Units, Scale, Maps & Compound Measures — AQA GCSE Mathematics

SPEC R1, R2, R11, R12 • RATIO, PROPORTION & RATES OF CHANGE

Compound Measures

Unit conversions, map scales, area & volume scaling, and the big three compound measures: speed, density and pressure.

Why This Topic Matters

Compound measures and unit conversions appear on every paper, at every tier — often worth 3–5 marks and frequently hidden inside multi-step worded questions (lorry loads, population density, concrete deliveries, petrol costs).

The Big Idea

A compound measure combines two quantities into a single rate using division. Speed, density, pressure, rates of pay, unit prices and population density are all built the same way. Master the pattern once and you've covered them all.

What You Need To Be Able To Do

Convert between standard units — length, mass, volume, time.

Convert area units (cm² ↔ m²) and volume units (cm³ ↔ m³).

Use map scales to convert between map and real distance.

Calculate and compare using speed, density and pressure.

Get the units right on the final answer — the examiner looks for them.

Standard Unit Conversions

Know these conversion factors by heart — they appear on every paper and are not given in the formulae sheet.

QuantityConversions

Length1 km = 1000 m • 1 m = 100 cm • 1 cm = 10 mm

Mass1 tonne = 1000 kg • 1 kg = 1000 g • 1 g = 1000 mg

Volume / Capacity1 litre = 1000 ml • 1 cm³ = 1 ml • 1 m³ = 1000 litres

Time1 hour = 60 min = 3600 s • 1 min = 60 s

How to Remember the Direction

Going from a bigger unit to a smaller unit → you multiply (more small ones fit inside a big one). Going the other way → you divide. Example: 3.5 km into metres → multiply by 1000 → 3500 m.

Area & Volume Scaling

This is where most marks are lost on unit conversion. When you convert length, you use the conversion factor once. When you convert area, you must square it. When you convert volume, you must cube it.

The Rule

If length scales by a factor of k, then area scales by k² and volume scales by k³.

1 m² = 10 000 cm² (since 100² = 10 000)

1 m³ = 1 000 000 cm³ (since 100³ = 1 000 000)

The Classic Error

Writing "1 m² = 100 cm²" — treating area like length. Always square (or cube) the length factor when converting.

Map Scales

A map scale like "1 : 25 000" means 1 unit on the map represents 25 000 of the same units in real life. The unit is whatever you choose — usually cm, since that's how rulers measure on a map.

The Two Steps

Multiply the map distance by the scale number → real distance in the same unit (usually cm).

Convert the real distance to a sensible unit (usually m or km).

Quick Example

On a 1 : 25 000 map, two villages are 6 cm apart. Real distance:

Multiply: 6 × 25 000 = 150 000 cm.

Convert: 150 000 cm ÷ 100 = 1 500 m = 1.5 km.

Common Scales in Exams

1 : 25 000 is Ordnance Survey's "Explorer" scale. 1 : 50 000 is the "Landranger" scale. 1 : 100 000 is sometimes used for regional maps. In all cases — multiply first, then convert units.

Reverse Direction — Real to Map

If the question gives you the real distance and asks for the map distance, work backwards: convert real distance to cm first, then divide by the scale number. Example: real distance 3 km on a 1 : 25 000 map → 3 km = 300 000 cm ÷ 25 000 = 12 cm on the map.

The Three Compound Measures

The spec names three key compound measures. They all share the same structure: one quantity on top, two quantities on the bottom, connected by division.

How to Read Each Triangle

Cover the quantity you want to find — what remains tells you the operation:

Cover the top → bottom two are multiplied (e.g. D = S × T).

Cover a bottom → top is divided by the other bottom (e.g. S = D ÷ T).

Other Compound Measures

The spec also names everyday compound measures that follow exactly the same pattern: one quantity divided by another. Their names and units are different, but the procedure is identical.

MeasureFormulaTypical Units

Rate of paytotal pay ÷ time worked£/hour

Unit pricetotal cost ÷ quantity£/kg, p/ml, £/litre

Population densitypopulation ÷ areapeople/km²

Flow ratevolume ÷ timelitres/minute, m³/s

Fuel economydistance ÷ fuel usedmiles per gallon, km/litre

Best-Buy Questions — Unit Pricing

To compare two offers, compute the unit price (price per unit) for each and pick the smaller number. Example: "500 g for £2.40" vs "800 g for £3.60" → 0.48 p/g vs 0.45 p/g — the 800 g pack is better value.

Reading the Units Tells You the Formula

If you see "£ per hour" you know the numerator is £ and the denominator is hours. The slash in the unit is the division. This works for every compound measure — read the unit and write the formula.

Speed

Definition

Speed is the distance travelled per unit of time.Speed = Distance ÷ Time

Units — Must Match

m/s (metres per second) — distance in m, time in s.

km/h (kilometres per hour) — distance in km, time in h.

mph (miles per hour) — UK road speeds.

