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Cambridge IGCSE Mathematics — 0580 Extended

Topic 1.11: Number — Rates

What is a rate?

A rate compares two quantities measured in different units. You divide one quantity by the other and write the answer with both units.

Examples: km/h (speed), people/km² (population density), litres/min (flow), g/cm³ (density).

Method

  1. Identify what the rate measures (e.g. speed, density).
  2. Write the formula or ratio: rate = first quantity \(\div\) second quantity.
  3. Convert units so they match what the question asks (especially time).
  4. Calculate and state the answer with units.

Average speed

Average speed is total distance divided by total time:

\[s = \dfrac{d}{t}\]

Rearrangements: \(d = s \times t\) and \(t = \dfrac{d}{s}\).

Which conversion?

Time given in hours and minutes

Convert to a single unit first — usually hours for km/h, or minutes for m/min.

Speed needed in m/s but distance in km

Convert km → m (\(\times 1000\)) and hours → seconds (\(\times 3600\)), or use km/h then convert.

A car travels 180 km in 2 h 30 min. Find the average speed in km/h.

Worked solution: 180 km in 2 hours 30 minutes is 72 km per hour
Convert 2 h 30 min to 2.5 h, then \(180 \div 2.5 = 72\) km/h

A runner covers 400 m in 50 s. Find the speed in m/s.

Worked solution: 400 m in 50 s is 8 m per second
\(400 \div 50 = 8\) m/s

Paper 2 (non-calculator)

Convert mixed times mentally: 1 h 15 min = 1.25 h; 45 min = 0.75 h. For m/s, divide distance by time — e.g. 300 m in 1 min 30 s → \(300 \div 90 = 3.33\ldots\) m/s.

Pay, exchange and flow rates

Hourly pay: total pay = hourly rate \(\times\) number of hours worked.

Exchange rate: multiply by the rate to convert one currency to another; divide to convert back.

Flow rate: volume (or mass) per unit time — e.g. litres/min, m³/s.

Fuel consumption: often km/l or litres/100 km — read the units carefully.

The exchange rate is £1 = 1.25 euros. Convert 240 euros to pounds.

Worked solution: 240 euros at 1.25 euros per pound is 192 pounds
Divide by the rate: \(240 \div 1.25 = £192\)

Try this

A tap delivers 9 litres in 6 minutes. What is the flow rate in litres/min?

Show answer
Answer

  1. Flow rate = volume \(\div\) time

    \[ \dfrac{9}{6} \]

  2. Final answer

    \[ = 1.5 \text{ litres/min} \]

Density, pressure and population density

In exams, the formula is usually given. You substitute and calculate — units matter.

RateFormula (given)Typical units
Density\(\text{density} = \dfrac{\text{mass}}{\text{volume}}\)g/cm³, kg/m³
Pressure\(\text{pressure} = \dfrac{\text{force}}{\text{area}}\)N/cm², Pa
Population density\(\text{density} = \dfrac{\text{population}}{\text{area}}\)people/km²

A block has mass 450 g and volume 150 cm³. Find the density in g/cm³.

Worked solution: density 450 g divided by 150 cm cubed equals 3 g per cm cubed
Density = mass \(\div\) volume: \(450 \div 150 = 3\) g/cm³

A force of 80 N acts on an area of 0.04 m². Find the pressure in N/m².

Worked solution: pressure 80 N divided by 0.04 m squared equals 2000 pascals
Pressure = force \(\div\) area: \(80 \div 0.04 = 2000\) N/m²

Exam Traps

  • Do not treat 2 h 30 min as 2.30 hours. 30 min is half an hour, so 2 h 30 min = 2.5 h.
  • If £1 = 1.25 euros, converting euros → pounds is a divide, not a multiply. Multiplying 240 by 1.25 gives euros, not pounds.

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