Percent vs percentage points: a small wording difference that changes the number

A central bank raises interest rates from 2% to 3%. One news outlet calls it “a one-point rise”; a mischievous one could call it “rates up 50%”. Both are arithmetically correct. They describe the same event with two different measures, and confusing them — accidentally or on purpose — is one of the most common ways numbers mislead.

The two measures

Percentage points measure the absolute gap between two percentages: 3% − 2% = 1 percentage point. Simple subtraction.

Percent change measures the relative jump: (3 − 2) ÷ 2 = 50%. The change is expressed as a fraction of where you started.

Both are legitimate. The trouble is that everyday language uses “percent” loosely for either, and the two numbers can be wildly different — the smaller the starting percentage, the more dramatic the relative version sounds:

From → to	Percentage points	Percent change
40% → 44%	+4 pp	+10%
10% → 14%	+4 pp	+40%
2% → 3%	+1 pp	+50%
0.1% → 0.2%	+0.1 pp	+100%

A risk “doubling” (+100%) may mean it went from one-in-a-thousand to two-in-a-thousand. Headlines about medical studies live in this gap: relative changes sound newsworthy, absolute changes tell you whether to care. Whenever a percentage change of a percentage is reported, the first question is: points, or percent?

Percentages are not symmetric

A related trap: a percentage change does not undo itself. Down 50% then up 50% does not return to the start — €100 drops to €50, then rises to €75. The reason is that the two percentages are of different bases: the drop was 50% of 100, the rise 50% of 50.

The recovery needed after a fall grows viciously with the size of the fall: after −20% you need +25% to break even; after −50%, +100%; after −90%, +900%. There is no error here, just an asymmetry hiding in “percent of what?”.

Percentages do not add

Two successive 10% increases are not a 20% increase — they compound: 1.10 × 1.10 = 1.21, so 21%. A 20% discount on top of a 30% discount is not 50% off but 1 − (0.80 × 0.70) = 44% off. Stores understand this; shoppers frequently do not, which is presumably why stacked-discount promotions are phrased the way they are.

The same logic explains why a 10% raise after a 10% pay cut leaves you below where you started (0.90 × 1.10 = 0.99), and why inflation of 3% a year for a decade is not “30%” but 1.03¹⁰ − 1 = 34.4%.

Markup is not margin

The business version of the base-confusion: adding 25% to cost is a 25% markup, but the resulting profit is only 20% of the selling price — a 20% margin. Cost €80, sell €100: the €20 profit is 25% of 80 and 20% of 100. Same transaction, two honest percentages, different bases.

Quoting a markup where a margin is expected overstates profitability every time, which is why the two get their own dedicated calculator here rather than a single ambiguous “profit %” field.

Reverse percentages: the subtraction trap

“Price including 21% VAT is €121 — so remove 21%?” No: 121 × 0.79 = 95.59, and the right answer is €100. Undoing a percentage increase requires dividing by 1.21, not subtracting 21%, because the 21% was calculated on the smaller pre-tax base. This single error appears in bookkeeping, discount reasoning and salary arithmetic alike; it has its own article with the full working.

A checklist for any percentage claim

1. Points or percent? If a percentage itself changed, insist on both numbers.
2. Percent of what? Identify the base. If two percentages have different bases, they cannot be added, compared or cancelled.
3. Sequential changes? Multiply factors (1.10 × 0.90…), never add the percentages.
4. Undoing an increase? Divide by (1 + rate); never subtract.

The percentage calculator handles the mechanical cases — X% of Y, percent change, “what percent is A of B” — with the formula shown for each, so you can see which base every number is standing on.