Ratio: additive vs multiplicative scaling

Here is a recipe-style question that mirrors the AQA NOV24 ratio surface and triggers the additive trap directly:

The additive answer is $9$ cups, found by reasoning "I added 6 to the flour, so I add 6 to the sugar": $2 + 6 = 8$, $3 + 6 = 9$. It feels symmetric. It is also wrong — the ratio $8 : 9$ is no longer "for every 2 of flour, 3 of sugar".

The correct answer is $12$ cups. Flour scaled from 2 to 8, which is $\times 4$. Sugar must also scale $\times 4$, giving $3 \times 4 = 12$. Check the new ratio: $8 : 12 = 2 : 3$. The relationship is preserved.

Ratios use the colon notation $2 : 3$, which looks like a pair of integers sitting side by side. The brain reads them as two independent numbers and reaches for the only tool it has for adjusting two independent numbers — adding the same amount to each. The colon, like the fraction bar, looks decorative. It isn't: it's telling you that the two numbers are in a fixed proportional relationship.

The same shortcut explains the "one-part-is-the-total" error. On a question like 120 people split in ratio 2 : 3 car to bus, the brain sees three numbers (120, 2, 3) and reaches for the most familiar operation — divide the big number by one of the small ones. 120 ÷ 2 = 60 looks plausible and arrives quickly, which is precisely why AQA examiners flagged it on 3F NOV24 Q15. The correct move is to recognise that 120 is the total of $2 + 3 = 5$ equal shares, so each share is 24 — and 2 × 24 = 48 went by car, not 60.

A ratio expresses a multiplicative relationship. Scaling preserves the ratio of parts only when both parts scale by the same factor. To scale a ratio, work out the multiplier from one part (the part you have information about), then apply that same multiplier to every other part. If a ratio has three parts, all three get the same multiplier. If a ratio has already been simplified, you can still scale up to match a real-world quantity — but multiplicatively, never by addition.

For sharing problems ("split £120 in the ratio 2 : 3") the same logic applies in reverse: add the parts to find how many equal shares the total is divided into $(2 + 3 = 5)$, then divide the total by that to find the value of one share, then multiply each part of the ratio by that share-value. The total is the sum of the parts — never one of them.

Scale the recipe $\text{flour} : \text{sugar} = 2 : 3$ to use 8 cups of flour.

1. Identify the part you have information about. Flour: 2 in the ratio, 8 in the scaled recipe.
2. Work out the multiplier. $8 \div 2 = 4$. Flour was multiplied by 4.
3. Apply the same multiplier to every other part. Sugar: $3 \times 4 = 12$ cups.
   $$\text{flour} : \text{sugar} = 2 : 3 \;\xrightarrow{\times 4}\; 8 : 12$$
4. Check the new ratio simplifies back. Divide both parts of $8 : 12$ by 4: you get $2 : 3$. The relationship is preserved, so the answer is right.

Compare with the additive answer 9. The ratio $8 : 9$ does not simplify to $2 : 3$ (it doesn't simplify at all — 8 and 9 share no common factor). That mismatch is the structural sign that addition has broken the proportional link.