Percentage change: why 'increase by 20%' is not '20% of'

Here is a classic Foundation percentage-change item.

The wrong answer many students write is $\pounds 16$, by computing $20\% \text{ of } \pounds 80 = \pounds 16$ and stopping there. But $\pounds 16$ is smaller than $\pounds 80$, so it cannot be the price after an increase. The $\pounds 16$ is only the change — the part you add on.

The correct answer is $\pounds 96$, because "increase by 20%" means the whole plus that part: $\pounds 80 + \pounds 16 = \pounds 96$ (equivalently $\pounds 80 \times 1.20 = \pounds 96$).

Primary-school percentage problems are almost all "percent of an amount" questions, where the answer is the part: "find 20% of £80" wants £16 and nothing more. The student arrives at GCSE with a deep habit of computing $p\% \text{ of } n$ and handing that back. When the question instead says "increase by 20%", the habit fires anyway: they produce the part and forget to add it to the original whole.

The base error is the same trap pointing the other way. To turn a change into a percentage you must divide by the original — the amount you started from. But the number most in view is often the new value or the bare money gap, so the brain divides by whichever is handy. A rise from £40 to £50 becomes "10%" (the gap read off raw) or "20%" ($10 \div 50$, the new value as base) instead of the correct $10 \div 40 = 25\%$. In every case the original amount, which is the true reference, has been quietly dropped.

Write down the original amount first, then decide whether the answer is the change (the part) or the new amount (the whole plus the part). "20% of £80" is the part on its own: $\pounds 16$. "Increase £80 by 20%" is the whole plus that part: $\pounds 80 + \pounds 16 = \pounds 96$. The percentage figure is always the size of the change, never the new amount by itself.

Going the other way, a percentage change divides by the original. Percentage change = $\dfrac{\text{change}}{\text{original}} \times 100$. From $\pounds 40$ to $\pounds 50$ the change is $\pounds 10$ and the original is $\pounds 40$, so $10 \div 40 \times 100 = 25\%$ — not the raw gap as "10%", and not $10 \div 50 = 20\%$ against the new value. The original is the base every time.

The decimal multiplier keeps both numbers in view: $\text{new} = \text{original} \times 1.20$ puts the original and the change in the same line, so you cannot lose the whole. Once you trust this going forwards, the two related skills are reversing a change to find the original and stacking two changes.

A coat costs $\pounds 80$; the price is increased by 20%. Find the new price.

1. Name the original. The amount we are changing is $\pounds 80$ — that is the base.
2. Work out the change (the part). $20\% \text{ of } \pounds 80 = 0.20 \times 80 = \pounds 16$.
3. Add the part to the whole. "Increase by" means the new price is the original plus the change:
   $$\pounds 80 + \pounds 16 = \pounds 96$$
4. Sanity check. $\pounds 96$ is bigger than $\pounds 80$, so it really is an increase. Equivalently $\pounds 80 \times 1.20 = \pounds 96$.

Now read a change backwards into a percentage: a price rises from $\pounds 40$ to $\pounds 50$. The change is $\pounds 10$, and the percentage divides by the original: $10 \div 40 \times 100 = 25\%$. The tempting wrong answers — 10% (the raw gap) or 20% ($10 \div 50$, against the new value) — both use the wrong base.