Why two discounts never add up
A sign reading "40% off, take an extra 10% at the register" is not a 50% sale. The extra 10% is taken from the already-reduced price, so you pay 90% of 60% of list, which is 54%. The combined discount is 46%, four points short of the number the signage plants in your head.
The gap follows a simple identity. For two discounts, the true combined discount is d₁ + d₂ − d₁·d₂. That last term — the product — is the amount the second discount fails to reach because the first one already removed it. At 40% and 10% the product is 0.04, exactly the four points. At 50% and 50% the product is 0.25, so the pair gives 75% off, not 100%: no stack of percentages below 100 will ever hand you a free item.
Dollar coupons behave differently. A flat $10 off is worth exactly $10 wherever it lands in the sequence — but the percentages applied after it shrink it. Take $10 off first and then 25% off, and your $10 coupon has become worth $7.50. Take the 25% first and the $10 last, and it is worth the full $10. That is why every retailer's terms specify the order, and why it is worth reading them.
This calculator applies whichever order you select, shows the running price after each step, and reports what the wrong order costs on your specific basket.
The formula, term by term
Start with the merchandise total L·q. Every percentage discount multiplies it by (1 − d_i), its survival factor — the fraction of the price that remains. Chaining discounts means multiplying survival factors, which is why the composition is a product and not a sum.
Read the product directly and the arithmetic gets easy. 40% off leaves 0.60; a further 10% leaves 0.90; 0.60 × 0.90 = 0.54, so you pay 54% and the equivalent discount is 46%. Three 20% discounts leave 0.8³ = 0.512, so 48.8% off. You can do most stacked-coupon math in your head this way, and it never produces the addition error.
The flat coupon. Subtracting C after the percentages gives L·q·Π − C. Subtracting it before gives (L·q − C)·Π = L·q·Π − C·Π. Since Π ≤ 1, the coupon-first price is the larger of the two, and the difference is exactly C·(1 − Π). On a 25% discount with a $10 coupon that is $10 × 0.25 = $2.50, which is the figure the calculator reports as the cost of the wrong order.
Tax. Sales tax is charged on the price actually paid for the goods, so it applies to the discounted merchandise, not to list. A store coupon reduces the taxable amount in every US state. Manufacturer coupons that the retailer is reimbursed for are treated differently in several states — there, tax is due on the pre-coupon price, because the merchant received full value.
Shipping is added last and excluded from the equivalent-discount figure, since a shipping charge is not a discount on the goods and folding it in would misstate how good the deal was.
Worked example: a $120 jacket, 40% off, extra 10%, $15 coupon
A jacket lists at $120. It is marked 40% off on the rack, the store is running an extra 10% off sale prices this weekend, and you have a $15 store coupon that the terms say applies after all other discounts. Sales tax is 7%.
- Merchandise total. $120.00 × 1 item = $120.00.
- Apply 40%. $120.00 × 0.60 = $72.00.
- Apply 10%. $72.00 × 0.90 = $64.80.
- Apply the coupon. $64.80 − $15.00 = $49.80.
- Equivalent single discount. 1 − 49.80 ÷ 120.00 = 1 − 0.4150 = 58.50% off.
- Tax. $49.80 × 7% = $3.486, which rounds to $3.49.
- Total paid. $49.80 + $3.49 = $53.29.
Now check what the coupon order was worth. Applied first, the sequence is $120.00 − $15.00 = $105.00, then × 0.60 = $63.00, then × 0.90 = $56.70. That is $6.90 more than $49.80 — and the identity predicts it exactly: $15.00 × (1 − 0.54) = $15.00 × 0.46 = $6.90.
Note also what the percentages alone achieved: 46% off, against the 50% that adding them would suggest. On this jacket the difference is $120 × 0.04 = $4.80, which is real money on one garment and compounds across a basket.
Reading the equivalent discount
The equivalent single discount is the number to compare across offers, because it is the only one stated on a common basis. "30% off plus an extra 20%" and "a flat 45% off" sound comparable; the first is 44% off and the second is 45%, so the flat offer wins. Convert every offer to its equivalent before deciding.
Where a dollar coupon is in play, the equivalent discount depends on the price, which is why the same coupon looks like a great deal on a cheap item and a rounding error on an expensive one. A $15 coupon on a $49.80 discounted price is 30% of it; on a $498 discounted price it is 3%. If you hold one flat coupon and a basket of items, spend it where the merchandise total is closest to the coupon's minimum-spend threshold.
