Stacked Discounts Do Not Add Up

Discounts multiply, they do not add

Think about what survives rather than what is taken away. A 20% discount leaves 80% of the price, which is a multiplication by 0.80. A further 10% leaves 90% of whatever is in front of it, a multiplication by 0.90. Applying both leaves 0.80 × 0.90 = 0.72 of the original, so the total discount is 1 − 0.72 = 0.28, twenty-eight percent. The second cut is smaller in absolute terms than the first because it operates on a smaller number — the missing two points are exactly 10% of the 20% already removed. That is the whole mechanism, and it generalises immediately: for any chain of discounts, the surviving fraction is the product of the individual surviving fractions, and the total discount is 1 − Π(1 − dᵢ).

The table above runs the common stacks. Notice how the error in the naive sum grows with the size of the cuts: 20% and 10% overstates by only two points, but 40% and 30% overstates by twelve, and a shop advertising 50% then 30% then 20% is nowhere near giving the goods away — the true figure is 72%, not 100%. In fact no chain of ordinary discounts can ever reach 100%, because each factor is strictly positive and their product is too. A stack of a hundred separate 10% cuts still leaves 0.9¹⁰⁰, which is small but not zero. That is a useful sanity check when reading an offer: if the advertised discounts add to more than 100%, the arithmetic is not being done the way you would do it.

Order is irrelevant for percentages and decisive for vouchers

Two percentage discounts commute, and you can prove it in one line: 0.80 × 0.90 and 0.90 × 0.80 are both 0.72. On an item at 120 you pay 86.40 whichever way round the till applies them, so any argument at the counter about which promotion should go first is wasted breath. This is not a coincidence of these particular numbers; multiplication is commutative for every pair, so no ordering of pure percentages can ever change the result. If a shop claims otherwise, one of the offers is not a percentage.

Add a fixed-amount voucher and the symmetry collapses, because subtracting a constant and multiplying by a factor do not commute. Take an item at 120 with a 25% promotion and a 20 voucher. Apply the percentage first: 120 × 0.75 = 90, then subtract the voucher for 70. Apply the voucher first: 120 − 20 = 100, then take 25% off for 75. Five units of real money separate the two paths, and the gap is not arbitrary — it is always the voucher multiplied by the discount rate, here 20 × 0.25 = 5. Push the numbers up and the stake grows: on a 200 item with 40% off and a 30 voucher, percentage-first gives 90 and voucher-first gives 102, a gap of 30 × 0.40 = 12. Percentage first is always better for the customer, and voucher first is always better for the shop, which is why the order is written into terms and conditions rather than left to chance.

What tax does to the answer — and it is not the same on both sides of the Atlantic

In the United States the shelf price excludes sales tax, which is added at the till, so the number on the tag is never the number you pay. Take a $120 item in a jurisdiction with a combined rate of 8.875%. At full price the bill is $130.65. With 20% off, the taxable subtotal drops to $96 and the bill is $104.52. The sticker promised you $24 off, but your bill actually fell by $26.13, because you also avoided the tax on the discounted portion. Order does not matter here: 120 × 0.80 × 1.08875 and 120 × 1.08875 × 0.80 both come to $104.52, since a percentage discount and a tax rate are both multiplications.

The interesting case in a tax-exclusive market is the difference between a coupon and a gift card, because they are treated differently and the gap is real money. A store coupon reduces the taxable price, so a $20 coupon on that $120 item leaves a taxable subtotal of $100 and a bill of $108.88. A $20 gift card is tender, not a price reduction, so tax is computed on the full $120 first — $130.65 — and the card is applied afterwards, leaving $110.65. The difference is $1.77, which is exactly $20 × 0.08875. Two pieces of paper with the same number printed on them are not worth the same amount, and which one you hand over first changes the total. Ask at the counter rather than assuming; the answer varies by state and by the type of promotion.

Working backwards: the single equivalent discount

The question shoppers actually ask is not what the stack does but what single discount it is worth, and that falls straight out of the same formula: multiply the surviving fractions and subtract from one. A stack of 20% and 10% is worth a single 28%. A stack of 25% and 15% is worth 36.25%. A stack of 50%, 30% and 20% is worth 72%. Reading offers this way makes them comparable at a glance, which is the only way to tell whether a competitor's flat 35% beats the shop across the road advertising 25% plus an extra 15%. It does: 35% against 36.25% is a loss by a fraction over a point.

The same formula answers the design question from the other side. If you want a two-step promotion that lands on exactly 50% off, you cannot use two 25% cuts — that gives 43.75%. Two equal discounts d that combine to 50% must satisfy (1 − d)² = 0.5, so d = 1 − √0.5 = 29.29%. And if you are constructing a headline stack, remember that the first cut does most of the work: after a 50% discount, an additional 20% only removes another ten points of the original price. The marketing gain from a long chain of small extra offers is much smaller than it looks on the poster, which is exactly why long chains appear on posters.

