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.

Number · Fractions & decimals

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.

Why it works

"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.

Watch what the fraction is a fraction OF. 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 \;\Rightarrow\; \text{remainder } 320; \qquad \tfrac{1}{4} \times 320 = 80 \;\Rightarrow\; \text{left } 240.

Track the running amount line by line; every "of" binds to the most recent whole the sentence names.

Chains of fractions multiply into one. "

\frac{2}{5}

don't walk, and

\frac{1}{4}

of those cycle" — the cyclists are

\frac{1}{4} \times \frac{2}{5} = \frac{1}{10}

of everyone. Collapsing the chain first turns a two-step problem into one clean equation.

Reversing: from the part back to the whole. If

\frac{2}{5}

of a number is 84, one fifth is

84 \div 2 = 42

, so the number is

5 \times 42 = 210

. Divide by the numerator to reach ONE unit fraction, multiply by the denominator to rebuild the whole — the same unitary move as reverse percentages.

One quantity as a fraction of another: same units first. 45 minutes as a fraction of 2 hours: convert —

\frac{45}{120} = \frac{3}{8}

. The of-quantity is the denominator; and minutes-over-hours without converting is meaningless.