What to enter
Choose Sale price when you know the price before a discount. Choose Original price when you know the discounted price and want to work backward. Enter the price in US dollars and the discount as a percentage. Open Extra discount if a second percentage applies after the first.
Choose Discount percentage when you know both prices. Enter an original price greater than zero and a sale price from zero up to the original price. This mode finds the overall percentage saved; it does not take an extra discount.
All prices are before tax. Shipping, fees, fixed-dollar coupons and purchase conditions are outside this calculation. Check the retailer’s terms to confirm that the two discounts can be used together.
Method and example
For an original price P, first discount d and extra discount e, with percentages divided by 100:
- Sale price =
P × (1 − d) × (1 − e). - Original price =
sale price ÷ [(1 − d) × (1 − e)]. - Effective discount =
[1 − (1 − d) × (1 − e)] × 100%. - Discount percentage from two prices =
(original price − sale price) ÷ original price × 100%.
For $200 with 20% off and then another 10% off, the first reduction leaves $160. The extra discount leaves $144, saving $56. The effective discount is 28%. To work backward, $144 ÷ (0.8 × 0.9) = $200.
If you only know the original price of $200 and the sale price of $144, the discount percentage is (200 − 144) ÷ 200 × 100 = 28%. A zero sale price gives a 100% discount; equal prices give a 0% discount.
The OpenStax explanation of discounts covers the sale-price formula, finding a discount percentage from two prices, recovering an original price and applying successive discounts to a reduced price. The combined formula above applies that single-discount method twice.
Rounding and limits
This model accepts prices from $0 to $1 trillion, with up to two decimal places, and discounts from 0% to 100%, with up to six decimal places. A blank extra discount means zero. Use a decimal point without commas, currency symbols or scientific notation. The calculated original price must also fit the price limit.
In Discount percentage mode, the original price must be positive and the sale price must not exceed it. Both entered prices are treated as exact. The calculated percentage is displayed and copied with up to 14 decimal places; repeating percentages are rounded.
Calculation uses decimal arithmetic without rounding the price between discounts. Displayed prices round to cents, with a half-cent rounded up. Savings equal the displayed original price minus the displayed sale price, so those amounts agree. The effective discount describes the percentage reduction before cent rounding. The displayed intermediate price is for explanation; its rounded value is not used for the next discount.
For example, $0.05 × 0.5 × 0.5 = $0.0125, which rounds to $0.01. Rounding after the first discount would produce a different answer. A store’s rounding policy may therefore change the final cent.
An original price cannot be recovered when either discount is 100%: every original price would produce a zero sale price. In reverse mode, the entered sale price is treated as exact. Since real sale prices may already be rounded, the recovered original price is an estimate. A zero price is allowed in the sale-price and original-price modes; its effective discount still describes the entered rates, while the money saved is zero.