Why is 50% off plus 20% off not 70% off?

Discounts multiply, they do not add

Take a hundred. Fifty percent off leaves fifty. Twenty percent off that leaves forty. You paid forty, so the total discount was sixty percent — not the seventy that adding the two numbers suggests.

The reason is that the second discount is applied to what remains, not to what you started with. Twenty percent of the original hundred would be twenty; twenty percent of the reduced fifty is only ten. The second offer is working on a smaller base, so it is worth less than its headline.

The general form is a multiplication rather than a sum: final = price × (1 − d₁) × (1 − d₂). Two thirty-percent discounts come to 51 percent, not 60. Twenty and ten come to 28, not 30. The gap widens as the discounts grow, because each one shrinks the base the next works on.

Nothing here is a trick played by the retailer — it is how sequential percentages behave, and applying them in the other order gives the identical result. But it does mean that "an extra 20% off sale prices" is systematically less generous than the arithmetic people do in their heads.

The single discount that would match

A more useful way to read a stacked offer is to ask what one discount would produce the same price, because that is the number you can compare against a competitor.

Multiply what remains after each discount and subtract from one. Fifty then twenty leaves 0.5 × 0.8 = 0.4, so you pay 40 percent and the equivalent single discount is 60. Thirty then thirty leaves 0.49, so 51 percent. Twenty then ten leaves 0.72, so 28 percent.

A quick approximation for small discounts: add them, then subtract their product as a percentage. Twenty and ten gives 30 minus 2, which is 28 — exact in this case. The approximation drifts as the numbers grow, but it is close enough to tell you whether an offer is worth crossing town for.

The comparison this enables is the point. A flat 55 percent off beats "50 percent plus an extra 20 at checkout", which most people would guess the other way round. Converting to a single equivalent discount is the only way to see that without a calculator.

Multi-buy offers convert the same way, by asking what fraction of the full price you end up paying. Buy one get one free means paying for one of two, so 50 percent off. Buy one get one half price means paying 1.5 for two, so 25 percent. Three for two is paying two for three, which is 33.3 percent. Setting them out like this makes an otherwise awkward comparison trivial — three for two beats "25 percent off everything", and buy one get one half price does not.

The catch with all three is that they price a quantity rather than an item. A 33 percent discount that requires buying three of something you wanted one of is not a saving, it is spending more for a lower unit price. That is a fine trade when the thing keeps and a bad one when it does not, and the percentage says nothing either way.

The order does not matter, but where tax lands does

Applying twenty percent then fifty gives the same answer as fifty then twenty, because multiplication is commutative. So a retailer arguing about which order to apply your voucher in is arguing about nothing — in the discount arithmetic itself.

Tax is different, and there the order is real money. A discount applied before tax reduces both the amount you pay and the tax charged on it. Applied after, it reduces only the total while leaving the declared tax based on the undiscounted price.

For a shopper this rarely matters, because consumer prices in the UK and the EU already include VAT and the discount comes off the gross figure. For anyone invoicing, it matters a great deal: discount first, then apply tax, or you have understated what you owe and overstated what the customer saved.

The place it bites shoppers is US sales tax, which is added at the register. A coupon applied before tax saves you the tax on the discounted amount as well; applied after, it does not. On a large purchase in a high-tax jurisdiction that difference is worth checking.

The percentage-up, percentage-down trap

A related asymmetry catches people constantly: raising a price by twenty percent and then discounting it by twenty percent does not return the original.

A hundred raised by twenty percent is 120. Twenty percent off 120 is 96. You are four short, and the shortfall is the twenty percent taken from the twenty that was added — the same tax-on-the-tax structure that makes removing VAT a division rather than a subtraction.

This is worth knowing because it is occasionally how a "sale" is constructed. A price marked up before a promotion and then discounted by the same percentage lands below where it started, so the discount is real — but a price marked up by more than the discount that follows ends up higher than it began while displaying a large percentage off. The advertised number is genuine and the saving is not.

The defence is the same in every case: look at the price you will actually pay, not at the percentage. Percentages compress and mislead in both directions, and the amount is the thing that leaves your account.

Reading an offer quickly

Three habits handle almost every case without arithmetic.

Convert to what you pay, not what you save. "Sixty percent off" is "you pay forty percent", and the second phrasing is much harder to be confused by when a second discount arrives. Stacking becomes multiplication of things you can hold in your head: pay 0.4, then pay 0.8 of that.

Check the reference price. A percentage is meaningless without knowing what it is a percentage of, and "was" prices are the weakest part of any offer — a discount from a price nothing sold at is not a discount. The useful comparison is against what the item costs elsewhere today.

And compare unit prices rather than package prices when quantity varies, because a discount on a larger pack can still be worse per unit than the small one at full price. That is the same failure of intuition as stacked discounts: the percentage is doing the persuading while the arithmetic is somewhere else.

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