Why is 2/5 of 1020 equal to 408 and not 40?

Because a fraction is an instruction, not a label. '2/5 of' means divide by 5, then multiply by 2: 1020 ÷ 5 = 204, then 204 × 2 = 408. Writing 40 reads 2/5 as '40%' and copies the digits. Size-check it: 2 out of every 5 is nearly half, so the answer should be close to half of 1020, which 408 is and 40 is not.

Why isn't 1/4 of 780 the same as 4% of 780?

Because 1/4 is the operator 'divide by 4', giving 780 ÷ 4 = 195. Reading 1/4 as '4%' gives 0.04 × 780 = 31.2, which is a tiny sliver of 780, not a quarter of it. AQA examiners report a minority of students computing 4% of the amount on exactly this kind of question.

What does the word 'of' mean in maths?

'Of' means multiply. For a fraction of a quantity, the safe method is: divide by the denominator (the bottom number) to make one equal part, then multiply by the numerator (the top number) to take that many parts. So 3/4 of £160 = 160 ÷ 4 × 3 = 40 × 3 = £120.

GCSE Maths Foundation · AQA · Fractions

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

On the calculator paper, "find a fraction of a quantity" trips students who read the fraction as a percentage label and write its digits. The AQA examiner's report for the "1020 books, $\dfrac{2}{5}$ are blue" question noted that "weaker responses ... worked out that it was 40% and gave the answer 40". On a separate paper, asking for $\dfrac{1}{4}$ of 780, the report flagged that "a very small minority incorrectly calculated 4% of the amount". One root cause, two surfaces: the fraction read as a label, not run as an instruction.

The thirty-second fix: a fraction is an operator, not a label. The word "of" means divide by the denominator, then multiply by the numerator. Divide to make one equal part, multiply to take how many you need. Then size-check: the answer must match "how many parts out of how many".

Ready to fix this? The Fractions, decimals & place value lesson works through this misconception and the others in Fractions, decimals & place value, one altitude at a time.

How to spot it in your own work

You answered $40$ for " $\dfrac{2}{5}$ of 1020 books" by reading $\dfrac{2}{5}$ as 40%.
You answered $31.2$ for " $\dfrac{1}{4}$ of 780" by computing 4% of 780.
You wrote the fraction's digits straight down as the answer instead of dividing and multiplying.
Your answer was far smaller than it should be, and you did not size-check it against "parts out of parts".

An exam question that triggers it

Here is the canonical AQA Foundation trigger, the shape of JUN23 Paper 2 Q10a:

There are 1020 books in a box. $\dfrac{2}{5}$ of the books are blue.

How many books are blue?

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".

Why students fall for this

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.

The fix: Divide by the bottom, multiply by the top

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.

Worked example

Work out $\dfrac{1}{4}$ of 780.

Read "of" as an instruction. The denominator is 4, so divide the quantity into 4 equal parts: $780 \div 4 = 195$ .
Multiply by the numerator. The numerator is 1, so take one of those parts: $195 \times 1 = 195$ .
Answer.
$\dfrac{1}{4} \text{ of } 780 = 780 \div 4 = 195$
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.

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Common questions

How do I work out a fraction of an amount on the calculator paper?: Divide the amount by the denominator (the bottom number), then multiply by the numerator (the top number). $\dfrac{3}{4}$ of £160 is $160 \div 4 = 40$ , then $40 \times 3 = 120$ , so £120. Never read the fraction's digits as the answer.
Why can't I just read $\dfrac{2}{5}$ as 40 and write that?: Because $\dfrac{2}{5}$ is an instruction, not a number you copy out. It means "divide by 5, then multiply by 2". The 40 comes from reading $\dfrac{2}{5}$ as the percentage 40% and writing its digits; AQAexaminers report exactly this on the 1020-books question. The instruction gives $1020 \div 5 \times 2 = 408$ .
How is a fraction of an amount different from a percentage of an amount?: They are the same idea in different costumes, but you must not swap the digits. $\dfrac{1}{4}$ of 780 means $780 \div 4 = 195$ . As a percentage that is 25% of 780, which is also 195. Reading $\dfrac{1}{4}$ as "4%" gives $31.2$ , a completely different and far too small number.
How do I know if my fraction-of answer is sensible?: Size-check it against "how many parts out of how many". $\dfrac{2}{5}$ is nearly half, so the answer should be close to half the quantity. $\dfrac{3}{8}$ is a bit under half. If your answer is a tiny sliver when the fraction is close to half, you have probably read the fraction as a label rather than an operator.

Related misconceptions

Adding fractions: why 1/2 + 1/3 is not 2/5The other big fraction trap: a fraction is one number, so you need a common denominator before adding.
Decimal place valueConverting fractions to decimals and ordering them: the next place column reasoning bites.
Error intervals and boundsRounding and bounds: another place where reading the instruction, not the surface, decides the answer.

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