Work out the final price and your savings from any percentage off, including stacked discounts and sales tax.
“Take an extra 10% off sale prices!” sounds like it turns a 20% sale into 30% off. It doesn’t. Stacked (successive) discounts multiply, and the result is always less than the sum of the percentages.
How stacked discounts work
Each discount applies to the price after the previous one:
Final price = Price × (1 − d₁) × (1 − d₂) × …
$100 with 20% off, then an extra 10% off:
- 20% off: $100 × 0.80 = $80
- Extra 10% off the $80: $80 × 0.90 = $72
You pay $72, an effective discount of 28%, not 30%. The second 10% only applied to $80, not $100.
The effective discount formula
Effective discount = 1 − (1 − d₁)(1 − d₂)
Or, for two discounts: d₁ + d₂ − d₁ × d₂
For 20% and 10%: 0.20 + 0.10 − 0.02 = 0.28 → 28%.
| First | Extra | Looks like | Actually |
|---|---|---|---|
| 10% | 10% | 20% | 19% |
| 20% | 10% | 30% | 28% |
| 25% | 15% | 40% | 36.25% |
| 30% | 20% | 50% | 44% |
| 50% | 20% | 70% | 60% |
| 50% | 50% | 100% (free!) | 75% |
The bigger the discounts, the bigger the gap. “50% off plus an extra 50%” is 75% off, not free.
Does the order matter?
No. 20% then 10% gives the same price as 10% then 20%, because multiplication is commutative: 0.8 × 0.9 = 0.9 × 0.8. What does matter is whether a coupon is a percentage or a fixed amount, and when it’s applied.
Fixed amounts change the picture
On a $100 item, compare “$15 off, then 20% off” with “20% off, then $15 off”:
- $15 first: (100 − 15) × 0.8 = $68
- 20% first: 100 × 0.8 − 15 = $65
When you can choose, apply the percentage discount first and the fixed-amount coupon last.
Comparing deals
Store A: 35% off. Store B: 25% off plus an extra 10% at checkout.
- A: you pay 65%.
- B: you pay 0.75 × 0.90 = 67.5%, an effective 32.5% off.
Store A is cheaper. Plug both into the discount calculator, which has an “extra discount” field and shows the effective discount for exactly this comparison.
Stacked discounts with tax
Sales tax is applied after all discounts, on the final price. $100 → 20% → extra 10% → $72, plus 8% tax = $77.76.
Related maths
Stacked discounts are a special case of successive percentage changes, the same reason a 10% fall followed by a 10% rise doesn’t return you to where you started. See Percentage Increase and Decrease: How to Calculate Change Correctly for more, and How to Calculate a Discount: Formulas, Mental Maths Tricks and Examples for single-discount shortcuts.
Work out the final price and your savings from any percentage off, including stacked discounts and sales tax.