Discount math, in short
A discount multiplies the original price by (1 − discount rate): $100 at 20% off is $100 × 0.80 = $80. The part that trips people up is stacking two discounts — 20% off followed by another 10% off comes to $72, a 28% effective discount, not the 30% simple addition would suggest, because the second discount applies to the already-reduced $80, not the original $100.
Key takeaways
- $100 at 20% off = $80 sale price, $20 saved.
- Stacking 20% off then 10% off on $100 lands at $72 — a 28% effective discount, not 30%.
- Paid $75 for something originally $100? That's a 25% discount: (100 − 75) ÷ 100.
- Paid $80 after a 20% discount? The original price was $100: $80 ÷ (1 − 0.20).
The core formula: sale price from a discount
Sale price = Original price × (1 − discount rate)
= $100 × (1 − 0.20) = $80
Savings are just the difference: $100 − $80 = $20, which is also exactly what you'd get multiplying the original price by the discount rate directly (100 × 0.20). Both routes always agree — they're the same calculation viewed from opposite ends.
Why stacked discounts don't just add up
Each discount in a stack applies to whatever price is left after the previous one, not the original price. On a $100 item with 20% off followed by 10% off:
After 20% off: $100 × 0.80 = $80
After 10% off that: $80 × 0.90 = $72
Effective discount: ($100 − $72) ÷ $100 = 28%
That 10% is taken off a smaller number than the original price, so it's worth less in absolute dollars than a flat 10% off $100 would be — which is exactly why the combined effect (28%) lands short of the naive sum (30%). The gap grows the more discounts get stacked, and the order doesn't actually change the final price — 10% then 20% lands at the same $72 as 20% then 10%.
Working backward: original price and discount percentage
The same relationship runs in reverse for the other two calculation modes:
Discount % = (Original − Sale) ÷ Original × 100 → $100 to $75 is a 25% discount
Original price = Sale price ÷ (1 − discount rate) → $80 paid after 20% off means the original was $100
The second formula is useful for spotting inflated "original" prices — if a tag claims a big discount off a suspiciously round sale price, working the math backward tells you what the pre-sale price would have needed to be for that percentage to be genuine.
Related calculators
- Sales Tax Calculator — add tax on top of a sale price to see the final out-the-door total.
- Percent Off Calculator — a quicker single-purpose tool for one straightforward percent-off calculation.
- VAT Calculator — figure value-added tax instead of a discount, for shoppers outside the US.