Compound measures

Compound measures combine two different measurements, usually by dividing one by the other. Speed is distance per unit of time (m/s, km/h), density is mass per unit of volume (g/cm³, kg/m³), and pressure is force per unit of area (N/m², or pascals). Rates such as litres per minute, pay per hour and price per 100 g are compound measures too.

The three main compound measures

The unit tells you the formula. "Kilometres per hour" means kilometres divided by hours; "grams per cubic centimetre" means grams divided by cubic centimetres.

Formula triangles

Cover the quantity you want. If the other two are side by side, multiply them; if one is above the other, divide. For speed: S = D ÷ T, D = S × T, and T = D ÷ S.

Speed

Example 1: average speed

A train travels 252 km in 1 hour 45 minutes. Find its average speed in km/h.

1 hour 45 minutes = 1.75 hours (45 ÷ 60 = 0.75). Speed = 252 ÷ 1.75 = 144 km/h.

Example 2: converting km/h to m/s

Convert 54 km/h to m/s.

54 km = 54 000 m and 1 hour = 3600 s, so 54 km/h = 54 000 ÷ 3600 = 15 m/s. A shortcut: divide km/h by 3.6 to get m/s, and multiply m/s by 3.6 to get km/h.

The speed equation guide has more speed problems.

Density

Example 3: the density of a block

A metal block measures 5 cm × 4 cm × 3 cm and has a mass of 540 g. Find its density.

Volume = 5 × 4 × 3 = 60 cm³. Density = 540 ÷ 60 = 9 g/cm³.

Converting density units

1 g/cm³ = 1000 kg/m³, because 1 m³ = 1 000 000 cm³ and 1 kg = 1000 g. So 9 g/cm³ = 9000 kg/m³. Water has a density of about 1 g/cm³, or 1000 kg/m³.

Example 4: mixing two liquids

200 cm³ of liquid A (density 0.8 g/cm³) is mixed with 300 cm³ of liquid B (density 1.2 g/cm³). Find the density of the mixture.

1. Mass of A = 0.8 × 200 = 160 g; mass of B = 1.2 × 300 = 360 g.
2. Total mass = 520 g; total volume = 500 cm³.
3. Density of the mixture = 520 ÷ 500 = 1.04 g/cm³.

You can't just average the two densities: there is more of liquid B, so the mixture is closer to B's density.

Pressure

Example 5: pressure on the ground

A crate weighs 900 N and rests on a base of area 0.3 m². Pressure = 900 ÷ 0.3 = 3000 Pa (N/m²).

If the area is in cm², the answer is in N/cm². Since 1 m² = 10 000 cm², 3000 N/m² = 0.3 N/cm².

Check your understanding.

Rates, rates of pay and best buys

Any "per" quantity is a rate. Flow rate is volume per unit time, rate of pay is money per hour, and unit price is cost per item or per gram.

Example 6: a flow rate

A tank holds 1500 litres and fills at 12 litres per minute. Time = volume ÷ rate = 1500 ÷ 12 = 125 minutes = 2 hours 5 minutes.

Example 7: best buy

Pack A: 500 g for £2.40. Pack B: 750 g for £3.45. Which is better value?

Find the price for the same amount of each. Pack A: £2.40 ÷ 5 = 48p per 100 g. Pack B: £3.45 ÷ 7.5 = 46p per 100 g. Pack B is better value. You could also compare grams per penny: the larger amount for each penny is the better buy.

Common mistakes

• Treating minutes as decimals of an hour. 2 hours 30 minutes is 2.5 hours.
• Mixing units. Make units match before dividing: km with hours, or m with seconds.
• Dividing the wrong way round. Use the units: g/cm³ means grams ÷ cm³.
• Averaging densities or speeds. Find total mass ÷ total volume, or total distance ÷ total time.
• Forgetting area conversions are squared. 1 m² = 10 000 cm², not 100 cm².

Practice questions

1. A cyclist rides 18 km in 45 minutes. Find the average speed in km/h.
2. Convert 30 m/s to km/h.
3. A stone has a mass of 260 g and a volume of 100 cm³. Find its density in kg/m³.
4. A force of 120 N acts on an area of 0.04 m². Find the pressure.
5. Rice costs £1.80 for 1 kg or £4.95 for 3 kg. Which is the better buy?

Answers: 1) 18 ÷ 0.75 = 24 km/h. 2) 30 × 3.6 = 108 km/h. 3) 2.6 g/cm³ = 2600 kg/m³. 4) 120 ÷ 0.04 = 3000 Pa. 5) 3 kg costs £1.65 per kg, cheaper than £1.80, so the 3 kg bag.

Practise with a short timed paper

Compound measures come up on both tiers of GCSE Maths, and in GCSE Science. The daily GCSE Maths Higher and GCSE Maths Foundation papers are short timed papers of 8 questions, with a new one every day.

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