Fraction of an amount: why 2/5 of 1020 is 408, not 40

Here is the canonical AQA Foundation trigger, identical in shape to JUN23 Paper 2 Q10a:

The misconception answer is $40$, found by reading $\dfrac{2}{5}$ as "40%" and writing the digits. Sanity check it: $\dfrac{2}{5}$ is 2 out of every 5, which is nearly half. So the blue books should be close to half of 1020, a big chunk, not 40. The label reading produces an answer that is far too small.

The correct answer is $408$. The fraction is an instruction: $1020 \div 5 = 204$ makes one fifth, then $204 \times 2 = 408$ takes two fifths. $408$ is just under half of 1020, which matches "2 out of every 5".

A fraction like $\dfrac{2}{5}$ is visually two small integers. A student under time pressure reaches for the fastest pattern they know: turn the fraction into a percentage they have half-memorised ($\dfrac{2}{5} = 40\%$), then read the digits straight off as the answer. The numerals 4 and 0 become "40 books". The instruction to divide and multiply never runs, so no arithmetic error is even made; the question was misread, not miscalculated.

The same shortcut produces the $\dfrac{1}{4}$ of 780 slip. Knowing $\dfrac{1}{4} = 25\%$ would at least be a percentage; but a student who reads the digits 1 and 4 as "4%" computes $0.04 \times 780 = 31.2$, a sliver, when a quarter of 780 is obviously a large part of it. The label hides the size; the operator reveals it.

A fraction is an operator, not a label. "Of" means divide by the denominator, then multiply by the numerator. The denominator names how many equal parts the whole is split into; the numerator says how many of those parts you take: $\dfrac{a}{b} \text{ of } N = (N \div b) \times a$.

For $\dfrac{2}{5}$ of 1020: divide first, $1020 \div 5 = 204$ (one fifth), then multiply, $204 \times 2 = 408$ (two fifths). Always finish with a size check: $\dfrac{2}{5}$ is nearly half, so 408 (just under half of 1020) is sensible and 40 is not.

Work out $\dfrac{1}{4}$ of 780 (AQA NOV24 Paper 3 Q1a).

1. Read "of" as an instruction. The denominator is 4, so divide the quantity into 4 equal parts: $780 \div 4 = 195$.
2. Multiply by the numerator. The numerator is 1, so take one of those parts: $195 \times 1 = 195$.
3. Answer.
   $$\dfrac{1}{4} \text{ of } 780 = 780 \div 4 = 195$$
4. Size check. A quarter of 780 should be a sizeable chunk, roughly 200. $195$ fits. The label answer $0.04 \times 780 = 31.2$ does not, so reading $\dfrac{1}{4}$ as "4%" must be wrong.

Notice the fraction's digits never became the answer. The bottom number told you what to divide by; the top number told you how many parts to take.