Rates
Section: Number 2 | Syllabus: Cambridge IGCSE Mathematics (0580)
Common Measures of Rate
A rate compares two different quantities, such as pay per hour or one currency against another. Many real-life problems chain two or more rates together.
- Common rates include hourly rates of pay, exchange rates between currencies, flow rates, and fuel consumption
- To find a total from a rate, multiply the rate by the amount, e.g. total pay = hourly rate × hours worked
- Currency conversions sometimes need two steps - converting through a third currency in between
Worked Example: Hourly Pay
- Question: Meera works 7 hours at a rate of 12.50 per hour. How much does she earn?
- Step 1: Multiply the rate by the hours worked: 12.50 × 7 = 87.50
- Answer: 87.50
Worked Example: Chained Currency Conversion
- Question: The exchange rate between pounds and euros is £1 = €1.15. The exchange rate between pounds and South African rand is £1 = 23 rand. A laptop costs €460. Calculate the cost of the laptop in rand.
- Step 1: Convert euros to pounds: 460 ÷ 1.15 = 400
- Step 2: Convert pounds to rand: 400 × 23 = 9200
- Answer: 9200 rand
Worked Example: Fuel Consumption
- Question: A car travels 340 km and uses 40 litres of fuel. Find its fuel consumption in km per litre.
- Step 1: Divide the distance by the fuel used: 340 ÷ 40 = 8.5
- Answer: 8.5 km per litre
Common Mistakes
MistakeMultiplying by both exchange rates in a two-step currency conversion, instead of dividing by one and multiplying by the other
Fixwork out which direction each conversion goes - dividing converts into the base currency (pounds), multiplying converts out of it
MistakeDividing by the hourly rate instead of multiplying, when finding total pay
Fixto find a total from a rate, multiply the rate by the amount: total pay = hourly rate × hours worked
Density, Pressure and Population Density
Density, pressure and population density all follow the same pattern: one quantity divided by another. The formula needed is always given in the question, so the key skill is substituting the right numbers into it correctly.
- Density = mass ÷ volume, typically in g/cm³ or kg/m³
- Pressure = force ÷ area, typically in N/cm² or N/m²
- Population density = population ÷ area, typically in people per km²
Worked Example: Density
- Question: A metal block has a mass of 316 g and a volume of 40 cm³. Calculate its density.
- Step 1: Density = (mass/volume) = (316/40) = 7.9
- Answer: 7.9 g/cm³
Worked Example: Population Density
- Question: A city has a population of 850 000 and covers an area of 340 km². Calculate its population density.
- Step 1: Population density = (850\,000/340) = 2500
- Answer: 2500 people per km²
Common Mistakes
MistakeDividing the quantities the wrong way round, e.g. working out volume ÷ mass instead of mass ÷ volume
Fixalways check the exact formula given in the question, and substitute each value into the correct position
Average Speed
Unlike density and pressure, the speed formula is not given in the exam - it must be recalled. Time given in hours and minutes needs converting to a decimal number of hours before it can be used.
- Speed = (distance/time) - this formula must be known, since it is not given in the exam
- Rearranged forms: distance = speed × time, time = (distance/speed)
- Minutes must be converted to a decimal fraction of an hour before use, e.g. 30 minutes = 0.5 hours
Worked Example: Finding Average Speed
- Question: A train travels 217 km in 3 hours 30 minutes. Calculate its average speed in km/h.
- Step 1: Convert the time to decimal hours: 3 h 30 min = 3 + (30/60) = 3.5 hours
- Step 2: Divide distance by time: 217 ÷ 3.5 = 62
- Answer: 62 km/h
Worked Example: Rearranging the Speed Formula
- Question: A car travels at an average speed of 80 km/h for 2 hours 15 minutes. Find the distance travelled.
- Step 1: Convert the time: 2 h 15 min = 2.25 hours
- Step 2: Multiply speed by time: 80 × 2.25 = 180
- Answer: 180 km
- Question: A cyclist travels 54 km at an average speed of 15 km/h. How long does the journey take, in hours and minutes?
- Step 1: Divide distance by speed: 54 ÷ 15 = 3.6 hours
- Step 2: Convert the decimal part to minutes: 0.6 × 60 = 36 minutes
- Answer: 3 hours 36 minutes
Common Mistakes
MistakeUsing hours and minutes directly in the formula without converting, e.g. dividing distance by "3.30" instead of 3.5 hours
Fixminutes are not decimal hours - divide the minutes by 60 first: 30 minutes = 30 ÷ 60 = 0.5 hours
MistakeConverting the decimal part of an hour to minutes by taking it as it appears, e.g. reading 3.6 hours as 3 hours 6 minutes
Fixmultiply the decimal part by 60 to convert to minutes: 0.6 hours = 0.6 × 60 = 36 minutes, not 6 minutes
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