Working with Proportion (Cambridge (CIE) IGCSE International Maths)
Revision Note
Written by: Naomi C
Reviewed by: Dan Finlay
Working with Proportion
What is direct proportion?
Direct proportion
As one quantity increases/decreases by a certain rate (factor)
The other quantity will increase/decrease by the same rate
The ratio of the two quantities is constant
E.g. 2 boxes of cereal is 800 g of cornflakes
Doubling the number of boxes of cereal (4 boxes) will double the amount of cornflakes (1600 g)
How do I solve direct proportion questions?
Read through wordy direct proportion questions carefully
Ensure that you understand the context of the question
Some questions may tell you the relationship between the two values as a ratio
Identify the two quantities involved
E.g. Hours worked and pay
Find the factor that you will be increasing/decreasing by
This may be given to you in the question, e.g. 'the amount is tripled'
The quantity is multiplied by three
Alternatively, find the factor by dividing the 'new' quantity by the 'old' quantity
Multiply the other quantity by this factor to find the required quantity
E.g. If three times as many hours are worked, the pay will be three times more in total
Give your final answer in context
Round and give units where appropriate
Examiner Tips and Tricks
You may have to round an answer to a whole number, but think carefully about the context of the question!
Rounding to the nearest whole number is often appropriate
Sometimes you need to round up to the next whole number even if it is not the nearest
E.g. If you need 1.3 tins of paint, round the number of tins required up to 2 to ensure that you have enough paint
What is the unitary method?
The unitary method means finding one of something (1 unit of something)
This can be a useful strategy
For example, find the weight of 7 boxes, if 8 boxes weigh 60 kg
Find the weight of 1 box (1 unit) using division
60 kg ÷ 8 boxes = 7.5 kg per box
Scale this unit up using multiplication
7.5 kg per box × 7 boxes = 52.5 kg
How do I find which item is the best value?
A common proportion question is to find which product is the best value for money
Which is best value for money?
1.5 kg of flour for $0.84
or 5 kg of flour for $2.65?
To do this, compare the prices of 1 kg (1 unit):
Divide the price in dollars by the weight this buys
$0.84 ÷ 1.5 kg = $0.56 per kg
$2.65 ÷ 5 kg = $0.53 per kg
Make sure you compare prices for the same size unit of both items
The lowest unit price is the best value (0.53 < 0.56)
5 kg of flour for $2.65 is better value for money than 1.5 kg of flour for $0.84
Examiner Tips and Tricks
Start a question by jotting down the key values involved
You'll be able to pick them out quickly and easily later on
Worked Example
The bonus received by an employee is directly proportional to the profit made by the company they work for.
Bonuses are paid at a rate of $250 per $3000 profit the company makes.
(i) Work out the bonus an employee receives if the company makes a profit of $18 000.
(ii) If the company makes less than $600 profit, no bonus is paid.
Find the lowest bonus an employee could receive.
(i) Identify the two quantities 'profit' and 'bonus'
Find the factor ('new' ÷ 'old') from the profit
Multiply the bonus by the factor
Answer in context with units
An employee should receive a bonus of $1500
(ii) We are still working with profit and bonus
The lowest bonus will be when the company makes exactly $600 profit
Find the factor using 'new' ÷ 'old'
Find the amount of bonus by multiplying by the factor
Answer in context with units
The lowest amount of bonus an employee could receive is $50
What is inverse proportion?
Inverse proportion
As one quantity increases by a certain rate (factor)
The other quantity will decrease by the same rate
This relationship applies vice versa too, if one quantity decreases the other increases
E.g. If 2 robots take 15 hours to build a car
Tripling the number of robots (6) would mean the time taken to build a car would be divided by 3 (5 hours)
How do I solve inverse proportion questions?
Read through wordy inverse proportion questions carefully
Ensure that you understand the context of the question
Some questions may tell you the relationship between the two values as a ratio
Identify the two quantities involved
Find the factor that you will be increasing/decreasing by
This may be given to you in the question, e.g. 'the amount is tripled'
Alternatively, find this by dividing the 'new' quantity by the 'old' quantity
Divide the other quantity by this factor to find the required quantity
Give your final answer in context
Round and give units where appropriate
Examiner Tips and Tricks
Think about the context to determine if a question is direct or inverse proportion
As the number of robots goes up, the time to build a car comes down (inverse proportion)
If you buy more boxes of cereal, the amount of cereal also increases (direct proportion)
Worked Example
The time taken to fill a swimming pool is inversely proportional to the number of pumps used to pump the water in.
If 3 pumps are used it will take 12 hours to fill the pool.
(i) Work out the amount of time required to fill the pool if 9 pumps are used.
(ii) If only 2 pumps are available find out how much extra time will be needed to fill the pool.
(i) Identify the two quantities, 'number of pumps' and 'time (hours)'
Find the factor ('new' ÷ 'old') from the number of pumps
Divide the time by the factor
Answer in context with units
If 9 pumps are used it will take 4 hours to fill the swimming pool
(ii) We are still working with 'pumps' and 'time'
Find the factor using 'new' ÷ 'old'
Avoid rounding, keep the exact value in your calculator (it will be stored under the ANS key)
Find the time taken by dividing by the factor
Answer in context with units
If only 2 pumps are available then it will take 18 hours to fill the swimming pool
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