The markup that undoes a discount is bigger than the discount

A 20% discount takes a price to 80% of itself. To return to the original you must divide by 0.80, which is a multiplication by 1.25 — a markup of 25%, not 20%. The general rule is that reversing a discount of d requires a markup of d ÷ (1 − d), and the two only converge for very small values. After a 25% discount you need 33.33%. After a 30% discount you need 42.86%. After a 50% discount you need to double the price. This asymmetry is not a curiosity; it is the reason a retailer who marks a cost of 100 up by 20% to 120, then advertises 20% off, sells at 96 and loses four percent on cost. Marking up by a percentage and discounting by the same percentage never breaks even.

This matters when a discount is negotiated in one unit and priced in another. If a buyer asks for 15% off and the seller has a 15% gross margin, granting it does not halve the profit — it eliminates it entirely and then some, because the margin was computed on the selling price while the discount is applied to the same selling price. The disciplined way to run these conversations is to convert everything to multipliers before agreeing to anything: cost times markup gives price, price times (1 − discount) gives the amount received, and the two are comparable only in that form. Percentages quoted against different bases will not add up, and in this particular case the arithmetic error comes straight off the bottom line.

Frequently asked questions

Is 20% off plus a further 10% off the same as 30% off?

No. It is 28% off. The first cut leaves 80% of the price and the second leaves 90% of that, so what survives is 0.80 × 0.90 = 0.72, and the total discount is 1 − 0.72 = 0.28. The two missing points are exactly 10% of the 20% that was already taken off, because the second discount is applied to a smaller number. The general rule is that the total is 1 minus the product of the individual surviving fractions, and the naive sum always overstates it — by more as the cuts get larger. 30% and 20% give 44% rather than 50%; 40% and 30% give 58% rather than 70%; 50%, then 30%, then 20% gives 72% rather than the 100% the addition suggests. No chain of ordinary percentage discounts can ever reach 100%, which is a useful check on any advertisement.

Does the order of discounts change the final price?

Not for percentages, and decisively for fixed-amount vouchers. Two percentage discounts commute because multiplication does: 0.80 × 0.90 and 0.90 × 0.80 are both 0.72, so an item at 120 costs 86.40 whichever way the till applies them. Put a voucher in the chain and the two paths separate. On an item at 120 with 25% off and a 20 voucher, taking the percentage first gives 120 × 0.75 = 90, then minus 20 for 70; taking the voucher first gives 120 − 20 = 100, then minus 25% for 75. The gap is always the voucher multiplied by the discount rate — here 20 × 0.25 = 5 — so it grows with both. On a 200 item at 40% off with a 30 voucher the gap is 30 × 0.40 = 12. Percentage first favours the customer, voucher first favours the shop, and reputable terms and conditions state which applies.

How does sales tax affect a discount?

In a tax-exclusive market like the United States, the shelf price is not what you pay, so the discount is worth more off your bill than the tag suggests. A $120 item at a combined rate of 8.875% costs $130.65 at full price. Take 20% off and the taxable subtotal falls to $96, so the bill is $104.52 — your bill dropped $26.13 even though the sticker promised $24, because you avoided the tax on the discounted portion too. Order does not matter for percentages, since a discount and a tax rate are both multiplications: 120 × 0.80 × 1.08875 and 120 × 1.08875 × 0.80 both give $104.52. Fixed amounts are the exception. A $20 store coupon reduces the taxable price, giving $108.88, whereas a $20 gift card is tender applied after tax, giving $110.65 — a real difference of $1.77, which is exactly $20 × 0.08875.

What single discount is equivalent to a stack?

Multiply the surviving fractions and subtract the result from one: the equivalent single discount is 1 − Π(1 − dᵢ). A stack of 20% and 10% is worth a single 28%; 25% and 15% is worth 36.25%; 30% and 20% is worth 44%; 50%, 30% and 20% together are worth 72%. Converting every offer to its single equivalent is the only reliable way to compare them, and it settles arguments quickly: a flat 35% loses to a 25% plus a further 15%, because the latter is 36.25%. The formula also works in reverse when you are designing a promotion. Two equal cuts that must combine to exactly 50% off are not 25% each — they satisfy (1 − d)² = 0.5, so d = 1 − √0.5 = 29.29%. Note that no chain of ordinary percentage discounts ever reaches 100%, because a product of strictly positive factors is strictly positive.

What markup reverses a 20% discount?

Twenty-five percent, not twenty. A 20% discount leaves 80% of the price, so restoring the original means dividing by 0.80, which is multiplying by 1.25. The general rule is that reversing a discount of d requires a markup of d ÷ (1 − d): after a 25% cut you need 33.33%, after 30% you need 42.86%, and after 50% you need to double the price. The asymmetry has a direct commercial consequence. A retailer who marks a cost of 100 up by 20% to reach 120, then advertises 20% off, sells at 96 and is four percent below cost — marking up and discounting by the same percentage never breaks even, because the two percentages are computed on different bases. If you negotiate discounts against a gross margin, convert everything to multipliers before agreeing: granting 15% off when your gross margin is 15% does not halve the profit, it wipes it out.