Converting km/h ↔ m/s

1 km/h = 1000 m / 3600 s = 1/3.6 m/s. So:

km/h → m/s: divide by 3.6

m/s → km/h: multiply by 3.6

Example: 90 km/h = 90 ÷ 3.6 = 25 m/s.

The Big Trap — Mixed Units

If distance is in km but time is in minutes, you cannot just divide — the answer would be meaningless. Convert one of them first so the units of speed are consistent (km/h or m/s).

Worked Example — Density with Unit Conversion01

A block of metal has a mass of 500 g and a volume of 200 cm³. Find the density of the metal in kg/m³. [3 marks]

M

Density = mass ÷ volume. Calculate in the given units (g/cm³), then convert to kg/m³.

C

Density = 500 ÷ 200 = 2.5 g/cm³.

Convert to kg/m³: 1 g/cm³ = 1000 kg/m³. So 2.5 × 1000 = 2500 kg/m³.

E

Density = 2500 kg/m³.

Why ×1000?

1 m³ = 10⁶ cm³ (divides density by 10⁶); 1 kg = 10³ g (divides density by 10³). Net effect: multiply by 10⁶ ÷ 10³ = 1000.

Density

Definition

Density is the mass per unit volume of a substance — how much matter is packed into a given space.Density = Mass ÷ Volume

Units — Two to Know

g/cm³ — mass in grams, volume in cm³ (usually for solids of small size).

kg/m³ — mass in kilograms, volume in m³ (bigger scale, including liquids and gases).

Conversion: 1 g/cm³ = 1000 kg/m³.

Useful Reference Densities

Water is 1 g/cm³ (= 1000 kg/m³). Anything denser than water sinks; less dense floats. Steel is about 8 g/cm³. Gold is about 19.3 g/cm³.

Irregular Shapes

When volume is found by water displacement ("a stone sinks and the water rises by 40 cm³"), that volume reading is the stone's volume. Use it directly in density = mass ÷ volume.

Pressure

Definition

Pressure is the force applied per unit area.Pressure = Force ÷ Area

Units

N/m² — also called pascals (Pa). Force in newtons, area in m².

N/cm² — for smaller scale contexts. Force in newtons, area in cm².

Intuition — Why Area Matters

The same force spread over a bigger area gives less pressure. That's why snowshoes work (big area, low pressure, no sinking) and why high heels dent floors (tiny area, huge pressure). Exam questions often exploit this — if the area changes, re-do the division.

Watch the Area Units

If force is in N and area is in cm², the answer is in N/cm² — not pascals. To get pascals (N/m²), convert the area to m² first (remember: 1 m² = 10 000 cm²).

02Worked Example — Pressure & Area Units

A rectangular box exerts a downward force of 600 N on the ground. The base of the box has area 400 cm².

(a) Calculate the pressure the box exerts on the ground, in N/cm². [2 marks]

(b) Give your answer to (a) in pascals (N/m²). [2 marks]

M (a)

Pressure = Force ÷ Area. Units match (N and cm²) — answer will be in N/cm².

C (a)

Pressure = 600 ÷ 400 = 1.5 N/cm².

M (b)

For pascals, the area must be in m². Convert: 400 cm² ÷ 10 000 = 0.04 m².

C (b)

Pressure = 600 ÷ 0.04 = 15 000 Pa.

Check: 1 N/cm² = 10 000 Pa, so 1.5 N/cm² × 10 000 = 15 000 Pa ✓.

E

(a) 1.5 N/cm² (b) 15 000 Pa.

03Worked Example — Speed & Unit Conversion

A car travels at a constant speed of 90 km/h.

(a) Convert this speed to m/s. [2 marks]

(b) At this speed, how far does the car travel in 45 seconds? Give your answer in metres. [2 marks]

M (a)

To convert km/h to m/s, divide by 3.6.

C (a)

90 ÷ 3.6 = 25 m/s.

M (b)

Distance = Speed × Time. Use the m/s value so the time units (seconds) match.

C (b)

Distance = 25 × 45 = 1125 m.

E

(a) 25 m/s (b) 1125 m.

04Worked Example — Distance, Speed & Time

A train travels at an average speed of 80 km/h.

(a) How far does the train travel in 45 minutes? Give your answer in km. [2 marks]

(b) How long, in hours and minutes, does it take to travel 180 km at this speed? [3 marks]

M (a)

D = S × T. Units must match — speed is in km/h, so convert 45 min to hours: 45/60 = 0.75 h.

C (a)

D = 80 × 0.75 = 60 km.

M (b)

T = D ÷ S. Both already in matching units (km and km/h).

C (b)

T = 180 ÷ 80 = 2.25 hours = 2 hours 15 minutes (0.25 h × 60 = 15 min).

