Ratio, proportion and percentage are the same idea three ways

Ratio, proportion, and percentage are taught as separate chapters, which hides the fact that they're three views of one thing: comparing quantities by division. Once you can slide between them, a huge class of word problems collapses into one method.

They're all fractions in disguise

• A ratio $3:2$ compares two parts — and it's just the fraction $\frac{3}{2}$.
• A proportion says two ratios are equal — two fractions that match.
• A percentage is a fraction with the bottom fixed at $100$. "$15\%$" means $\frac{15}{100}$.

So $15\%$, $0.15$, and $\frac{3}{20}$ are the same number wearing three outfits. Being able to swap between them on sight is most of the skill.

Percentage of an amount

"$15\%$ of $240$" is just multiplication once you've converted the percent to a fraction or decimal:

Sharing in a ratio — the "parts" method

Share £$40$ in the ratio $3:2$. Don't guess — count the parts.

1. Total parts: $3 + 2 = 5$.
2. One part: $40 \div 5 = 8$.
3. Each share: $3\times8 = 24$ and $2\times8 = 16$. Check: $24 + 16 = 40$. ✓

Direct proportion — the unitary method

If $4$ pens cost £$3$, what do $10$ cost? Find the value of one, then scale up.

Always find "one" first. Cost of one item, length of one step, mass of one litre — the unitary method turns nearly every proportion question into two easy lines, and it never lets you down.

Last revised 12 November 2024.