How To Calculate A Discount

A discount is a reduction applied to the pre-tax price of an item, and the arithmetic is one multiplication followed by one subtraction. Take an original price of 120 with a 25 per cent discount: the reduction is 120 x 0.25 = 30.0, so the sale price is 120 - 30.0 = 90.0. The percentage always attaches to the price before tax, because tax is levied on what is actually paid rather than on money that was never charged. Doing the two steps in that order is what keeps the numbers consistent, and reversing them is the single most common way a discount calculation goes quietly wrong. The headline rate, the money saved and the price paid are three different quantities, and only the first is what the tag advertises.

Tax is added back on top of the discounted price, not on top of the original one. With an 8 per cent sales tax, the 90.0 sale price becomes 90.0 x 1.08 = 97.2 at the till, so the customer pays 97.2 rather than the 90.0 the shelf label implies. The discount saved 30.0 in the same transaction, and the two figures answer different questions: 30.0 is what was avoided, while 97.2 is what is handed over. Because tax is charged on 90.0, the tax itself is 7.2, not the 9.6 that 8 per cent of 120 would have produced. Discounting before tax therefore reduces the tax as well as the price, which is a detail that surprises people who assume tax is a fixed add-on.

Running the calculation backwards recovers the rate from the two prices actually visible on a receipt. If the original was 120 and the sale price is 90.0, the money saved is 30.0 and the rate is 30.0 / 120 = 0.25, which is 25 per cent. Written as a formula, (original - sale) / original gives the effective rate directly, without ever needing the tag. This is the check to run whenever a discount is quoted in currency rather than as a percentage, and it is the only way to compare two offers that were never labelled alike. The same division also exposes a discount that has been exaggerated, because a claimed rate of 25 per cent demands a saving of exactly 30.0 on a 120 item.

The phrase '25 per cent off' and the phrase '25 per cent of the final bill' describe genuinely different claims. On the 120 item, 25 per cent off means paying 90.0, which is 75 per cent of the original price. Saying the bill is 25 per cent of the original would mean paying 30.0, a saving of 90.0 instead of 30.0, and the two answers differ by a factor of three. The percentage attaches to the reduction in the first case and to the amount paid in the second, and swapping them turns a modest saving into a dramatic one. Read the wording first, then decide which quantity the percentage is a fraction of before touching the arithmetic.

Stacked discounts behave differently again, because successive percentages multiply rather than add. Two reductions of 20 per cent and 10 per cent are not 30 per cent off; they leave 0.8 x 0.9 = 0.72 of the price, which is an effective discount of 0.28. The second discount applies to an already-reduced price, so it bites on a smaller base than the first. That gap widens as the rates grow, and on large rates the difference between the naive sum and the true figure becomes substantial. The page on how to calculate stacked discounts works the full example; here it is enough to remember that a single 25 per cent discount and a pair of smaller ones are not interchangeable.

A discount quoted in currency rather than as a percentage is not comparable across items. Saving 30.0 on a 120 item is a quarter of the price, but saving the same 30.0 on a 300 item is only a tenth, even though both receipts show an identical figure. The absolute amount looks reassuring because it is large in the hand, while the proportion it represents may be trivial. Whenever two discounts are expressed as money, convert both back to rates with (original - sale) / original before deciding which is the better deal. Only the rate survives a change of base, which is exactly why retailers prefer to advertise the amount.

The effective discount rate is the single number worth keeping, because it folds the whole transaction into one figure. For the 120 item the effective rate is 0.25, and it stays 0.25 whether you look before or after tax. Adding 8 per cent tax changes the money paid to 97.2 but not the discount itself, since tax is charged proportionally on the reduced price. Blending a discount rate and a tax rate into one percentage produces a number that means nothing and cannot be checked against anything. Keep the two rates separate, and report the discount as a fraction of the original price every time.

Rounding is harmless when it is done last and dangerous when it is done first. A 25 per cent discount is easy to estimate by halving twice: half of 120 is 60, half of 60 is 30, and 120 minus 30 gives 90.0, matching the exact answer. Adding 8 per cent tax on top reaches 97.2 whether the intermediate figures are exact or slightly rounded. On a larger purchase, though, rounding the discount early and then compounding tax on the rounded price drifts the bill by several units. Carry the exact figure through both steps and round once, at the very end.

Discounts are quoted so often that it pays to know what the tag leaves out. A '25 per cent off' sign says nothing about whether the pre-discount price was itself inflated, nor about the 8 per cent tax that will be added to reach 97.2. The only figures that settle the question are the original price, the sale price and the tax rate, and from those three everything else follows. When a retailer quotes a saving of 30.0, ask what the original was, because the same 30.0 means 25 per cent on a 120 item and just 10 per cent on a 300 item. The percentage, not the amount, is what tells you whether the offer is worth taking.