Working with RatiosCambridge IGCSE Maths: Revision notes

Section 1

What is a ratio and how do I simplify it?

A ratio compares two or more quantities of the same kind. The ratio $20:30:40$ simplifies to $2:3:4$ by dividing every term by the highest common factor (here, 10).

Ratios have no units once simplified — convert all quantities to the same unit first (e.g. cm to cm, not cm to m)
A ratio in simplest form has no common factor left between all terms
$a:b$ is equivalent to $ka:kb$ for any non-zero $k$

Key termsratiosimplest form

Common mistake

Forgetting to convert to matching units before simplifying — e.g. 50 cm : 2 m must become 50:200 = 1:4, not 50:2.

Section 2

How do I divide a quantity in a given ratio?

To split an amount in a given ratio:

Add the parts of the ratio together to find the total number of shares
Divide the total quantity by the total shares to find the value of one share
Multiply each ratio part by the value of one share

Example: share $60 in the ratio$ 2:3:1 $. Total shares$ = 6 $, so one share$ = $10 $. The three amounts are$ $20, $30, $10$.

Key termsshare

Example

Divide 45 kg of fruit in the ratio 4:5. Total shares = 9, one share = 5 kg, so the amounts are 20 kg and 25 kg.

Section 3

How do I use ratios and proportional reasoning in context?

Ratios and proportion appear throughout real-world problems:

Recipes: scale ingredient quantities up or down keeping the same ratio between ingredients
Map scales: a scale of $1:25000$ means 1 cm on the map represents 25 000 cm (250 m) in real life
Best value: compare unit prices (price per gram, per item) to decide which product offers better value for money

Proportional reasoning means recognising that if one quantity doubles, a directly linked quantity also doubles.

Key termsscalebest valueunit price

Exam tip

When comparing best value, always divide price by quantity to get the SAME unit (e.g. pence per 100 g) for both products before comparing.

Section 4

What is direct proportion?

Two quantities are in direct proportion if their ratio stays constant — as one increases, the other increases at the same rate. This is written algebraically as $y = kx$ , where $k$ is the constant of proportionality.

Extended candidates also meet non-linear direct proportion:

$y \propto x^2$ means $y = kx^2$
$y \propto x^3$ means $y = kx^3$
$y \propto \sqrt{x}$ means $y = k\sqrt{x}$
$y \propto \sqrt[3]{x}$ means $y = k\sqrt[3]{x}$

To solve: substitute a known pair of values to find $k$ , then use the formula to find any unknown quantity.

Key termsdirect proportionconstant of proportionality

Example

If $y \propto x^2$ and $y = 18$ when $x = 3$ , then $18 = k(9)$ , so $k = 2$ and $y = 2x^2$ .

Section 5

What is inverse proportion?

Two quantities are in inverse proportion if one increases at the same rate as the other decreases — their product stays constant. This is written as $y = \frac{k}{x}$ .

As with direct proportion, substitute a known pair of values to find $k$ first, then use the formula for any other value.

Key termsinverse proportion

Common mistake

Mixing up direct and inverse proportion formulas — always check the wording: 'proportional to' with no mention of inverse means direct ( $y=kx$ ); 'inversely proportional to' means $y = k/x$ .

Must Know

Simplify a ratio by dividing all terms by their highest common factor, after matching units
To divide a quantity in a ratio: find total shares, find one share, multiply
Best value comparisons need the same unit for price (e.g. price per 100 g)
Direct proportion: $y = kx$ (or $kx^2$ , $kx^3$ , $k\sqrt{x}$ , $k\sqrt[3]{x}$ for Extended)
Inverse proportion: $y = \frac{k}{x}$
Always find $k$ first using given values before answering the question

That's the notes covered.

Carry on to the next subtopic.