided by another.
Distance is found by multiplying speed by time: Distance = Speed × Time. In the formula triangle, covering Distance leaves Speed × Time, and the units must be consistent, such as km/h with hours.
A unit conversion is checked by asking whether the unit is getting smaller or larger. Moving to a smaller unit, such as km to m, makes the number bigger, and moving to a larger unit, such as cm to m, makes the number smaller.