Ratio to a fraction: why won : lost = 5 : 9 is 5/14, not 5/9

Here is the canonical AQA Foundation trigger (Jun24 P1 Q13b shape, non-calculator):

The misconception answer is $\tfrac{5}{9}$ — won over lost. But that measures the wins against the losses, not against all the games.

Add the parts to get the whole: $5 + 9 = 14$ games, so the fraction won is $\tfrac{5}{14}$.

A ratio gives two numbers side by side, and the most visible way to make a fraction from two numbers is to stack one on the other — so won : lost = $5 : 9$ becomes $\tfrac{5}{9}$. But the bottom of a fraction is the whole, everyone, and 9 is only the games lost — a part. The whole is the sum of the parts: $5 + 9 = 14$, so the fraction won is $\tfrac{5}{14}$. The giveaway is the size: a fraction of a part can never exceed 1, so an answer like $\tfrac{4}{3}$ for green : red $= 4 : 3$ must be wrong.

The same slip grows with a three-part ratio, where students put the first part over the rest: lemon : lime : orange = $1 : 2 : 2$ becomes $\tfrac{1}{4}$ instead of $\tfrac{1}{5}$. The whole still includes the part you are asking about, so the denominator is $1 + 2 + 2 = 5$.

A close cousin reads a ratio number straight as a percentage — boys : girls = $9 : 11$ read as “9% are boys”. You must find the fraction of the whole first, $\tfrac{9}{20}$, then convert: $\tfrac{9}{20} = 45\%$. AQA Foundation papers exploit every face — won : lost = 5 : 9 (Jun24 P1 Q13b), glasses 3 : 8 (Jun24 P3 Q17b), and green : red 4 : 3 (Jun23 P2 Q10b).

Add the parts to get the whole. For won : lost = $5 : 9$, the whole is $5 + 9 = 14$, so the fraction won is $\tfrac{5}{14}$ — never $\tfrac{5}{9}$.

It works for any number of parts. The whole always includes the part you are asking about: $1 : 2 : 2$ totals $1 + 2 + 2 = 5$, so the first part is $\tfrac{1}{5}$, not $\tfrac{1}{4}$.

A ratio number is not a percentage. For $9 : 11$ the boys are $\tfrac{9}{20} = 45\%$ of the class — find the fraction of the whole first, then convert.

In a class, glasses : no glasses = 3 : 8. What fraction wear glasses?

1. Add the parts to get the whole. Everyone in the class is $3 + 8 = 11$ parts.
2. Put the part over the whole.
   $$\text{fraction wearing glasses} = \frac{3}{3 + 8} = \frac{3}{11}$$
3. Sense-check the size. $\tfrac{3}{11}$ is less than 1, as a part of the whole must be; the trap $\tfrac{3}{8}$ measured glasses against the no-glasses group, not against everyone.

The same habit handles green : red = $4 : 3$: $4 + 3 = 7$, so green is $\tfrac{4}{7}$ and red is $\tfrac{3}{7}$ — the trap $\tfrac{4}{3}$ is bigger than 1, which a part can never be. Add the parts, then put the part over the total.