E

(a) 60 km (b) 2 hours 15 minutes.

Common Errors to Avoid

These five errors account for the majority of marks lost on compound measure questions. Learn them so you don't repeat them.

Students often writeCorrect versionWhy it matters

60 km/h ÷ 60 = 1 m/s60 ÷ 3.6 ≈ 16.67 m/sConfusing "divide by 60" (minutes) with "÷ 3.6" (km/h to m/s)

1 m² = 100 cm²1 m² = 10 000 cm²Square the length factor (100²)

1 m³ = 1000 cm³1 m³ = 1 000 000 cm³Cube the length factor (100³)

Speed = km ÷ minutes (mixed units)Convert one unit first so the pair matchesMixed units give a meaningless answer

Answer given with no unitsAlways include units on the final lineExaminer looks for the unit as part of the mark

Examiner Alerts

Examiner Alert — Unit Mismatch in Compound Measures

The single biggest source of lost marks: computing with mismatched units. If you see "kilograms" in the mass and "cm³" in the volume, decide before dividing whether to convert the mass to grams or the volume to m³. Don't just divide and hope.

Examiner Alert — "Tonnes" in Worded Questions

Worded questions often give one quantity in tonnes and another in kilograms. Students routinely divide the two numbers as given, forgetting that 1 tonne = 1000 kg. Convert to one consistent unit first.

Showing Full Working

For 3–4 mark compound measure questions, show: (1) the formula, (2) the substituted values with units, (3) the intermediate result, (4) any unit conversion, (5) the final answer with units. Method marks live in each line.

Practice Question 1

A car travels 240 km in 3 hours. Work out the average speed of the car in km/h. [2 marks]

M

Speed = Distance ÷ Time.

C

Speed = 240 ÷ 3 = 80 km/h.

Practice Question 2

A small gold bar has mass 1158 g and volume 60 cm³. Work out the density of gold in g/cm³. [2 marks]

M

Density = Mass ÷ Volume. Units already match (g and cm³).

C

Density = 1158 ÷ 60 = 19.3 g/cm³.

Practice Question 3 — Map Scale

A map has a scale of 1 : 50 000. Two towns are 8 cm apart on the map. Work out the real distance between the two towns in kilometres. [3 marks]

M

Scale 1 : 50 000 means 1 cm on map = 50 000 cm in real life. Multiply, then convert cm to km.

C

Real distance = 8 × 50 000 = 400 000 cm.

Convert: 400 000 ÷ 100 = 4 000 m = 4 km.

E

4 km.

Practice Question 4 — Population Density

Town A has a population of 18 200 and an area of 520 km².

Town B has a population of 12 800 and an area of 320 km².

Which town has the greater population density? Show your working. [3 marks]

M

Population density = population ÷ area. Compute for each town, then compare.

C

Town A: 18 200 ÷ 520 = 35 people per km².

Town B: 12 800 ÷ 320 = 40 people per km².

E

Town B has the greater population density (40 > 35).

Practice Question 5 — Multi-leg Journey

A delivery driver's journey has two legs:

Leg 1: 30 km in 20 minutes.

Leg 2: 45 km in 40 minutes.

Work out the driver's average speed for the whole journey, in km/h. [4 marks]

M

Average speed = total distance ÷ total time — not the average of the two speeds. Convert all times to hours first.

C

Total distance = 30 + 45 = 75 km.

Total time = 20 + 40 = 60 minutes = 1 hour.

Average speed = 75 ÷ 1 = 75 km/h.

E

75 km/h.

Trap

Averaging the two speeds (90 km/h and 67.5 km/h) gives 78.75 km/h — wrong. Always use total ÷ total.

Summary

Unit Conversions to Memorise

Length: 1 km = 1000 m; 1 m = 100 cm; 1 cm = 10 mm.

Mass: 1 tonne = 1000 kg; 1 kg = 1000 g.

Volume: 1 litre = 1000 ml; 1 cm³ = 1 ml; 1 m³ = 1000 litres.

Time: 1 hour = 3600 s.

Area: 1 m² = 10 000 cm² (square the length factor).

Volume: 1 m³ = 1 000 000 cm³ (cube the length factor).

The Three Compound Measures

MeasureFormulaUnits

SpeedDistance ÷ Timem/s, km/h

DensityMass ÷ Volumeg/cm³, kg/m³

PressureForce ÷ AreaN/m² (= Pa), N/cm²

Key Conversions

km/h → m/s: divide by 3.6 . m/s → km/h: multiply by 3.6 .

g/cm³ → kg/m³: multiply by 1000 .

Map scale 1 : n → multiply map distance by n, then convert units.

The Three Traps

Mixed units: never divide km by minutes — convert one first.

Area/volume scaling: square (or cube) the length factor. Don't write "1 m² = 100 cm²".

Average speed: use total distance ÷ total time — don't average two speeds.