Scope and assumptions
Two sequential percentage discounts off the same item; each applies to the new running price.
Twenty percent off, then another ten. The instinct says thirty, and the register says twenty-eight. The gap isn't a rounding artefact — it's structural. Each discount is applied to the price that's left, and after the first one, there's less left to discount.
Getting this right is mostly about refusing to add the percentages.
Chain the survivors
Apply each rate to the running price, in sequence:
Final = Price × (1 − r₁) × (1 − r₂) × …
Each bracket is the share you keep at that step. Multiplying survivors is the same as applying discounts one after another, and it's more reliable than tracking the price through several steps.
Twenty then ten
| Step | Price | What happened |
|---|---|---|
| Start | $100.00 | — |
| 20% off | $80.00 | × 0.80 |
| 10% off | $72.00 | × 0.90 |
You paid $72, so you saved $28 — a 28% discount, not 30%. The two points that went missing are exactly the 10% of the $20 that the first discount already removed.
The equivalent single rate
Any stack collapses into one number:
Combined = 1 − (1 − r₁)(1 − r₂)
- 20% + 10% → 1 − 0.72 = 28%
- 15% + 15% → 1 − 0.7225 = 27.75%
- 25% + 20% → 1 − 0.60 = 40%
Notice that 15% + 15% doesn't even reach 28%. Two equal rates always fall short of their sum, and the shortfall grows with the size of the rates.
Three or more
The rule chains without modification: 10% three times is 1 − 0.9³ = 27.1%, not 30%. Each additional ten percent shaves a smaller slice than the one before, because the base keeps shrinking. This is also why a long chain of small discounts never reaches 100% — the price approaches zero without arriving.
Order is irrelevant
Multiplication commutes, so 20% then 10% lands at the same $72 as 10% then 20%. Sequence only matters when one discount is conditional — a coupon that applies "only to items over $50," for instance, where the running price decides eligibility.
Why Retail Stacked Promos Mislead Shoppers
Adding the rates instead of multiplying the survivors; applying both discounts to the original price; assuming the second discount removes the same dollars as the first; rounding between steps so the error compounds; expecting order to change the outcome; and ignoring caps or exclusions, which decide whether the discounts apply at all.
Related discount guides
Stacked promos chain the survivors. For the single-discount price-to-pay method see Sale Price After Percent Off, and for the full formula index start at the hub: Discount Formula.
FAQ
Why don't two 50% discounts add up to 100% off? Because the second applies to half the original, not the full price. 0.5 × 0.5 = 0.25, so you still pay a quarter — a 75% total reduction.
How do I find the equivalent single discount? Use 1 − (1 − r₁)(1 − r₂). For 20% and 10%: 1 − 0.8 × 0.9 = 0.28, so a single 28% discount is equivalent.
How do I reverse a stacked discount to find the original price? Divide by the product of the survivors. Sale ÷ (0.8 × 0.9) = sale ÷ 0.72.
Do a store coupon and a sale discount always stack? Not necessarily. Many coupons exclude sale items or apply to the post-sale price. The arithmetic is trivial; the terms decide whether it applies at all.
Sources: Khan Academy, Percent word problems; Maths is Fun, Percentages.
Disclaimer: Illustrative. Some promos exclude stacking or cap combined savings; check terms. Informational only.
Sources and references
- Khan Academy — Percent word problemshttps://www.khanacademy.org/math
- Maths is Fun — Percentageshttps://www.mathsisfun.com/percentage.html
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