Percent Off Calculator

The easy part first

Sale price = Original price × (1 - Percent off / 100)

A jacket at 4,500 with 20 percent off comes to 3,600, and you save 900.

Nobody needs a calculator for that one, and if the whole of discount arithmetic worked that cleanly this page would end here.

It does not, for three reasons. Discounts get stacked on top of each other and do not combine the way people assume. Marking a price up and then taking the same percentage off does not return you to where you started. And the original price you are being shown a discount from is itself a claim, one that is regulated in some countries and worth checking in all of them.

Those three are the rest of this page.

Two discounts do not add up

The sale is 20 percent off. At the till there is a further 10 percent off sale items.

That is not 30 percent off.

Follow it through on our 4,500 jacket. Twenty percent off brings it to 3,600. The extra ten percent applies to 3,600, not to the original 4,500, so it takes off 360 rather than 450, giving 3,240.

Total saved: 1,260. Which on an original of 4,500 is 28 percent, not 30.

The reason is that the second discount only ever applies to what survived the first one. So discounts multiply rather than add.

Combined discount = 1 - (1 - first) × (1 - second)

On 20 and 10 percent that is 1 minus 0.8 times 0.9, which is 0.28.

The shortfall grows as the discounts get bigger:

Two useful consequences.

The order does not matter. Twenty then ten gives the same answer as ten then twenty, because multiplication does not care about sequence. So a shop offering to apply them in a particular order is not doing you a favour.

Stacked discounts can never reach 100 percent. Fifty percent off four times running is 93.75 percent off, not 200. Each one only removes a share of what is left, so the price approaches zero without ever arriving.

To use this calculator for stacked discounts, run it twice. Take the sale price from the first run and enter it as the original price for the second.

Why a markup and an equal discount do not cancel

The same asymmetry runs the other way, and it catches out sellers rather than buyers.

Take something priced at 100. Mark it up 20 percent and it is 120. Now take 20 percent off, and you get 96.

Not 100. Four percent below where you started.

Because the 20 percent added was 20 percent of 100, and the 20 percent removed was 20 percent of 120. The second percentage was applied to a bigger number, so it took away more than the first put on.

To genuinely reverse a 20 percent markup you need 16.67 percent off, which is the same margin-and-markup relationship that runs through our profit calculator. The formula:

Discount needed = Markup / (1 + Markup)

This matters commercially. A shop that inflates prices by 20 percent before a sale and then advertises 20 percent off is selling below its normal price, not at it. And a shop trying to clear stock at cost after a 50 percent markup needs to take 33.33 percent off, not 50, or it will sell at a loss.

The general rule, stated once: percentages are not symmetrical, because they are always a percentage of something, and the something changes.

What the original price is supposed to mean

Everything above assumes the original price is real. In the United States, that is not merely an assumption, it is a legal requirement, and the reasoning behind it is worth knowing wherever you shop.

The Federal Trade Commission's Guides Against Deceptive Pricing address exactly this. A discount from a former price is legitimate when that former price was the actual, bona fide price at which the article was offered to the public on a regular basis for a reasonably substantial period of time.

When it was not, the guides are blunt about what has happened. If an artificial, inflated price was established for the purpose of enabling a subsequent offer of a large reduction, the bargain is a false one, and the reduced price is in reality probably just the seller's regular price.

A few specifics from the same rules that are useful as a shopper.

A former price need not have produced sales to be legitimate, but it must have been openly and actively offered, in the recent regular course of business, honestly and in good faith. And an advertiser should avoid implying a former price was a selling price rather than an asking price, using wording like formerly sold at, unless substantial sales at that price actually happened.

List price and manufacturer's suggested price claims are covered too. A suggested retail price used for comparison must correspond to a price at which a substantial number of sales were actually made in the advertiser's trade area. The guides give the example of pens advertised at a retail value of fifteen dollars reduced to seven fifty, when only a few small suburban outlets charged fifteen and every large one charged around seven fifty. That comparison is deceptive because the fifteen was not a prevailing price for the shoppers being addressed.

Trivial reductions are called out specifically. The guides note that describing something as reduced to 9.99 when the former price was 10 is misleading.

None of this makes a calculator unnecessary. It makes one more useful, because the only reliable way to judge a discount is on the price you will actually pay rather than the percentage on the sign. Two practical habits follow. Compare final prices between sellers rather than discount percentages, since a 40 percent discount off an inflated price loses to a 15 percent discount off a keen one. And check what the item has cost over the past few months rather than what the sign says it used to cost.

Questions people ask

How do I work out two discounts together?

Run the calculator twice, using the first sale price as the second original price. Or use the formula: one minus the two remaining fractions multiplied together. Twenty then ten percent gives 28 percent, not 30.

Does it matter which discount is applied first?

No. Multiplication gives the same answer in either order, so twenty then ten equals ten then twenty.

How much discount reverses a markup?

The markup divided by one plus the markup. A 20 percent markup is reversed by 16.67 percent off, and a 50 percent markup by 33.33 percent off.

Can a stacked discount reach 100 percent?

No. Each one removes a share of whatever remains, so the price approaches zero without reaching it. Four successive 50 percent discounts come to 93.75 percent off.

How do I know the original price is real?

You often cannot from the sign alone. In the United States a former price used in advertising must have been genuinely offered on a regular basis for a reasonably substantial period. Comparing final prices between sellers is more reliable than comparing discount percentages.

Is tax applied before or after the discount?

After, normally, so tax is charged on the discounted price rather than the original. This calculator works with pre-tax figures.

I know the amount off, not the percentage.

Divide the amount saved by the original price and multiply by 100. Nine hundred off 4,500 is 20 percent.

References

A note on the sources. The section on what an advertised original price must mean is drawn directly from the Federal Trade Commission's Guides Against Deceptive Pricing, which are federal rules rather than guidance from a retailer or a consumer blog, and which set out in their own words when a former price comparison is legitimate and when the bargain is a false one. The examples quoted, including the pens advertised at a fictitious retail value and the reduction from ten to 9.99, are the regulation's own illustrations. These rules apply in the United States, and equivalent protections vary considerably elsewhere.

1. Federal Trade Commission, 16 CFR § 233.1, Former price comparisons, on bona fide former prices, fictitious inflated prices, and the distinction between an asking price and a selling price. https://www.ecfr.gov/current/title-16/chapter-I/subchapter-B/part-233/section-233.1
2. Federal Trade Commission, 16 CFR Part 233, Guides Against Deceptive Pricing, covering former price comparisons, comparable value comparisons, manufacturer suggested prices, bargain offers conditioned on other purchases, and miscellaneous price comparisons. https://www.ecfr.gov/current/title-16/chapter-I/subchapter-B/part-233
3. Cornell Law School, Legal Information Institute, 16 CFR § 233.1, Former price comparisons. https://www.law.cornell.edu/cfr/text/16/233.1