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

The Eduqas ratio content asks you to solve direct and inverse proportion problems, using the unitary method at both tiers and, at Higher tier, forming a proportion equation with a constant of proportionality $k$. Direct proportion (two quantities rising together in a fixed ratio) and inverse proportion (one rising as the other falls) model recipes, currency, speed-time and work-rate problems, so they are heavily tested. The Higher-tier algebraic version, writing $y = kx$ or $y = \dfrac{k}{x}$ and finding $k$, is a dependable multi-mark question.

Direct proportion and the unitary method

Two quantities are in direct proportion when their ratio is constant, so a graph of one against the other is a straight line through the origin.

If $3$ kg of apples cost $2.40$ pounds, then $1$ kg costs $2.40 \div 3 = 0.80$ pounds, so $7$ kg cost $0.80 \times 7 = 5.60$ pounds. The unitary method works without any algebra and is the expected approach at Foundation, while the equation $y = kx$ packages the same idea for Higher.

Inverse proportion

In inverse proportion the two quantities change in opposite directions, but their product stays constant.

A classic context is workers and time: if $4$ workers take $6$ days to do a job, the total work is $4 \times 6 = 24$ worker-days, so $3$ workers take $24 \div 3 = 8$ days. The product being constant ($xy = 24$) is the engine of every inverse-proportion calculation.

Forming a proportion equation (Higher)

At Higher tier you set up and use an explicit equation, which also covers proportion to a power.

The same routine handles proportion to a cube ($y = kx^3$) or to a square root ($y = k\sqrt{x}$): identify the form from the wording, find $k$ from one data pair, then use the equation.

Recognising direct versus inverse

The wording tells you which model to use. "Directly proportional" or "in proportion" means $y = kx$; "inversely proportional" or "varies inversely" means $y = \dfrac{k}{x}$. A quick test: if making one quantity bigger should make the other bigger (more items, more cost) it is direct; if bigger should make the other smaller (more workers, less time) it is inverse. Choosing the right model is exactly the reasoning Eduqas rewards, so read the relationship carefully before calculating.

Proportion graphs

The two kinds of proportion look different when graphed, which is a useful check. Direct proportion $y = kx$ is a straight line through the origin, with gradient equal to the constant $k$, so a graph that is straight and passes through $(0, 0)$ confirms direct proportion. Inverse proportion $y = \dfrac{k}{x}$ is a reciprocal curve with two branches that approach but never touch the axes, falling steeply at first and then levelling off. Eduqas sometimes gives the graph and asks you to identify the type of proportion or to read off the constant, so recognising these shapes lets you connect the algebra to the picture quickly.

Why proportion matters

Proportion is one of the most widely applied ideas in the whole qualification because so many real relationships are proportional. Currency exchange, scaling a recipe, fuel used against distance, and the time a task takes against the number of people doing it are all proportion problems, and they recur in the compound-measures and percentage work too. The unitary method gives a reliable route at Foundation, while the equation method at Higher generalises to powers and gives the cleaner reasoning that examiners reward in longer questions.