Tip: which base, and whether it matters
A tip percentage has to be applied to something. On a $100 bill with 8.5% tax, an 18% tip on the pre-tax subtotal is $18.00, while 18% on the taxed total of $108.50 is $19.53. Neither is wrong, but the answers differ by $1.53, and a page that does not say which one it used is only accidentally right.
US tipping guidance generally uses the pre-tax subtotal: tax is not part of the service being tipped. Other places quote tips on the total, because that is the number printed on the receipt. The page computes one and reports what the other would have been, so the question is answerable from the page rather than from an argument at the table.
Rounding is the second decision. Splitting $71.50 four ways gives $17.875 each, which is not money. Rounding up to the next dollar means four payments of $18.00 and $72.00 collected against a $71.50 bill; the page prints the 50 cents of difference rather than quietly keeping it or quietly losing it.
Why the shares are split in cents, not by division
Dividing a total by the number of people gives fractions of a cent, and rounding each share on its own loses or invents money that somebody has to cover. Splitting whole cents instead - a base share for everybody, then the remainder handed out one cent at a time - guarantees that the parts add up to exactly the amount collected.
That is why $10.01 between two people is $5.01 and $5.00 rather than two shares of $5.005, and why the page names who pays the odd cent when the shares are not identical. It is the same arithmetic a till does when it splits a bill, and it is why the column on this page adds up when you check it.
Stacked discounts: a smaller base each time
When a shop says "an extra 15% off the sale price", the 15% comes off the reduced price rather than the original. So 10% then 15% is 0.9 × 0.85 = 0.765, which is 23.5% off - not the 25% that adding the two percentages together suggests. The gap grows with longer chains: 20%, 10% and 5% come to 31.6% rather than 35%.
The page shows the price after each step, so the shrinking base is visible, and it prints three readings side by side: the first discount alone, the real sequence, and the tempting sum. Prices are where the "add the percentages" mistake costs the most, because it is the buyer who makes it.
One thing the arithmetic cannot check is whether the original price was ever charged. A discount from an inflated reference price is not a discount, and no calculator can tell that from the numbers it is given.
Margin against markup
Both are profit, measured against different things. Margin is profit divided by the selling price; markup is profit divided by the cost. Buy for 40 and sell for 60 and the profit is 20, so the margin is 20 ÷ 60 = 33.3% and the markup is 20 ÷ 40 = 50%.
That is not a technicality. A shop that wants a 40% margin on a cost of 40 has to charge 66.67, because 40% of the price has to be profit; charging 56 is a 40% markup and only a 28.6% margin. The difference is more than 10 of price per unit, and it is the most common pricing error in small businesses.
The page prints the price needed for a target margin, the cost allowed at a given price, and the conversion between the two measures in both directions - a 50% markup is a 33.3% margin, a 50% margin is a 100% markup, and the two meet only at a zero cost.
What this page does not decide
It does not tell you how much to tip: that depends on the country, the service and whether the bill already carries a service charge, and a calculator that states a social norm as a fact is worse than no calculator. It does not know whether a service charge reaches the staff, and it does not treat one as a tip.
It does not know your tax rate, your card fees, your shipping or your overheads. Gross margin on a unit is not profit, and a pricing decision made here still has to survive those numbers - the page says so rather than presenting a margin as a living.