The three questions that get confused
Take a rate that moves from 40 to 50. Three sentences describe it, all true, all different.
| Question | Arithmetic | Answer |
|---|---|---|
| 40 is what percent of 50 | 40 ÷ 50 × 100 | 80 percent |
| How much did it change | (50 − 40) ÷ 40 × 100 | up 25 percent |
| How many points did it move | 50 − 40 | 10 percentage points |
The first has 50 as the denominator, the second has 40, and the third has no denominator at all — it is plain subtraction. Reporting the wrong one is not a rounding disagreement, it is a different claim: "support rose 25 percent" and "support rose 10 points" describe the same poll and will be read as very different news. Percentage points are the honest unit whenever both numbers are already rates. Percent change is the right unit when the numbers are quantities: dollars, units, visits.
Increases and decreases do not cancel
Raise 100 by 20 percent and you have 120. Lower 120 by 20 percent and you have 96, not 100. The rise was measured against 100 and the fall against 120, so the second move is worth more in absolute terms. In multiples, 1.20 × 0.80 = 0.96, and the order does not matter: 0.80 × 1.20 is also 0.96. To actually undo a 20 percent rise you have to take off 16.67 percent, because that is 20 divided by 120.
The same arithmetic governs stacked discounts. Thirty percent off followed by twenty percent off is not fifty percent off. It is 0.70 × 0.80 = 0.56, a 44 percent discount. And it governs recovery from a loss: a portfolio down 50 percent needs a 100 percent gain to get back to even. The "raise or lower" mode prints the undo figure underneath the result for exactly this reason.
Margin and markup are the same profit over different denominators
Buy at $8.00, sell at $10.00, and the gross profit is $2.00. Divide that $2.00 by the $10.00 price and you get a 20 percent gross margin. Divide it by the $8.00 cost and you get a 25 percent markup. Retail and reporting conventions usually mean margin on price; wholesale and shop-floor pricing conventions often mean markup on cost. Both get spoken aloud as "twenty percent".
The direction of the arithmetic differs too. To hit a 30 percent gross margin on a $7.00 cost you divide by 0.70 and price at $10.00. To apply a 30 percent markup you multiply by 1.30 and price at $9.10. Ninety cents an item, on every item, from one word left unspecified. The margin mode prints both figures whichever one you asked for, so the gap is on screen before you quote anything.
What this page will not settle for you
It does arithmetic on the numbers you supply, which means it inherits whatever they mean. It does not know whether your two values are rates or quantities, so it prints both the percent change and the point difference and leaves the choice with you. It does not know your tax jurisdiction — combined rates vary by address down to the district, and groceries, clothing and services are treated differently from state to state, so the rate field is yours to fill from an actual rate lookup. And gross margin here is price less cost, which is not net margin: fees, freight, shrinkage and returns all sit below the line this page draws.
Questions people ask
Which mode gives me a discount percentage?
If you have the list price and the sale price and want the discount, use percent change with the list price as A and the sale price as B. The result comes out negative and its magnitude is the discount. If you have the list price and the discount rate and want the sale price, use the raise-or-lower mode set to lower. Two coupons applied one after the other need that mode run twice; adding the two rates together and applying them once takes off more than the register will.
Why can I not just subtract the tax rate to get the pre-tax price?
Because the tax was calculated on the smaller number. At 10 percent, a $100.00 item becomes a $110.00 total. Taking 10 percent off $110.00 removes $11.00 and leaves $99.00, which is a dollar short, because 10 percent of the total is more than 10 percent of the pre-tax amount. Dividing the total by 1.10 recovers the original $100.00 exactly. As a share of the total, tax at a 10 percent rate accounts for about 9.09 percent.
Should I report a change in percent or in percentage points?
If both numbers are already percentages — a conversion rate, an interest rate, a market share, an unemployment figure — report percentage points, and say "points" out loud. If the numbers are counts or amounts, report percent change. When both are informative, give the points figure as the headline and the relative change in support, which is the convention financial and statistical writing has settled on. The percent change mode prints both so you can quote either.
The result is not what I expected. What went wrong?
Usually A and B are swapped, or the mode is not the one you think. Every result carries a "which fields were read" block naming the mode and listing exactly which inputs were used and which were ignored. The margin mode reads the selling price from its own separate field rather than from B, and the sales tax mode reads only A and the rate. Check those two lines before checking the arithmetic.