Two thresholds are worth carrying in your head. Two equal discounts of d give 2d − d², so twin 25s give 43.75% and twin 30s give 51%. And any pair reaches 50% off only when d₁ + d₂ − d₁·d₂ ≥ 0.5 — for a 20% first discount that requires a second discount of 37.5%, which is why "20% plus an extra 30%" still leaves you paying 56 cents on the dollar.
Finally, treat the Cost of the wrong coupon order output as a reason to read the terms rather than assume. It equals the coupon amount times the combined percentage discount, so it grows with both. On a deep clearance stack it can exceed half the coupon's face value.
What two stacked percentage discounts actually give you
| First / second | 10% | 15% | 20% | 25% | 30% | 50% |
|---|---|---|---|---|---|---|
| 10% | 19.0% | 23.5% | 28.0% | 32.5% | 37.0% | 55.0% |
| 20% | 28.0% | 32.0% | 36.0% | 40.0% | 44.0% | 60.0% |
| 25% | 32.5% | 36.25% | 40.0% | 43.75% | 47.5% | 62.5% |
| 30% | 37.0% | 40.5% | 44.0% | 47.5% | 51.0% | 65.0% |
| 40% | 46.0% | 49.0% | 52.0% | 55.0% | 58.0% | 70.0% |
| 50% | 55.0% | 57.5% | 60.0% | 62.5% | 65.0% | 75.0% |
| 70% | 73.0% | 74.5% | 76.0% | 77.5% | 79.0% | 85.0% |
Every cell is below the sum of its row and column headings, and the shortfall is the product of the two rates. The gap is widest in the middle of the table — twin 50s lose a full 25 points — and narrowest at the extremes.
Traps in stacked-discount offers
- Adding the percentages. The error is always in the retailer's favour and always equals the product of the rates. On 40% plus 10% it is 4 points; on 50% plus 50% it is 25 points.
- Assuming coupons stack at all. Most terms say one coupon per transaction or exclude clearance. Two codes in a cart field frequently means the second silently replaces the first — check the order summary, not the code box.
- Letting the flat coupon go first. Where the terms permit either order, applying the dollar amount last is worth the coupon times the combined percentage discount. Where the terms fix the order, this calculator will at least tell you what it cost.
- Reading a percentage off an inflated "was" price. The discount is only as real as the reference price. If the item has never sold at list, the equivalent discount is arithmetic about a fiction.
- Losing free shipping to the discount. Free-shipping thresholds are usually assessed on the post-discount merchandise total, so a coupon can drop you back under the threshold and cost more than it saved.
- Forgetting that tax follows the discount. A discount saves you the tax on the amount discounted too, so the true saving on a 7% tax rate is 1.07 times the merchandise saving.
Store coupons and manufacturer coupons are taxed differently
When a retailer discounts an item out of its own margin, sales tax applies to the reduced price, because that is the consideration the buyer paid. When a manufacturer's coupon is redeemed and the retailer is reimbursed by the manufacturer, several US states treat the full pre-coupon price as the taxable amount, since the merchant received full value for the goods. That is why a grocery receipt sometimes shows tax that looks too high relative to the total. This calculator taxes the discounted price, which is correct for store discounts and store coupons — the common case in apparel and general merchandise.
Where this fits among the other purchase-decision tools
The equivalent discount answers "how good is this offer" but not "should I buy this". Those are different questions, and a deep discount on something you would not otherwise own is not a saving.
For unit-price comparisons across pack sizes, work the price per ounce or per item rather than the headline discount — the calculator's all-in price per item output is designed for that, since it includes tax and spreads shipping across the quantity. When a discount is a percentage off a subscription rather than an item, the number that matters is the price after the promotional period, which is what the streaming vs cable cost calculator and the cell phone plan comparison calculator are built to expose.
Where a discount is really a rebate paid back on a card, the value depends on the redemption terms rather than the percentage; the credit card cash back calculator handles the recurring version of that. And where you are splitting a discounted purchase with other people, the uneven bill split calculator allocates a whole-order discount in proportion to what each person is taking.
One structural limit: this calculator composes discounts that apply to the same merchandise. It does not model category-restricted stacks — "20% off shoes, 30% off outerwear" — where each percentage applies to a different subtotal. Run those as separate calculations and add the totals, because composing them as a stack would apply each discount to items it does not cover.