How to calculate a discount
A percent-off discount takes a fraction of the original price away. Convert the percent to a decimal by dividing by 100, multiply by the price to get the amount saved, and subtract it. Equivalently, multiply the price by (1 - discount) to get the sale price in one step:
you save = price x discount / 100 sale price = price - you save = price x (1 - discount / 100)
A second discount on top of the first (a sale item with an extra coupon, or a clearance markdown on an already reduced price) applies to the reduced price, not the original, so the two multiply. Sales tax is charged last, on the price you actually pay:
sale price = price x (1 - first / 100) x (1 - second / 100) price with tax = sale price x (1 + tax / 100)
Worked example
The form defaults are an $80 item at 25% off:
you save = 80 x 25 / 100 = $20.00 sale price = 80 - $20.00 = $60.00
Add an extra 10% coupon and 8% sales tax. The coupon takes 10% of $60.00, not of $80, and the tax is charged on the final $54.00:
after coupon = $60.00 x 0.90 = $54.00 total discount = (80 - $54.00) / 80 = 32.5% (not 35%) sales tax = $54.00 x 0.08 = $4.32 price with tax = $58.32
Percent off a $100 price
Because the item is $100, the "you save" column is also the discount in dollars per $100, so you can scale it to any price: 30% off a $250 item saves 2.5 x $30 = $75.
| Percent off | Sale price | You save | Multiply the price by |
|---|---|---|---|
| 5% | $95.00 | $5.00 | 0.95 |
| 10% | $90.00 | $10.00 | 0.9 |
| 15% | $85.00 | $15.00 | 0.85 |
| 20% | $80.00 | $20.00 | 0.8 |
| 25% | $75.00 | $25.00 | 0.75 |
| 30% | $70.00 | $30.00 | 0.7 |
| 33% | $67.00 | $33.00 | 0.67 |
| 40% | $60.00 | $40.00 | 0.6 |
| 50% | $50.00 | $50.00 | 0.5 |
| 60% | $40.00 | $60.00 | 0.4 |
| 70% | $30.00 | $70.00 | 0.3 |
| 75% | $25.00 | $75.00 | 0.25 |
Stacked discounts are less than their sum
"Extra 20% off sale prices" sounds like it adds to the sale percentage, but it multiplies. The table shows the true combined discount for common pairs on a $100 item.
| First discount | Extra discount | Final price | Combined discount | If they simply added |
|---|---|---|---|---|
| 10% | 10% | $81.00 | 19% | 20% |
| 20% | 10% | $72.00 | 28% | 30% |
| 25% | 10% | $67.50 | 32.5% | 35% |
| 30% | 20% | $56.00 | 44% | 50% |
| 50% | 10% | $45.00 | 55% | 60% |
| 50% | 20% | $40.00 | 60% | 70% |
| 50% | 50% | $25.00 | 75% | 100% |
The gap grows with the size of the discounts: 10% and 10% lose only one point, while 50% and 50% is 75% off rather than 100%. The order of the two discounts does not matter, since multiplication commutes.
Mental math for discounts
- 10%: move the decimal point one place left. 10% of $64.90 is $6.49.
- 5%: half of 10%. 15% is 10% plus 5%; 20% is 10% doubled; 30% is 10% tripled.
- 25%: divide by 4. 50%: halve it. 75% off: a quarter of the price remains.
- Sale price directly: 30% off means you pay 70%, so multiply by 0.7. For $80 that is 8 x 7 = $56.
- Sales tax: at 8%, add 10% and take away a fifth of that: $54 + $5.40 - $1.08 = $58.32.
Discount versus markdown versus markup
A discount and a markdown are the same arithmetic: a percentage of the current price removed. A markup goes the other way, a percentage of the cost added, and the percentages are not symmetric. A 25% markup followed by a 25% discount does not return to the starting price: 1.25 x 0.75 = 0.9375, a net 6.25% loss. The markup calculator covers that side of pricing.