What a percent off actually means
A percentage discount is a multiplication, not a subtraction. When a sign says 30% off, the store is telling you that you pay 70% of the tag: the discount rate d defines a keep fraction of 1 − d, and the sale price is the original price times that fraction. Everything else on this page follows from that one idea.
Thinking in keep fractions rather than in percentages off is what stops the two errors shoppers make most. First, it makes stacking obvious — two coupons multiply their keep fractions, so 20% off followed by an extra 15% off leaves 0.80 × 0.85 = 0.68 of the price, a 32% discount rather than 35%. Second, it makes reversing trivial: if you know you paid 68% of the original, you divide by 0.68 to get back to the original price.
Judge a deal on the effective discount: one minus the price you end up paying divided by the price you would otherwise have paid. It absorbs every coupon, stacked code and flat reward certificate into one figure you can hold against another store's. This calculator reports it in every mode, including the two where you never type a percentage at all.
The formula, and why the order of coupons changes the answer
Start with the price of one item, P. Apply each percentage coupon as a keep fraction, then subtract any flat dollars-off amount, and floor the result at zero:
S = P × (1 − d₁) × (1 − d₂) − F
The percentages commute with each other — 20% then 15% gives the same answer as 15% then 20%, because multiplication does not care about order. The flat amount does not commute with them unless every percentage is zero. Taking $5 off a $50 item first and then 10% off gives (50 − 5) × 0.90 = $40.50, while taking the 10% first and then the $5 gives 50 × 0.90 − 5 = $40.00. Do it symbolically and the gap is exact: flat-first costs (P − F)·k, percentage-first costs P·k − F, and subtracting one from the other leaves F(1 − k), where k is the combined keep fraction and neither price has hit the zero floor. Here that is $5 × (1 − 0.90) = $0.50. The flat coupon loses value in proportion to whatever discount comes after it, and at k = 1 — no percentage at all — the two orders give the same price.
That is not a trivia point, because store policy decides the order and the coupon's own terms are where that policy is written down. Read them, set the checkbox in the coupon group to match, and price it both ways when the wording is ambiguous so you know which number to expect at the register.
Two more relationships close the system. Rearranging for P gives P = (S + F) / ((1 − d₁)(1 − d₂)), which is how the calculator recovers an original price from a sale tag. And the effective discount is defined directly from the two prices, d_eff = 1 − S / P, with no reference to any coupon at all — which is why it still works when a store simply prints two numbers and never tells you the percentage. If you want to practise that operation on its own, the percentage calculator and the percentage change calculator handle the bare arithmetic.
Finally, quantity and tax sit outside all of it: Total = S × q × (1 + t). Sales tax is charged on what you actually pay, so a discount reduces your tax bill in proportion — one of the few places where a saving compounds slightly in your favour.
Worked example: a $89.99 jacket, two coupons and a $5 certificate
The jacket is marked $89.99. The sign says 30% off, your email code gives an extra 15% off sale items, and you have a $5 reward certificate. Sales tax is 7.25% and you are buying two.
- Turn each percentage into a keep fraction. 30% off keeps 1 − 0.30 = 0.70. The extra 15% keeps 1 − 0.15 = 0.85.
- Multiply them. 0.70 × 0.85 = 0.595. You pay 59.5% of the tag, so the two coupons together are worth 40.5% off — not the 45% you get by adding the signs.
- Apply the percentages. 89.99 × 0.595 = $53.54.
- Subtract the flat certificate. 53.544 − 5 = $48.54 per jacket.
- Work out the effective discount. 48.544 ÷ 89.99 = 0.5394, so d_eff = 1 − 0.5394 = 46.06% off.
- Find the saving. 89.99 − 48.544 = $41.45 per jacket; across two jackets that is $82.89.
- Add quantity and tax. Subtotal 48.544 × 2 = $97.09. Tax at 7.25% is 97.088 × 0.0725 = $7.04. Total at the register = $104.13.
Two things are worth noticing. The effective 46.06% is higher than the 40.5% from the percentages alone, because the flat $5 is a bigger slice of an already-reduced price than it would be of the original. And had the store taken the $5 off first, you would have paid (89.99 − 5) × 0.595 = $50.57 per jacket — $2.03 more each, which is precisely 0.405 × $5 of certificate value handed back to the store.
How to read the result
Read the effective discount and the sale price together, because neither settles a comparison alone. The effective discount tells you how much of its own reference price a store has given up; the sale price tells you what you hand over. If one store quotes 40% off and another 25% off, but the second store's regular price is lower, only the two sale prices decide it — which is why the calculator reports both.
Be sceptical of the original price itself. An effective discount is only as meaningful as the number it is measured against, and a "was" price that nothing ever sold at inflates every percentage derived from it. Use the reverse-solve mode on any tag that quotes both a percentage and a price: if the arithmetic does not reconcile, the sign and the tag were not produced from the same original.
