The idea

Fractions of amounts

Why "of" means multiply, why the base matters when a problem says "of the remainder", and how to reverse from a fractional part back to the whole.

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Number · Fractions & decimals

Why "of" means multiply, why the base matters when a problem says "of the remainder", and how to reverse from a fractional part back to the whole.

"Of" is multiplication, run through the denominator first.

\frac{3}{5}

of 240: the denominator cuts the amount into 5 equal parts (

240 \div 5 = 48

), the numerator takes 3 of them (

3 \times 48 = 144

). Divide by the bottom, multiply by the top — swapping the two roles (÷3, ×5) is the standard slip, and it fails the sense check:

\frac{3}{5}

of something must be less than it.

The single biggest trap in word problems: "Amir spends

\frac{1}{3}

of £480 on rent, then

\frac{1}{4}

of the remainder on food." That second fraction acts on the remainder (£320), not the original £480:

\tfrac{1}{3} \times 480 = 160, \qquad \text{then} \qquad \tfrac{1}{4} \times 320 = 80

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