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Cambridge IGCSE International Maths — Extended
Compound Measures & Speed
A compound measure is just a "per" idea — like speed being distance per unit of time — and once you can spot the "per" in any unit, you can always figure out the formula and rearrange it, no memorising required.
Summary — What's in This Chapter
Speed = distance ÷ time, and it can be rearranged to find distance or time instead.
A formula triangle (with Distance on top, Speed and Time on the bottom) helps you rearrange without algebra — cover up what you want, and the triangle shows you the calculation.
Speed problems almost always involve a unit conversion — km ↔ m, hours ↔ minutes ↔ seconds. This is where most marks are lost.
A compound measure is any quantity built from two other measurements — speed, flow rate, population density, fuel consumption, and density are all examples.
You can work out the formula for any compound measure just by reading its units. If the unit is "a per b", the formula is (quantity a) ÷ (quantity b).
The same formula-triangle trick works for any compound measure — not just speed. Density and pressure use the exact same structure.
1. Speed
What speed actually means
Speed tells you how much distance is covered for every unit of time. That's it — that's the whole concept. If a car travels at 60 km/h, it means "for every hour that passes, this car covers 60 km." The unit itself is basically a mini-sentence: "kilometres per hour" = km ÷ h.
In real journeys, speed changes constantly (you speed up, slow down, stop at lights). So when a question says "speed," unless it specifies otherwise, it almost always means average speed — the overall rate for the whole journey, smoothing out all those changes.
Speed = Distance ÷ Time
Rearranged: Distance = Speed × Time, and Time = Distance ÷ Speed
Common units for speed: metres per second (m/s) for short, fast things (sprinters, cars in physics problems) and kilometres per hour (km/h) for longer journeys (road trips, flights).
The formula triangle — your rearranging shortcut
If rearranging Speed = Distance ÷ Time to solve for Time or Distance feels shaky, this diagram removes the guesswork completely.
Distance
Speed
Time
How to use it: cover up the letter you want to find with your finger. Whatever's left tells you the calculation.
Cover Distance (top) → left with Speed and Time side by side → multiply them: Distance = Speed × Time
Cover Speed (bottom-left) → left with Distance over Time → divide: Speed = Distance ÷ Time
Cover Time (bottom-right) → left with Distance over Speed → divide: Time = Distance ÷ Speed
Why this works
Anything on the top of the triangle is found by multiplying the two things on the bottom. Anything on the bottom is found by dividing the top by the other bottom value. This exact logic works for every compound measure triangle in this chapter — learn the pattern once, use it everywhere.
The unit-conversion trap
Here's the part that actually catches students out — not the speed formula itself, but converting units before you use it. Exam questions love mixing units on purpose: giving you a distance in km but wanting an answer in m/s, or giving you "three quarters of an hour" instead of a clean number of minutes.
Your conversion toolkit
1 km = 1000 m • 1 hour = 60 minutes • 1 minute = 60 seconds • 1 hour = 3600 seconds
Always ask first: "What units does my final answer need to be in?" Then convert everything to match before you plug numbers into the formula — not after.
Worked Example
A runner completes a 10 km race in three quarters of an hour. Find the average speed in metres per second, to 3 significant figures.
Step 1 — spot the target units: we need m/s, but we've been given km and a fraction of an hour. Convert both.
Step 4 — apply the formula:Speed = 10 000 ÷ 2700 = 3.703703… m/s
Step 5 — round to 3 s.f.:Speed = 3.70 m/s
Worked Example
A sprinter's average speed running 100 m is 8.85 m/s. Find the total time taken, to the nearest hundredth of a second.
Step 1 — check units: already in metres and m/s — no conversion needed here. That's a relief!
Step 2 — set up the equation:8.85 = 100 ÷ Time
Step 3 — rearrange (multiply both sides by Time, then divide by 8.85): Time = 100 ÷ 8.85 = 11.29943503…
Step 4 — round to 2 d.p.:Time = 11.30 s
Q1. A cyclist rides 24 km in 1 hour 20 minutes. Calculate her average speed in km/h.
Q2. A car travels at a constant speed of 45 m/s for 2 minutes. How far does it travel, in kilometres?
2. Compound Measures
What makes a measure "compound"?
A compound measure is simply any quantity built by combining two other measurements — usually by dividing one by the other. Speed is one example (distance ÷ time), but it's not special: the exact same idea applies to lots of other real-world rates.
