Four everyday percentage calculations in one place: percent of a number, what percent X is of Y, percent change, and discounts.
Formulas: Y% of X = X×Y÷100 · X as a share of Y = X÷Y×100 · change = (new−old)÷old×100 · sale price = original×(1−discount÷100).
A percentage expresses a value as a share of one hundred. The word comes from the Latin per centum, and the % sign is a worn-down way of writing 100.
Every percentage calculation therefore starts by deciding what counts as the hundred. Change the base and the answer changes with it — and nearly every percentage mistake is a mistake about the base.
Percent of a total: 15% of 200 = 200 × 0.15 = 30.
What percent one number is of another: 30 of 200 = (30 ÷ 200) × 100 = 15%.
Percentage change: from 50 to 60 is (60 − 50) ÷ 50 × 100 = 20% increase. The denominator is always the earlier value.
Reversing it: if 15% of a number is 30, the number is 30 ÷ 0.15 = 200.
Add 20% to 10,000 and you get 12,000. Take 20% off that and you get 12,000 × 0.8 = 9,600 — four hundred short of where you started.
The base moved. Going up you added 20% of 10,000 (2,000); coming down you subtracted 20% of 12,000 (2,400).
To undo a 20% increase you need 1 ÷ 1.2 = 0.8333, a cut of about 16.67%. The same arithmetic explains why "50% off, then another 50% off" is a 75% discount: 0.5 × 0.5 = 0.25 of the original price remains.
When support rises from 30% to 40%, both "up 10 percentage points" and "up 33 percent" are correct — they measure different things.
Percentage points are the plain difference between two rates: 40 − 30 = 10 pp.
Percent measures how much the rate itself changed: (40 − 30) ÷ 30 × 100 ≈ 33%.
For quantities already expressed in percent — interest rates, unemployment, market share — dropping the distinction inflates or deflates the claim threefold. "Rates rose 2%" is genuinely ambiguous between 2 pp and a relative 2%.
Stacked discounts multiply, they don't add. A 30% coupon plus a further 10% is not 40% off: 0.7 × 0.9 = 0.63, so you pay 63% — a 37% discount.
Margin and markup differ. Buying at 100 and selling at 150 is a 50% markup on cost but a margin of (150 − 100) ÷ 150 ≈ 33% on the sale price. The base is cost in one case and revenue in the other.
Backing tax out of an inclusive price: at 10% VAT, the tax inside 11,000 is 1,000, not 1,100, because 11,000 ÷ 1.1 = 10,000 is the pre-tax figure.
A return of +50% one year and −50% the next averages to 0% arithmetically, but you have lost money: 100 → 150 → 75.
Successive rates average geometrically: √(1.5 × 0.5) − 1 ≈ −13.4% a year.
Anything that compounds — investment returns, population growth, inflation — must not be averaged arithmetically.
(New value − old value) ÷ old value × 100. Going from 50 to 60 is a 20% increase.
Multiply by 0.7. The discount mode shows both the sale price and how much you save.
Percentage points are the simple difference between two rates (30%→40% is +10pp); percent measures the relative change (that same move is a 33% increase).
What this tool bases its numbers on, and how far those numbers go.
Percent of a total: value × rate ÷ 100. Share: part ÷ whole × 100. Change: (new − old) ÷ old × 100. Reverse: value ÷ (rate ÷ 100).50 to 60 is (60 − 50) ÷ 50 × 100 = 20% up; undoing it needs (60 − 50) ÷ 60 × 100 ≈ 16.67% down.