How the result is calculated
A discount is a percentage of a price, taken away from it. Rather than working out the saving and subtracting it, it is quicker to multiply by what is left: 25% off leaves 75% of the price, so a price of 80 becomes 80 × 0.75 = 60.
SP × (1 − d₁) × (1 − d₂) − v
- P
- the full price before any offer
- d
- each percentage discount as a decimal, 0.25 for 25%
- v
- any fixed amount off, such as a voucher
The calculator works through the offers one at a time, as a till does, and rounds each step to the smallest coin: cents, pence, or whole yen. That is why its steps table can differ by a cent from multiplying the factors together in one go. Turned round, the same relationship gives the discount between two prices, (P − S) ÷ P, and the full price behind a sale price, (S + v) ÷ ((1 − d₁)(1 − d₂)).
Tax comes last. Where it is added at the till, as with US sales tax, the rate is applied to the discounted price. Where prices already include it, as with VAT, the tax inside a price is the rate divided by 100 plus the rate, so at 20% it is one sixth of whatever you pay.
Worked example
| Item | Amount | How it is obtained |
|---|---|---|
| Full price | $80.00 | Full price |
| 25% off | −$20.00 | 80 × 0.25 |
| Sale price | $60.00 | 80 × 0.75 |
| You save | $20.00 | 25% of the full price |
| Item | Amount | How it is obtained |
|---|---|---|
| Full price | $100.00 | Full price |
| 20% off | −$20.00 | leaves $80.00 |
| Extra 10% off | −$8.00 | 10% of $80, not of $100 |
| Sale price | $72.00 | 28% off in all |
| Sales tax at 8.875% | $6.39 | $72 × 0.08875 = $6.39 |
| You pay | $78.39 | $72.00 + $6.39 |
The extra 10% saves $8, not $10, because it is taken from the price the first discount left. 8.875% is the combined state and city rate in New York City.
Reading the result
The headline is what you asked for: the price you pay, the discount between two prices, or the full price behind a sale price. When you add a tax rate and your prices do not include it, the headline is the amount you pay at the till, with the tax shown beside it. The "discount in all" figure is the single percentage your stack of offers comes to, and it is the number to compare with a rival's single offer.
The chart splits the full price into what you pay, what you save and any tax, and the steps table shows each offer applied in turn. The table under it repeats the sale price at common discount rates, which is a fast way to see what the next markdown would do.
What changes the result
Stacking order matters only for fixed amounts. Two percentage discounts give the same price in either order, because multiplication does not care about order, but a voucher does not. On a price of 100 with 20% off, a voucher for 10 taken after the discount leaves 70; taken before it, the 20% applies to 90 and leaves 72, so the voucher is in effect worth only 8. The calculator takes vouchers last. If your shop's terms deduct them first, your price will be slightly higher.
Where you shop changes how tax meets the discount. In New York, a store coupon reduces the taxable price, but a manufacturer's coupon does not: the store is reimbursed for it, so tax is due on the price before the coupon. Rules on coupons vary between US states. With VAT, the tax is simply a fixed share of what you pay, so a bigger discount means less tax.
What is not included
The calculator does not know whether the full price was genuine. A discount measured from a price the item never really sold at is not a saving, and both the US Federal Trade Commission's pricing guides and UK consumer law treat a misleading "was" price as a deceptive practice. Compare the sale price with what other shops charge, not with the crossed-out figure.
It also leaves out delivery, card surcharges, loyalty points, cashback and multi-buy offers such as three for two, each of which changes what you really pay per item. The rounding follows the common half-up rule; a particular till may round differently, by a cent at most per step.
Using the result
Before buying, compare the final price with tax, not the size of the discount: 40% off a high price can cost more than 10% off a low one. When a shop offers "an extra 20% off sale prices", use the extra discount field to see the true combined saving. After buying, the discount mode checks a receipt: enter the full price and what you paid, and see whether the percentage matches the sign. For the arithmetic of percentages in general, including working a price back from one that includes tax, the percentage calculator has a mode for each question.