Compound Measure
What it tells you
Speed
how much distance changes per unit of time
Flow rate
how much volume changes per unit of time
Population density
how many people there are per unit of area
Fuel consumption
how much fuel is used per unit of distance travelled
Density
how much mass is packed into a unit of volume
Pressure
how much force is applied per unit of area
The trick: read the units, find the formula
This is genuinely one of the most useful tricks in this whole chapter. You almost never need to memorise a compound measure's formula — you can derive it directly from its unit, because the unit is written as "a per b," and that literally means (quantity a) ÷ (quantity b).
Read it like a sentence
A unit written as a/b is pronounced "a per b" — and "per" always means divide. So km/min means "kilometres per minute," which tells you straight away that the formula is distance ÷ time, measured in those specific units.
Speed = Distance ÷ Time → unit: km/min or m/s
Flow rate = Volume ÷ Time → unit: m³/min
Worked-out example of the trick: if you're told density is measured in kg/cm³, break the unit apart: "kg" measures mass, "cm³" measures volume. So the formula must be:
Density = Mass ÷ Volume
You didn't need to recall this from memory at all — the unit told you exactly how to build the formula.
Formula triangles for other compound measures
Because density and pressure follow the exact same "one thing per another" structure as speed, they use identical triangles:
Mass
Density
Volume
Density, Mass, Volume
Force
Pressure
Area
Pressure, Force, Area
Cover the letter you want, and the triangle does the rearranging for you — for example, cover Area in the pressure triangle, and you're left with Force over Pressure, meaning Area = Force ÷ Pressure.
Worked Example
A racing car has average fuel consumption of 3 km per litre. One lap is 5.9 km. Calculate the volume of fuel used to complete 15 laps.
Step 1 — derive the formula from the unit: "km per litre" → fuel consumption = distance ÷ volume
Step 2 — find total distance:15 × 5.9 = 88.5 km
Step 3 — set up the equation:3 = 88.5 ÷ V
Step 4 — rearrange (multiply both sides by V, divide by 3): V = 88.5 ÷ 3 = 29.5
Answer:29.5 litres of fuel
Worked Example
A fuel pump has a flow rate of 720 litres per minute. Fuel is pumped for 3.1 seconds. Calculate the volume of fuel pumped, in litres.
Step 1 — spot the unit mismatch: flow rate is per minute, but time is given in seconds. Convert first!
Step 2 — convert the rate: 720 ÷ 60 = 12 litres per second
Step 3 — set up the equation:12 = volume ÷ 3.1
Step 4 — rearrange:volume = 12 × 3.1 = 37.2
Answer:37.2 litres
Q3. A block of metal has a mass of 540 g and a volume of 60 cm³. Work out the density, and state its units.
Q4. A city has a population of 1.2 million people living in an area of 400 km². What is the population density? What formula did you use, and how did you know?
What to Memorise
Term / Formula
Meaning
Speed = Distance ÷ Time
average speed over a journey
Distance = Speed × Time
rearranged version
Time = Distance ÷ Speed
rearranged version
Density = Mass ÷ Volume
how tightly packed matter is
Pressure = Force ÷ Area
force spread over a surface
Flow rate = Volume ÷ Time
how fast a liquid/gas moves through something
"a per b"
always means a ÷ b — the fastest way to find any compound formula
1 hour
= 60 minutes = 3600 seconds
1 km
= 1000 m
Concepts Checklist
Exam Tips & Common Mistakes
Forgetting to convert units before calculatingThe single biggest way marks are lost in this topic. If distance is in km but the answer needs m/s, convert distance to m and time to seconds first — never plug mismatched units straight into the formula.
Mishandling "hours and minutes" as if it were a decimal1 hour 20 minutes is NOT 1.20 hours — the minutes need to be divided by 60 first (20 ÷ 60 = 0.333), giving 1.333... hours. Mixing up decimal time and clock time is a classic trap.
Not reading the unit to find the formulaIf a question gives you an unfamiliar compound measure (e.g. "seats per row" or "cost per kg"), don't panic — read the unit as "a per b" and the formula is simply a ÷ b.
Rounding too earlyKeep full calculator accuracy throughout your working, and only round at the very last step, to the precision the question asks for (e.g. 3 significant figures or 2 decimal places).
Rearranging incorrectly by handIf algebraic rearranging feels risky under exam pressure, draw the formula triangle every time — it takes 5 seconds and removes the chance of dividing the wrong way round.
Examiner's eye
Examiners specifically design questions with mixed units (e.g. minutes and "half an hour" in the same question, or metres and km together) to test whether you convert correctly — always scan the whole question for units before you start calculating, not just the numbers you'll plug in.