How Successive Discounts Stack

A shop offers 20% off, and then a further 10% off the already reduced price. It is tempting to add the two discounts and call the deal 30% off. That is not how successive percentage discounts work. On a $100 item, 20% off leaves $80, and 10% off $80 leaves $72. The total saving is $28, which is 28% off the original, not 30%.

The same pattern appears in stacked coupons, employee discounts applied after a sale, and “extra 15% off already reduced prices.” This article shows how to combine the discounts, why the order of two simple percentage-off steps does not change the final price, and when adding percentages would be the wrong model. To compute a single discount from an original price, use the discount calculator or the percentage off calculator.

One discount at a time

A single discount of P% leaves a remaining fraction of 1 − P/100 of the original price.

• 20% off leaves 0.80 of the price.
• 10% off leaves 0.90 of the price.
• 50% off leaves 0.50 of the price.

A further discount is applied to the *current* price, not to the original, unless the fine print says otherwise. So two discounts in a row multiply the remaining fractions:

The combined remaining fraction is the product of the individual remaining fractions. The combined percentage off is 1 minus that product, then multiplied by 100.

Example: 20% off, then 10% off, on $100

• After 20% off: 100 × 0.80 = 80.
• After a further 10% off: 80 × 0.90 = 72.
• Amount saved: 100 − 72 = 28.
• Combined percentage off: 28%.

In one line: 1 − (0.80 × 0.90) = 1 − 0.72 = 0.28, so 28% off. The missing 2 percentage points, compared with a naive 30%, are 10% of the $20 already taken off — in other words, the second discount never applies to the portion the first discount already removed.

Example: the same discounts in the opposite order

• After 10% off: 100 × 0.90 = 90.
• After a further 20% off: 90 × 0.80 = 72.

The final price is again $72, and the combined discount is again 28%. For ordinary “P% off the current price” steps, multiplication of the remaining fractions is commutative: 0.80 × 0.90 = 0.90 × 0.80. The intermediate prices differ (80 versus 90), but the end point does not. That is useful to know when two coupons can be applied in either order and both are simple percentage-off reductions.

This does not hold if one of the offers is a fixed dollar amount, a percentage of the *original* ticket price, or a discount that is capped. Those rules do not all multiply in the same way, and the order can then matter.

Example: two 50% discounts are not 100% off

• After the first 50% off a $100 item: $50.
• After the second 50% off: $25.

The combined remaining fraction is 0.50 × 0.50 = 0.25, so the combined discount is 75% off, not 100%. You would need a remaining fraction of zero — a 100% off step at some point — to take the price to zero. Stacking two equal discounts never adds to a complete reduction, because each step still leaves half of what is in front of it.

A compact formula for the combined percentage off

For two discounts, P and Q:

For 20 and 10: 20 + 10 − (20 × 10)/100 = 30 − 2 = 28. For 50 and 50: 50 + 50 − (50 × 50)/100 = 100 − 25 = 75. The extra term (P × Q)/100 is the overlap that you must not count twice.

For three or more discounts, multiply all of the remaining fractions, then subtract the product from 1 and multiply by 100. For 20%, 10%, and 5% off in succession: 0.80 × 0.90 × 0.95 = 0.684, so the final price is 68.4% of the original and the combined off is 31.6%.

Why “take 20% of the sale price and add it back” also fails in reverse

The same stacking idea appears when you try to recover an original price. If a ticket shows $72 after 20% off and then 10% off, the original is 72 ÷ (0.80 × 0.90) = 72 ÷ 0.72 = 100. Adding 20% and 10% of 72 back to 72 would not return 100. Reverse percentages after a single discount are explained in How reverse percentages work. After several discounts, divide by the *product* of the remaining fractions.

When adding the percentages is the right thing to do

Adding is correct when the percentages are all taken on the *same* original base, and they are not applied one after another to a shrinking (or growing) amount. For example, if a store takes 20% off the original price and, separately, a coupon takes another 10% off the *original* price (not off the reduced price), the total off is 30% of the original, and the final price is $70 on a $100 item. That is a different offer from “20% off, then 10% off the reduced price.”

Read the wording:

• “10% off the already reduced price” or “an extra 10% off” usually means successive (multiplicative) discounts.
• “10% off the original price, plus 20% off the original price” means the percentages share the same base and can be added, up to a 100% cap if both are genuine reductions from that base.

If the text is unclear, compute both ways and see how large the gap is. On $100, the gap between 28% off and 30% off is $2. On a $1,000 item it is $20.

Taxes, fees, and other percentages after a discount

If a sales tax is added after the discount, apply the tax to the discounted price, not to the original, unless the local rule says otherwise. A 20% discount on $100 leaves $80; a 5% tax on $80 is $4, and the total is $84. That 5% is not a “discount,” and you should not fold it into the combined-off formula as if it were another reduction. It is a separate percentage of a later base. The same care applies to service fees and to shipping that is calculated as a percentage of the basket.

Common mistakes

• Adding successive “off the reduced price” discounts. 20% + 10% is not 30% in that setting.
• Assuming that two 50% discounts make the item free.
• Reversing a stacked discount by adding percentages of the final price.
• Ignoring order when one offer is a flat amount. $10 off then 20% off is not always the same as 20% off then $10 off.

Frequently asked questions