Direct and inverse proportion: the unitary method, recognising and using proportion relationships, and forming and using proportion equations with a constant of proportionality (Higher tier).

What this dot point is asking

Two quantities are in proportion when they change in a linked way. Edexcel expects you to solve direct proportion problems (as one increases, so does the other) and inverse proportion problems (as one increases, the other decreases), using the unitary method at Foundation tier and proportion equations with a constant of proportionality at Higher tier. Proportion is the engine behind recipes, currency conversion, best-buy comparisons and speed problems.

Direct proportion and the unitary method

In direct proportion, the two quantities keep a constant ratio: if one triples, so does the other. The unitary method is the reliable approach: find the value of a single unit, then multiply.

For $4$ pens costing $£3.20$, one pen costs $£3.20 \div 4 = £0.80$, so $7$ pens cost $7 \times £0.80 = £5.60$. The same method handles recipes (scaling ingredients), currency conversion (per unit of currency) and best-buy questions (cost per gram or per item).

Inverse proportion

In inverse proportion, the product of the two quantities stays constant: as one increases, the other decreases in step. If $6$ workers take $10$ days to do a job, the total work is $6 \times 10 = 60$ worker-days, so $4$ workers take $60 \div 4 = 15$ days. More workers means fewer days.

Proportion equations (Higher)

At Higher tier the relationship may involve powers, and you must form the equation, find the constant, and use it.

The four common forms are $y = kx$ (direct), $y = kx^2$ or $y = kx^3$ (direct to a power), $y = \dfrac{k}{x}$ (inverse) and $y = \dfrac{k}{x^2}$ (inverse square). Reading the wording carefully to pick the right form is the most important step, because the rest is routine substitution.

Recognising proportion from data

A table of values can reveal which kind of proportion is present. If dividing $y$ by $x$ gives the same value every time, it is direct proportion. If multiplying $x$ by $y$ gives a constant, it is inverse proportion. This check is useful when a question gives data and asks you to identify or justify the relationship.

Best-buy and conversion problems

Two everyday applications of direct proportion are best-buy comparisons and unit conversions. For a best buy, find the cost per unit (per gram, per item or per millilitre) for each option, then compare; the lower unit cost is the better value. For example, $500\,\text{g}$ for $£2.00$ is $0.4\,\text{p/g}$, while $750\,\text{g}$ for $£2.70$ is $0.36\,\text{p/g}$, so the larger pack is better value. Currency conversion is direct proportion through the exchange rate: if £1 = \1.25$, then$£40 = 40 \times 1.25 = \$50$, and converting back divides by the rate. The unitary method underlies both.

Try this

Q1. $6$ apples cost $£1.44$. Work out the cost of $10$ apples. [2 marks]

• Cue. One apple costs $£1.44 \div 6 = £0.24$, so $10$ cost $£2.40$.

Q2. $y$ is inversely proportional to $x$. When $x = 5$, $y = 12$. Find $y$ when $x = 4$. [3 marks]

• Cue. $y = \dfrac{k}{x}$, and $k = 5 \times 12 = 60$, so when $x = 4$, $y = \dfrac{60}{4} = 15$.