Converting Compound Units (GCSE Maths)

What is a compound unit?

A compound unit involves two (or more) base units combined by division or multiplication. The word "per" signals division:

• metres per second (m/s) = metres ÷ seconds — a unit of speed.
• grams per cubic centimetre (g/cm³) = grams ÷ cm³ — a unit of density.
• pounds per square metre (£/m²) = pounds ÷ m² — a unit of cost per area.

Converting a compound unit means replacing each base unit with its equivalent in the new unit system.

How do you convert m/s to km/h?

The unit fractions method:

Write the conversion you want to make as a fraction equal to 1, then multiply.

To convert m/s → km/h, you need two conversions:

• 1 km = 1000 m, so 1 m = 1/1000 km.
• 1 hour = 3600 seconds, so 1 s = 1/3600 h.

Worked example: Convert 20 m/s to km/h.

1. Replace metres: 20 m/s × (1 km / 1000 m) = 20/1000 km/s = 0.02 km/s.
2. Replace seconds: 0.02 km/s ÷ (1 h / 3600 s) = 0.02 × 3600 km/h = 72 km/h.

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

How do you convert g/cm³ to kg/m³?

Density is mass ÷ volume. Converting density from g/cm³ to kg/m³ requires converting both mass (grams → kilograms) and volume (cm³ → m³) separately.

Key facts:

• 1 kg = 1000 g, so 1 g = 1/1000 kg.
• 1 m = 100 cm, so 1 m³ = (100)³ cm³ = 1 000 000 cm³.

Worked example: Convert 8.9 g/cm³ to kg/m³.

1. Convert grams to kilograms: 8.9 g/cm³ × (1 kg / 1000 g) = 0.0089 kg/cm³.
2. Convert cm³ to m³: 0.0089 kg/cm³ × (1 000 000 cm³ / 1 m³) = 8 900 kg/m³.

Shortcut: To convert g/cm³ to kg/m³, multiply by 1000. (The factor of 1 000 000 from the volume cancels with the 1/1000 from the mass to leave × 1000.)

What is the general method for converting any compound unit?

Step-by-step method:

1. Identify each base unit that needs changing.
2. Write the conversion factor for each base unit.
3. Decide whether to multiply or divide: ask "am I going to a larger or smaller unit?"
   • Going to a smaller unit (e.g., km → m): multiply the numerator or divide the denominator.
   • Going to a larger unit (e.g., g → kg): divide the numerator or multiply the denominator.
4. Combine the conversions.

Example: Convert 5 km/h to m/min.

1. Base units: km (length) and h (time).
2. km → m: multiply by 1000. h → min: divide by 60 (1 h = 60 min).
3. 5 km/h = 5 × 1000 m ÷ 60 min = 5000/60 m/min ≈ 83.3 m/min.

How do compound unit conversions appear in GCSE problems?

GCSE questions often embed compound unit conversions inside a larger problem, for example:

Worked example: A car travels at 54 km/h. Convert this speed to m/s. Then find how far the car travels in 5 seconds.

1. Convert 54 km/h to m/s: 54 ÷ 3.6 = 15 m/s.
2. Distance = speed × time = 15 × 5 = 75 m.

Recognising that a conversion is needed — and choosing the right direction — is the key skill. Always check your answer makes sense: 15 m/s is a reasonable car speed; 15 m in 5 s feels right for city driving.

Frequently asked questions

Do I need to memorise the 3.6 shortcut for m/s and km/h?

It is worth knowing because it comes up very often, but understanding where it comes from is better than memorising it blindly. 3600 seconds in an hour ÷ 1000 metres in a kilometre = 3.6. If you forget the shortcut mid-exam, you can re-derive it using the unit fractions method.

Why does multiplying by 1 000 000 come into the density conversion?

Because 1 m³ is much larger than 1 cm³. Since 1 m = 100 cm, one cubic metre contains 100 × 100 × 100 = 1 000 000 cubic centimetres. Forgetting this cube is the most common error in density unit conversions.

How do I know whether to multiply or divide when converting?

Ask: "Is the new unit bigger or smaller than the old unit?" If you are moving to a bigger unit (e.g., cm to m), you divide — the number gets smaller. If you are moving to a smaller unit (e.g., m to cm), you multiply — the number gets bigger. Apply this to each base unit in the compound unit separately.

Can compound unit conversions appear in the non-calculator paper?

Yes. The conversions are designed to give exact or manageable numbers. Practise without a calculator using common conversions: 1 km = 1000 m, 1 hour = 60 min = 3600 s, 1 kg = 1000 g, 1 litre = 1000 cm³.

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Professor Pi can walk you through unit conversion problems with hints at every step — visit aitutors.me.