In direct proportion, both quantities increase (or decrease) together at the same rate — double one, double the other. In inverse proportion, as one quantity increases, the other decreases — double one, halve the other.

Find the value for one unit by dividing, then multiply up to the amount you need.

Example: 8 boxes weigh 60 kg. Find the weight of 7 boxes.

1. One box: $60 \div 8 = 7.5$ kg.
2. Seven boxes: $7.5 \times 7 = 52.5$ kg.

This is exactly the method used for scaling recipes: find the amount of one ingredient needed per single portion, then multiply by however many portions you need.

Instead of going via "one unit," find the multiplier linking the given amount directly to the new amount, then apply it once.

Example: A bonus is proportional to profit. A profit of \18{,}000$is 6 times a base profit of$\$3{,}000$, so the bonus is also $6$ times the base bonus.

Here you multiply to find the constant "total work," then divide to scale to the new number of units — the opposite order to direct proportion.

Example: 6 painters take 54 minutes to paint a room. How long would 12 painters take?

1. Total "painter-minutes" of work: $6 \times 54 = 324$.
2. Time for 12 painters: $324 \div 12 = 27$ minutes.

Many proportion questions are two-step: first divide a quantity using a ratio (e.g. apples:bananas $= 2:3$), then apply direct proportion to find a cost or weight for each share. "Best buy" questions use direct proportion too — find the cost (or weight) per single unit for each product, then compare like-for-like to see which is better value.

A recipe for 4 people needs 300 g of flour and takes one painter-independent task: 3 cooks can prepare it in 40 minutes.

(a) How much flour is needed for 10 people?

(b) How long would it take 5 cooks, at the same rate?

(a) Direct proportion, unitary method: one person needs $300 \div 4 = 75$ g. For 10 people: $75 \times 10 = 750$ g.

(b) Inverse proportion: total "cook-minutes" $= 3 \times 40 = 120$. For 5 cooks: $120 \div 5 = 24$ minutes.