There is no universal threshold for a good discount, and any single figure quoted as one is a category average wearing a disguise. Markdown depth is set by gross margin: categories that carry a wide margin — apparel, furniture, jewellery — have room to advertise deep percentages, while categories that turn on a thin one, such as grocery staples and consumer electronics, do not. A 40% sign therefore means something quite different in two different aisles. The reliable test is comparative, not absolute: check the sale price against a price you can verify somewhere else, and against what the item is likely to cost if you wait for the next promotion.
Finally, judge the total, not the saving. A 60% discount on something you will not use costs you the full sale price; the saving is only real relative to a purchase you were going to make anyway. If the item is a recurring cost rather than a one-off, price it over a year — the subscription cost audit calculator does that job for anything billed monthly.
What two stacked coupons are really worth
| Headline markdown | +10% code | +15% code | +20% code | +25% code |
|---|---|---|---|---|
| 10% off | 19.0% | 23.5% | 28.0% | 32.5% |
| 20% off | 28.0% | 32.0% | 36.0% | 40.0% |
| 25% off | 32.5% | 36.3% | 40.0% | 43.8% |
| 30% off | 37.0% | 40.5% | 44.0% | 47.5% |
| 40% off | 46.0% | 49.0% | 52.0% | 55.0% |
| 50% off | 55.0% | 57.5% | 60.0% | 62.5% |
| 70% off | 73.0% | 74.5% | 76.0% | 77.5% |
Every cell is below the sum of its row and column headings, and the shortfall equals a × b — the discount the second coupon does not get to take because the first one already removed it.
What a "regular price" is supposed to mean
In the United States, comparative price advertising is governed by the Federal Trade Commission's Guides Against Deceptive Pricing, 16 CFR Part 233. The guides treat a former-price comparison as deceptive when the higher figure is not the price at which the article was actually and openly offered for a reasonably substantial period of time in the recent regular course of business — an artificial inflated price used only to make a markdown look bigger is the specific practice they address. Several states add their own reference-price rules on top. Any percentage this calculator reports inherits the honesty of the original price you feed it, so treat a suspicious "was" figure as the thing to verify first.
Mistakes that make a discount look better than it is
- Adding stacked percentages. 30% plus an extra 20% is 44% off, not 50%. The gap is the product of the two rates, so it widens as the coupons get bigger.
- Assuming a markup and a markdown cancel. A price raised 20% and then cut 20% lands at 1.20 × 0.80 = 0.96 of where it started — 4% below, not level. Percentages taken off different bases never cancel.
- Forgetting sales tax on the comparison. Tax scales with price, so a cheaper sale price also carries less tax. Compare totals, not shelf prices, when you are weighing an in-store deal against an online one with a different tax treatment.
- Applying a per-basket coupon per item. This calculator treats the dollars-off amount as applying to each item. If your $10 code is one per order, enter it with a quantity of 1 and add the remaining items separately.
- Ignoring what you give up to get the coupon. A threshold code that needs $75 in the basket is only a discount if the extra items were things you wanted.
- Reading the percentage instead of the price. The percentage measures the store's generosity relative to its own reference price; the price measures your outcome.
Where this sits among the other price calculations
Discount arithmetic is the consumer-facing half of a pair. The retailer's half is markup and margin, and the two are not mirror images: a 50% margin corresponds to a 100% markup on cost, because margin is measured against the selling price and markup against the cost. When you reverse-solve an original price here, you are recovering the store's reference price, not its cost — nothing on a tag tells you the cost.
Sales tax deserves its own treatment once more than one rate is involved, and a separate sales tax calculator handles combined state, county and city rates and tax-inclusive back-outs. For the underlying operations — taking a percentage of a number, or expressing one number as a change from another — the percentage increase and decrease calculator is the general-purpose tool this page specialises.
One tax subtlety shows up on receipts. Under the definition of "sales price" used by Streamlined Sales and Use Tax Agreement member states, a discount the seller grants and is not reimbursed for reduces the taxable amount, while a coupon the retailer is reimbursed for by a manufacturer generally does not. This calculator assumes the seller-discount case, which covers ordinary markdowns and store codes; rules vary by state, so check yours if a receipt disagrees.
When the cost is shared, hand the discounted total to the split the bill calculator, and any gratuity on it to the tip calculator.
Key terms
- Keep fraction
- What is left after a discount, written as a decimal: 1 − d. A 35% discount has a keep fraction of 0.65. Stacked coupons multiply their keep fractions.
- Effective discount
- The single percentage that would produce your final pre-tax price in one step: 1 − S / P. The only discount figure worth comparing across stores.
- Stacking
- Applying more than one offer to the same item. Percentage coupons stack multiplicatively; a flat dollars-off coupon stacks by subtraction and its value depends on where in the sequence the register applies it.
- Former price
- The reference price a markdown is advertised against. Under the FTC guides it is expected to be a price at which the item was actually and openly offered, not a figure invented to size the discount.
- Sales price
- The tax base. In Streamlined Sales Tax member states it excludes discounts the seller grants and is not reimbursed for, which is why a store markdown normally reduces the tax you pay.