Stacked Discount Calculator — what successive percentages really cost

The stacked discount calculator above exists to settle one persistent piece of mental arithmetic that almost everybody gets wrong. Thirty per cent off, then a further twenty per cent off, is not fifty per cent off. It is forty-four per cent off. The gap is not a rounding artefact or a retailer trick — it is what happens when a percentage is taken from a number that has already been reduced.

Arb Digital builds free calculators that show the working rather than just the answer, because the reasoning here is more useful than the number. Once you can see why the second discount is smaller in money terms than the first, you can do this in your head at the till, and "extra 20% off sale prices" stops looking like a bigger offer than it is.

What This Stacked Discount Calculator Does

It applies up to three percentage discounts in sequence to an original price, then a fixed-amount coupon either before or after them, then an optional sales tax. The headline is the final price. Around it sit the four numbers that make the result interpretable.

Total saved is the plain money difference. The single equivalent discount is the one percentage that would have produced the same final price in a single step — the number to quote when you want to compare an offer against a competitor's straightforward reduction. The third item shows what the price would have been if the percentages had simply added together, which is the answer most people expect and the size of the gap between expectation and reality. The fourth shows what happens if the coupon is applied in the opposite order, because that one genuinely does change the total.

A boundary worth stating. Our discount calculator handles a single reduction and does it well: one price, one percentage, what you pay and what you save. This page is for the case that one cannot do in a single pass — several reductions applied in sequence, plus a fixed voucher whose position in the sequence matters. Our discount percentage calculator runs the question backwards, working out what percentage a known price cut represents.

How to Use It

1. Enter the original ticket price — the figure before any reduction, not the already-reduced sale price.
2. Enter each percentage discount in its own field. Use zero for any you do not have. The order does not matter, and you can prove that to yourself by swapping two of them and watching the answer stay put.
3. Enter a fixed-amount coupon if you have one, and choose whether the retailer applies it before or after the percentages. Most apply it after.
4. Add sales tax or VAT if it is charged on top of the displayed price rather than included in it.
5. Read the single equivalent discount. That is the number to compare against any competing offer.

The Formula and How It Is Calculated

Each percentage discount is a multiplier. A thirty per cent reduction multiplies the price by 0.70; a twenty per cent reduction multiplies it by 0.80. Applying both means multiplying by both:

Final = Price × (1 − d₁) × (1 − d₂) × (1 − d₃)

and the single equivalent discount is 1 − (1 − d₁)(1 − d₂)(1 − d₃).

Work through the defaults. A $200 item at 30 per cent off becomes 200 × 0.70 = $140. A further 20 per cent off that gives 140 × 0.80 = $112. Had the percentages added to 50 per cent, the price would have been $100. The $12 difference is the whole point: the second discount was taken from $140, not from $200, so it was worth $28 rather than $40.

The single equivalent is 1 − 0.70 × 0.80 = 1 − 0.56 = 44 per cent. So "30% off, then an extra 20% off" is a 44 per cent offer, and a competitor advertising a flat 45 per cent is offering more.

Then the coupon. Applied after the percentages, $112 − $15 = $97, which against the original $200 is a total saving of 51.5 per cent. Applied first, the sequence runs 200 − 15 = 185, then × 0.70 = 129.50, then × 0.80 = $103.60. Same three reductions, same order of percentages, $6.60 apart — because a fixed amount taken off early is itself subsequently discounted away.

Order Independence, Stated Properly

Here is the result that surprises people once they have accepted that percentages multiply: the order of percentage discounts makes no difference at all to the final price. Thirty per cent then twenty gives 200 × 0.70 × 0.80 = 112. Twenty per cent then thirty gives 200 × 0.80 × 0.70 = 112. Multiplication is commutative, so the sequence cannot matter.

That is worth knowing because retailers occasionally imply otherwise, and shoppers sometimes argue about which discount "should" go first. For pure percentages, there is nothing to argue about. If a cashier applies your loyalty discount before or after the seasonal sale, the total is identical.

Fixed amounts break the symmetry immediately, and this is the single most useful thing on the page. A $15 voucher is worth its full $15 only if it is the last thing applied. Applied before a 30 per cent and then a 20 per cent reduction, it is worth 15 × 0.70 × 0.80 = $8.40 — because everything applied after it discounts the voucher too. That is why almost every retailer's terms specify that fixed vouchers come off the reduced subtotal, and why the direction is always the one that favours the seller.

