One formula, three questions
Every basic percentage question is the same equation — part = (percent ÷ 100) × whole — asked with a different piece missing. "What is 20% of 50?" gives you the percent and the whole, and asks for the part (20 ÷ 100 × 50 = 10). "10 is what percent of 40?" gives you the part and the whole, and asks for the percent (10 ÷ 40 × 100 = 25%). "10 is 20% of what number?" gives you the part and the percent, and asks for the whole (10 ÷ (20 ÷ 100) = 50). The "Basic percentage" mode above is exactly this triangle: pick which piece you are solving for, and the other two become the inputs.
The word itself tells you why the formula works: "percent" comes from the Latin per centum, "by the hundred." The BIPM's own style guide for the SI confirms the modern usage: the % symbol is internationally recognized and, in running text, "generally takes the meaning of parts per hundred." A percentage is not a unit of anything on its own; it is always a percentage of some specific base number, which is the "whole" in the formula above.
Percentage change is not percentage difference
These two get mixed up constantly because they look almost identical, but they answer different questions and can give different numbers for the same pair of values. Percentage change — (new − old) ÷ |old| × 100 — always treats one value as the starting point. A stock at $80 that closes at $60 is down (60 − 80) ÷ 80 × 100 = −25%; the $80 is the anchor, so going back up from $60 to $80 would be a +33.3% change, not +25%, because the anchor itself moved from $80 to $60.
Percentage difference — |a − b| ÷ ((a + b) ÷ 2) × 100 — has no anchor. It is used to compare two independent measurements when neither one is the "correct" answer, so it divides by their average instead of by either value alone. Two lab scales reading 10 g and 12 g for the same object differ by |10 − 12| ÷ ((10 + 12) ÷ 2) × 100 = 18.18%, and it would come out exactly the same if the scales had reported 12 g and 10 g in the opposite order. Swap a and b in a percentage-change calculation, by contrast, and the answer changes sign and magnitude. If one of those two readings is actually the accepted or true value rather than a second independent measurement, that is a third, related calculation, percent error, which divides by the accepted value instead of the average: (12 − 10) ÷ 10 × 100 = 20%, a different number again for the same two readings.
Why a 50% drop and a 50% rise do not cancel out
This is the most common percentage mistake, and it is a direct consequence of "percentage change always uses the old value as its base." Take $100, apply a 50% decrease: $100 × (1 − 0.50) = $50. Apply a 50% increase to that new number: $50 × (1 + 0.50) = $75. The two 50%-magnitude moves do not cancel. You end up at $75, a net change of (75 − 100) ÷ 100 × 100 = −25% from where you started. Each percentage move is calculated against whatever the current value is at that moment, not against the original number, so a loss always needs a larger percentage gain to fully recover: recovering from that $50 back to $100 requires a 100% increase, not 50%.
Reading a discount tag correctly
"Percent off" mode applies a straightforward version of the same base formula: amount saved = original price × (discount ÷ 100), and sale price = original price − amount saved. A $79.99 item marked 30% off saves 79.99 × 0.30 = $23.997, which rounds to $24.00 for display; the sale price works out to 79.99 − 23.997 = $55.993, or $55.99 at the register. Because the original price is already a whole number of cents, rounding the savings first and then subtracting, or subtracting first and then rounding, always lands on the same cent, so the two figures never drift apart. A store's "you saved $X!" badge that looks rounded more loosely (to the nearest dollar rather than the nearest cent) is a marketing choice, not a sign that the underlying math disagrees with a hand calculation.
Stacked discounts do not add. A 20%-off sale on top of an already-30%-off clearance item works out to 20% off the already-discounted price, not 50% off the original price. $100 at 30% off is $70; take another 20% off that $70 and you get $56, a total discount of 44%, not 50%.