Solve problems involving direct and inverse proportion, including using the unitary method and forming proportion equations of the form $y = kx$ or $y = \dfrac{k}{x}$ with a constant of proportionality (Higher tier).

What this dot point is asking

OCR references R10 and R13 cover direct and inverse proportion: solving problems by the unitary method and, at Higher tier, forming proportion equations with a constant of proportionality $k$. Proportion is the formal version of ratio thinking and underpins compound measures, scale, and growth and decay. The Higher-tier algebraic proportion ($y = kx$, $y = kx^2$, $y = \tfrac{k}{x}$) is a reliable source of multi-mark questions and tests AO1 and AO2 together.

Direct proportion

Two quantities are directly proportional if their ratio stays constant.

The unitary method handles everyday cases: if $6$ apples cost £$1.50$, then one apple costs £$0.25$, so $10$ apples cost £$2.50$. The key step is finding the value of a single unit, which then scales to any amount. Recipes, currency conversion and "best buy" comparisons are all direct-proportion contexts.

Inverse proportion

In inverse proportion the product of the two quantities is constant.

The classic context is workers and time: if a job takes $4$ people $9$ days, the total work is $4 \times 9 = 36$ person-days, so $6$ people take $36 \div 6 = 6$ days. The product stays fixed while the two quantities trade off. Recognising inverse proportion is half the battle: more of one thing should mean less of the other.

Other inverse-proportion contexts include speed and journey time (a faster speed gives a shorter time for a fixed distance), and the number of items and the cost each for a fixed budget. The test is always whether increasing one quantity should decrease the other. If it should, the product is constant and you work with $xy = k$; if both should increase together, it is direct proportion and the ratio $\tfrac{y}{x} = k$ is constant instead. Deciding which model applies is the AO2 reasoning step that unlocks the rest of the question.

Forming proportion equations (Higher)

At Higher tier you build an algebraic model.

The same four-step structure (write the relationship, find $k$, write the equation, substitute) handles direct or inverse proportion to any power. Read the wording carefully: "proportional to the square" means $x^2$, "proportional to the square root" means $\sqrt{x}$, and "inversely" puts the variable on the bottom.

Why proportion matters

Proportion is the mathematics of scaling, and OCR sets it in pricing, mixing, physics-style and geometry contexts. The unitary method is a transferable problem-solving tool, while the Higher algebraic form connects proportion to graphs and to the constant of proportionality that appears throughout science. Stating the proportion equation explicitly earns method marks and makes the reasoning auditable.