ProportionCambridge IGCSE Maths: Revision notes

Section 1

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, and using the symbol $y \propto x$ .

To find unknown quantities, first find $k$ using one known pair of values, then use it to find the other unknown.

Example: if $y \propto x$ and $y=15$ when $x=3$ , then $k = \frac{15}{3} = 5$ , so $y = 5x$ . When $x = 8$ , $y = 40$ .

Key termsdirect proportionconstant of proportionality

Exam tip

Always find $k$ first using the given pair of values before answering the rest of the question — this is usually worth a method mark on its own.

Section 2

What is inverse proportion?

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

Example: if $y \propto \frac{1}{x}$ and $y=4$ when $x=5$ , then $k = 4 \times 5 = 20$ , so $y = \frac{20}{x}$ . When $x=2$ , $y=10$ .

Key termsinverse proportion

Common mistake

Don't confuse direct and inverse proportion — in direct proportion $y \div x$ is constant; in inverse proportion $y \times x$ is constant.

Section 3

What other types of proportion appear at IGCSE?

Proportion isn't limited to $x$ and $y$ directly — the same method applies when $y$ is proportional to a power or root of $x$ :

Relationship	Equation
$y$ proportional to $x^2$	$y = kx^2$
$y$ proportional to $x^3$	$y = kx^3$
$y$ proportional to $\sqrt{x}$	$y = k\sqrt{x}$
$y$ proportional to $\sqrt[3]{x}$	$y = k\sqrt[3]{x}$
$y$ inversely proportional to $x^2$	$y = \frac{k}{x^2}$

The method is identical: substitute the known pair to find $k$ , then use the formula to find any other unknown.

Example

If $y$ is proportional to $x^2$ and $y=50$ when $x=5$ , then $k = \frac{50}{25} = 2$ , so $y = 2x^2$ . When $x=4$ , $y=32$ .

Section 4

How do you use ratios and proportional reasoning in real-life contexts?

Proportional reasoning also underpins ratio problems in context, such as adapting recipes, using map scales, or determining best value.

Simplest form: divide all parts of a ratio by their HCF, e.g. $20:30:40 = 2:3:4$
Dividing a quantity: split into the total number of parts, then multiply by each share, e.g. share £60 in ratio 2:3:5 → total 10 parts → £6 per part → £12 : £18 : £30
Best value: compare unit prices (cost per single item or per unit of quantity) across different pack sizes

Key termssimplest formunit price

Must Know

Direct proportion: $y = kx$ (ratio $y \div x$ constant); inverse proportion: $y = \frac{k}{x}$ (product $xy$ constant)
Always find $k$ first from a known pair of values before answering the rest of a question
The same method applies to $y \propto x^2, x^3, \sqrt{x}, \sqrt[3]{x}$ — substitute to find $k$ , then use the formula
Use $\propto$ to denote proportionality: $y \propto x$
Simplify ratios by dividing by the HCF; divide quantities by finding the value of one part first

That's the notes covered.

Carry on to the next subtopic.