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

A central bank raises interest rates from 2% to 3%. One newspaper calls it a one-point rise. A more mischievous one could run “rates up 50%” and still be telling the truth.

Both descriptions are arithmetically correct. They measure the same event two different ways, and mixing them up, by accident or on purpose, is one of the most reliable ways numbers mislead people.

The two measures

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

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

Both are legitimate. The problem is that ordinary English uses “percent” loosely for either one, and the gap between them can be enormous. 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%

So a risk that “doubles” (+100%) might have gone from one in a thousand to two in a thousand. Headlines about medical studies live in exactly this gap: the relative change is what sounds newsworthy, the absolute change is what tells you whether to care.

Any time a percentage change gets reported for a number that is itself a percentage, ask the same question first. Points, or percent?

Percentages aren’t symmetric

Related trap: a percentage change doesn’t undo itself. Down 50% then up 50% won’t get you home. €100 falls to €50, then climbs to €75.

The two percentages came off different bases. The fall was 50% of 100; the rise was 50% of 50.

And the recovery you need grows viciously with the size of the fall. After −20% you need +25% to break even. After −50%, +100%. After −90%, +900%. Nothing has gone wrong here. It’s an asymmetry hiding inside the words “percent of what?”.

Percentages don’t add

Two successive 10% increases don’t make 20%. They compound: 1.10 × 1.10 = 1.21, so 21%. Stack a 20% discount on a 30% discount and you don’t get 50% off, you get 1 − (0.80 × 0.70) = 44% off. Shops know this perfectly well. Shoppers often don’t, which is presumably why stacked-discount promotions are worded the way they are.

Same logic, less cheerful: a 10% raise after a 10% pay cut leaves you slightly below where you started (0.90 × 1.10 = 0.99). And 3% inflation a year for a decade isn’t 30%, it’s 1.03¹⁰ − 1 = 34.4%.

Markup isn’t margin

The business version of the same base confusion. Add 25% to your cost and that’s a 25% markup, but the profit is only 20% of the selling price, which makes it a 20% margin. Cost €80, sell for €100: the €20 profit is 25% of 80 and 20% of 100. One transaction, two honest percentages, two different bases.

Quote a markup where somebody expected a margin and you overstate profitability every time. That’s reason enough for the two to get their own calculator here instead of one ambiguous “profit %” box.

Reverse percentages: the subtraction trap

“The price including 21% VAT is €121, so I take off 21%?” No. 121 × 0.79 = 95.59, and the right answer is €100.

Undoing a percentage increase means dividing by 1.21, not subtracting 21%, because the 21% was worked out on the smaller pre-tax base. The same error turns up in bookkeeping, discount reasoning and salary arithmetic alike, and it has its own article with the working spelled out.

A checklist for any percentage claim

1. Points or percent? If the thing that changed was itself a percentage, insist on both numbers.
2. Percent of what? Find the base. Two percentages with different bases can’t be added, compared or cancelled.
3. Sequential changes? Multiply the factors (1.10 × 0.90…). Never add the percentages.
4. Undoing an increase? Divide by (1 + rate). Never subtract.

The percentage calculator covers the mechanical cases (X% of Y, percent change, what percent A is of B) and shows the formula each time, so you can see which base every number is standing on.