Percent Off Calculator

Background.

What is 25% off $80? Sixty dollars. A discount tag tells you what share of the price disappears, but the quicker way to think is the share you still pay: 25% off means paying 75 cents of every dollar, and 80 × 0.75 = 60. In the same way 20% off is 80% of the price, 35% off is 65%, and 70% off leaves 30% to pay.

Stacked discounts are where the mental arithmetic goes wrong. A clearance rail says 30% off, the checkout adds an extra 20% coupon, and it feels like half price. It isn't. The coupon comes off a price that has already been cut, so it removes 20% of the remaining 70%, which is 14 points of the original rather than 20. The pair together take 44% off, and a $200 coat costs $112, not $100. The shortfall grows as the discounts get bigger: two 50% discounts in a row leave a quarter of the price to pay, which is 75% off, nowhere near free.

Enter the original price and the advertised discount, plus a second one if something stacks on top, and three figures come back: what you pay, how many dollars come off, and the single percentage the whole stack is worth. That last figure is the one to compare offers with. A flat 45% off beats 30% plus an extra 20%, even though the second sounds like more.

The arithmetic stops before sales tax. The separate discount calculator adds a tax rate if you need the till total; the discount itself works out the same either way.

What no calculator can check is the starting price. In the United States, the Federal Trade Commission's Guides Against Deceptive Pricing (16 CFR 233.1) treat a 'was' price as a fair basis for a percent-off claim only when it is the actual, bona fide price the item was offered at to the public, regularly, for a reasonably substantial period. A reference price invented the week before a sale makes the percentage look bigger without making the item any cheaper. Judge the deal by the final price, not by the size of the number on the sign.

What is percent off calculator?

Percent off is a discount stated as a share of the original price, in hundredths. A discount of d% leaves you paying (100 − d)% of the price, so the sale price is the original multiplied by (1 − d/100). The wording matters: "30% off" and "30% of" name opposite slices of the same price, so 30% off $200 leaves $140 to pay while 30% of $200 is the $60 that comes off. Percent off is the retail form of a markdown, the same proportional reasoning school math applies to tax, tips and commission. When a second percentage discount follows the first, it acts on what is left after the first rather than on the original, so the two multiply instead of adding. The single discount that does the same job as a stack of two is 1 − (1 − d₁/100)(1 − d₂/100), which simplifies to d₁ + d₂ − d₁d₂/100 percent: always a little less than the plain sum, and equal to it only when one of the two discounts is zero.

How to use this calculator.

1. 01Enter the original price as the tag or listing shows it before any discount. If the tag already shows a reduced price, either enter the original with both discounts, or enter the reduced price with only the extra discount: $200 with 30% and 20%, or $140 with 20%, both come to $112.
2. 02Enter the advertised discount as a whole number of percent: 25 for 25% off, not 0.25.
3. 03If a second percentage stacks on top, such as a checkout coupon, a loyalty discount or an 'extra 20% off sale items', put it in the second field. Leave it at 0 when there is only one discount.
4. 04Keep dollar-amount coupons out of the percentage fields. A '$10 off' coupon is not a percentage of anything until you know which price it comes off; subtract it yourself at the point the store applies it.
5. 05Read the final price, the dollar saving and the combined discount. Use the combined figure when comparing the stack against another store's single discount, and add sales tax afterwards if your purchase is taxed.

The formula.

Final = P × (1 − d₁/100) × (1 − d₂/100), where P is the original price, d₁ the first discount and d₂ the optional second one, both in percent. Each bracket is the fraction of the price you keep after that discount: 0.70 for 30% off, 0.80 for 20% off. The dollar saving is P − Final, and the combined discount re-expresses that saving as a share of the original price: (P − Final) ÷ P × 100.

Expanding the brackets shows where the 'missing' saving goes. The combined discount equals d₁ + d₂ − d₁d₂/100. The subtracted term is the second discount's share of the money the first discount already removed, money you can't be discounted on twice. With 30% and 20% that term is 30 × 20 / 100 = 6 points, which is why the pair is worth 44% rather than 50%. The term is small when the discounts are small (10% and 5% combine to 14.5%) and large when they are large (70% and 20% combine to 76%, not 90%).

Thinking in the share you keep also shows how much a deeper discount is really worth. Going from 20% off to 30% off is 10 percentage points on the sign, but the price you pay falls from 80% to 70% of the original, a cut of one eighth (12.5%). Going from 40% to 50% off cuts what you pay by a sixth. Each extra point of discount matters more as the discount grows.

Because multiplication can be done in any order, swapping the two discounts gives exactly the same answer. The price must be more than zero and each discount between 0 and 100%; a 100% discount makes the item free and leaves the second discount nothing to act on. The final price and the saving are shown to the cent and the combined discount to two decimal places, so a $59.99 item at 15% off works out to $50.9915 and is shown as $50.99, the price a till would charge.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Common Core State Standards for Mathematics, Grade 7, Ratios & Proportional Relationships, Standard 7.RP.A.3 — proportional relationships applied to multistep ratio and percent problems, explicitly including markups and markdowns.
2. [2]16 CFR Part 233.1 — Federal Trade Commission Guides Against Deceptive Pricing, governing when a former-price / percentage-off comparison is a lawful advertising claim.
3. [3]Common Core State Standards for Mathematics, Grade 6, Ratios & Proportional Relationships, Standard 6.RP.A.3(c) — finding a percent of a quantity, the foundational skill percent-off math builds on.
4. [4]NIST Guide to the SI, Special Publication 811, §7.10.2 — rules and conventions for expressing relative differences and ratios as dimensionless percentages.
5. [5]NIST Digital Library of Mathematical Functions, §4.2 — formal definitions of powers and multiplicative relationships underlying compounded-rate arithmetic such as stacked percentage discounts.