Syllabus Edition

First teaching 2021

Last exams 2024

Direct & Inverse Proportion (CIE IGCSE Maths: Core)

Revision Note

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Paul

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Maths

Proportion is a way of talking about how two quantities are related to each other.
There are two types of proportion to consider - direct proportion and inverse proportion.

Direct Proportion

What is direct proportion?

Direct proportion means that as one quantity increases/decreases by a certain rate, the other quantity will increase/decrease by that rate too (same factor)
- this means that the ratio of the two quantities is always the same
- e.g. if buying 2 boxes of cereal means you get 800 g of cornflakes then doubling the amount of boxes of cereal purchased (4 boxes) will double the amount of cornflakes (1600 g)

How do I solve direct proportion questions?

Direct proportion questions can be wordy, so spend some time reading through so the context of the question is clear

STEP 1

Identify the two quantities involved

e.g. "boxes of cereal" and "grams of cereal"

STEP 2

If not given in the question, find the factor that you will be increasing/decreasing by
Find this by dividing the 'new' quantity by the 'old' quantity

e.g. "doubling" - so the factor is 2. If this hadn't been given in words, work out "4 ÷ 2" (= 2)

STEP 3

Multiply the other quantity by this factor to find the required quantity

e.g. "800 x 2 = 1600"

STEP 4

Give your final answer in context, rounding and giving units where appropriate

e.g. "4 boxes of cereal would contain 1600 g of cornflakes"

Direct proportion questions may tell you the relationship between the two values as a ratio
e.g. "the ratio of boxes of cereal to grams of cornflakes is 2 : 800" (which is 1 : 400 simplified)

Exam Tip

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.

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.

STEP 1
Identify the two quantities

'profit' and 'bonus'

STEP 2
Find the factor ('new' ÷ 'old') from the profit

\frac{18 000}{3000} = 6

STEP 3
Multiply the bonus by the factor

250 \times 6 = 1500

STEP 4
Answer in context with units

An employee should receive $1500 bonus.

ii)

We are still working with profit and bonus; find the factor using 'new' ÷ 'old'

$\frac{600}{3000} = \frac{1}{5}$

Find the amount of bonus by multiplying by the factor

$250 \times \frac{1}{5} = 50$

Answer in context with units

The lowest amount of bonus an employee could receive is $50.

Inverse Proportion

What is inverse proportion?

Inverse proportion means that as one quantity increases by a certain rate, the other quantity will decrease by that rate too (same factor)
- this applies vice versa too, if one quantity decreases the other increases
- e.g. if 2 robots take 15 hours to build car then 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?

Inverse proportion questions can be wordy, so spend some time reading through so the context of the question is clear

STEP 1

Identify the two quantities involved

e.g. "number of robots" and "hours to build a car"

STEP 2

If not given in the question, find the factor that you will be increasing (or decreasing) by
Find this by dividing the 'new' quantity by the 'old' quantity

e.g. "tripling" - so the factor is 3. If this hadn't been given in words, work out "6 ÷ 2" (= 3)

STEP 3

Divide the other quantity by this factor to find the required quantity

e.g. "15 ÷ 3 = 5"

STEP 4

Give your final answer in context, rounding and giving units where appropriate

e.g. "6 robots would take 5 hours to build a car"

Some inverse proportion questions may tell you the relationship between the two values as a ratio
e.g. "the ratio of the number of robots to the hours taken to build a car is 2 : 15"

Exam Tip

Use common sense to determine if a question is direct or inverse proportion
- You'd expect more robots to be quicker at building a car
  - as the number of robots goes up, the time to build a car comes down
  - one quantity goes up, the other goes down - inverse proportion question
- Similarly, you'd expect more grams of cornflakes if you purchased more boxes of cereal
  - both quantities are going up - direct proportion

Worked example

The time taken to fill a swimming pool is inversely proportional to the number of pumps used to.
If 3 pumps are used it will take 12 hours to fill the pool.

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

STEP 1
Identify the two quantities

'number of pumps' and 'time (hours)'

STEP 2
Find the factor ('new' ÷ 'old') from the number of pumps

\frac{9}{3} = 3

STEP 3
Divide the time by the factor

12 \div 3 = 4

STEP 4
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'

$\frac{2}{3} (= 0.666 666 . . .)$

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

$12 \div \frac{2}{3} = 18$

Answer in context with units

If only 2 pumps are available then it will take 18 hours to fill the swimming pool.

Check your answer is sensible - less pumps would mean more time!

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