How discounts are calculated
Final price = Price × (1 − d₁) × (1 − d₂) Effective discount = 1 − (1 − d₁) × (1 − d₂)
Where d₁ is the first discount and d₂ the extra discount, both as decimals. The extra discount is applied to the already-reduced price, which is how stores usually apply "an extra X% off" sale prices.
How to calculate a discount by hand
A jacket is $129.99, marked 35% off, with an extra 15% off at the register.
- Turn each discount into the share you still pay: 1 − 0.35 = 0.65 and 1 − 0.15 = 0.85.
- Apply the first discount: 129.99 × 0.65 ≈ $84.49.
- Apply the extra discount to that price: 84.49 × 0.85 ≈ $71.82.
- Work out the saving: 129.99 − 71.82 = $58.17.
- Effective discount: 58.17 ÷ 129.99 ≈ 44.75%, not the 50% you get by adding 35 and 15.
Stacked discounts don't simply add up
20% off and then an extra 10% off is not 30% off. On a $100 item, 20% off gives $80, and 10% off $80 gives $72. You save $28, an effective discount of 28%.
| First discount | Extra discount | Adds up to | Real total discount |
|---|---|---|---|
| 10% | 10% | 20% | 19% |
| 15% | 15% | 30% | 27.75% |
| 20% | 10% | 30% | 28% |
| 25% | 10% | 35% | 32.5% |
| 30% | 20% | 50% | 44% |
| 40% | 20% | 60% | 52% |
| 50% | 50% | 100% | 75% |
The last row shows why adding fails: 50% off plus an extra 50% off would be free if the percentages added, but you still pay a quarter of the original price.
Other discount questions you can answer
- Original price from a sale price: divide by the share you pay. A $60 item after 25% off was 60 ÷ 0.75 = $80.
- Percent off from two prices: (original − sale) ÷ original. A $240 item selling for $151.20 is 88.80 ÷ 240 = 37% off, which is what 30% plus an extra 10% gives.
- Fixed-amount coupons: subtract them separately. A $10 coupon is worth more on a cheap item than on an expensive one in percentage terms.