Stacked discounts: why 20% off then 10% off is 28%, not 30%
Two discounts in a row do not add up to their sum. 20% off, then 10% off the reduced price, is 28% off the original, not 30%, because the second discount applies to a smaller amount. Multiply the two remaining fractions, 0.80 × 0.90 = 0.72, and take the result from 1 to get the total discount.
Discount calculatorWhat’s the sale price?OpenWhy the second discount is smaller
Take an item at 80 with 20% off, then 10% off the sale price:
| Step | Calculation | Price |
|---|---|---|
| Start | 80.00 | |
| 20% off | 80.00 × 0.80 | 64.00 |
| 10% off that | 64.00 × 0.90 | 57.60 |
The first discount saves 16.00. The second saves 6.40, because it is taken from 64 rather than from 80. The total saving is 22.40, which is 28% of 80. Adding the percentages gives 30%, or 24 off, so the gap is 1.60. That is 2% of 80, the overlap of the two discounts: 20% of 10% of 80 is 1.60, and it is the amount counted twice when the percentages are added.
One formula for two discounts
For discounts a and b, written as fractions, the total discount is a + b − ab. With 20% and 10%, that is 0.20 + 0.10 − 0.02 = 0.28. The ab term is the overlap. Three successive discounts of 10% give 1 − 0.9 × 0.9 × 0.9 = 27.1% off, not 30%.
Working out the sale price
Multiply the price by the remaining fraction for each discount in turn. 80 × 0.80 × 0.90 = 57.60. The order does not matter, because 80 × 0.90 × 0.80 is also 57.60. The discount calculator does this for you and shows each step under the answer.
Working out the percentage off
Start with the two prices. The percentage off is (original − sale) ÷ original × 100. From 100 to 80 is (100 − 80) ÷ 100 × 100, which is 20%. If the sale price came from two stacked discounts, this gives the total, and the two prices alone cannot tell you which discounts were applied. Any pair of discounts that takes 100 down to 72 is a 28% total.
For why up then down is not zero, see the percentage change guide.
Comparing two offers
Stacked percentages can make a headline look better than it is. Take an item at 80 with three offers:
| Offer | Calculation | Final price |
|---|---|---|
| 30% off, once | 80.00 × 0.70 | 56.00 |
| 20% off, then 10% off | 80.00 × 0.80 × 0.90 | 57.60 |
| 10% off three times | 80.00 × 0.90 × 0.90 × 0.90 | 58.32 |
The single 30% offer is the cheapest. The three 10% offers sound generous, but together they take off only 27.1%. Compare the final price, not the figures in the headline. The discount calculator gives the final price for any combination you want to check.
Where sales tax goes
This calculator adds sales tax after the discount, on the sale price. For the 57.60 example at 10% tax, the tax is 5.76 and the total to pay is 63.36. Tax rules differ by place, and some sellers show tax differently on receipts, so the receipt is the final word. The percentage calculator can check the tax step on its own.
Discounts on prices that include tax
In many places the shelf price already includes sales tax. A percentage off then reduces that tax-inclusive price, and the tax is not added a second time at the till. Other places add tax at checkout, as this calculator does. The rules for how a discount and tax interact vary, so the receipt is the reliable record.
Rounding
The calculator works in exact values and rounds the final price to the cent. A shop may round after each discount, and the last cent can then differ. If a receipt is a cent out, check where each step was rounded before assuming an error.
Edge cases
- 100% off makes the price zero. The calculator accepts it.
- More than 100% would make the price negative, so the calculator refuses it.
- Rises and falls. A 20% rise followed by a 20% fall ends at 96, not 100, for the same reason: each change applies to a different base.
- A coupon after a sale. A coupon taken off a sale price is a second discount, so the same rule applies. Multiply the fractions rather than adding the percentages, and the result is the same whichever comes first.
Quick answers
Is 20% off plus 10% off the same as 30% off? No. It is 28% off the original price.
Does the order of two discounts change the price? No. Multiplying the fractions gives the same result in either order.
Why does the second discount remove less than its percentage of the original? It is taken from the price after the first discount, which is already smaller.
How can I check a sale price quickly? Add the amount saved to the sale price, and the total should be the original. For the example, 57.60 plus 22.40 is 80.00.