The same logic tells you how to read mixed promotions. Whenever a mixture of percentages and fixed amounts is on offer, you want percentages first and fixed amounts last. Whether you get to choose is a matter of the retailer's terms, not arithmetic.

Reading Real Offers Without a Calculator

Three shortcuts cover most situations. First, for two discounts, the single equivalent is the sum minus their product: 30 and 20 gives 50 − 6 = 44 per cent. The product term is what people forget, and it grows as the discounts grow. Two 50 per cent discounts are 100 − 25 = 75 per cent, not free.

Second, small discounts add almost correctly. Five per cent and then another five is 9.75 per cent, near enough to ten for a rough judgement. It is only at large percentages that the error becomes material.

Third, a second discount is always worth less in money than the same percentage applied first, and the more that came before it, the less it is worth. An "extra 20% off" on top of a 70 per cent clearance sounds enormous but only removes six per cent of the original price. Understanding that is what keeps a deep-clearance headline in proportion.

There is a consumer-protection dimension too. Advertised reductions have to be genuine, and both the US and UK have published rules about it. The Federal Trade Commission's Guides Against Deceptive Pricing cover former-price claims, retail price and value comparisons, and bundle offers. In the UK, the Competition and Markets Authority's guidance on price transparency addresses what has to be shown in an invitation to purchase, including drip pricing and partitioned pricing where the component parts of a price are given but the total the customer would actually pay is not. Neither is a substitute for checking the arithmetic yourself, which is what this page is for.

Where Stacking Goes Wrong in Practice

The most common real-world failure is not arithmetic at all — it is that the discounts do not stack. Terms frequently exclude sale items from a further percentage, cap the combined reduction, or allow only one promotional code per order. Working out that two offers combine to 44 per cent is academic if the checkout accepts only one of them.

The second is the reference price. A 60 per cent reduction from a price nobody has ever paid is not a 60 per cent reduction, which is precisely what the deceptive-pricing rules above are aimed at. If the same item sits at the "sale" price most weeks, the honest baseline is that price, not the ticket.

The third is what the discount applies to. Some codes exclude shipping, some exclude tax, some apply only to certain lines in a basket. This calculator applies the percentages to the whole price you enter, so if a code covers only part of the order, run it on that part separately.

The fourth is tax. Where tax is added at checkout rather than included in the ticket, it is charged on the discounted amount, which is what the tax field here does. If you need that side in more depth, the sales tax calculator handles it directly and the reverse sales tax calculator extracts the tax from a tax-inclusive total.

The Same Maths, Pointed the Other Way

Successive percentages behave identically when they go up rather than down, and the asymmetry catches people out just as often. A price raised by 20 per cent and then cut by 20 per cent does not return to where it started: 100 × 1.20 × 0.80 = 96. A four per cent loss appears from nowhere. Our percentage change calculator covers that general case, and the percentage calculator handles the basic operations.

For sellers, the mirror image is margin. A retailer stacking promotions is compounding reductions against a cost base, and the markup calculator shows how quickly stacked discounts eat into a margin that looked comfortable at full price. When comparing options across different pack sizes or bundles, the unit price comparison calculator is usually the better test of value than any percentage.

Common Mistakes to Avoid

• Adding the percentages together. 30 per cent and 20 per cent is 44 per cent, not 50. The larger the discounts, the larger the error.
• Assuming a fixed voucher is always worth its face value. Applied before percentage discounts, a $15 voucher on a 30-then-20 stack is worth $8.40.
• Arguing about the order of percentage discounts. There is nothing to argue about: multiplication is commutative and the final price is the same either way.
• Comparing offers by headline percentage. Convert each to a single equivalent discount first, then compare. An "extra 20%" on a deep clearance removes far less than it sounds like.
• Ignoring the terms. Exclusions, one-code-per-order rules and caps are more likely to decide the outcome than the arithmetic is.

Related Free Tools From Arb Digital

For a single reduction use the discount calculator, and to work backwards from a price cut to its percentage use the discount percentage calculator. The percentage calculator and percentage change calculator cover the general arithmetic, the sales tax calculator and reverse sales tax calculator handle tax at checkout, and the unit price comparison calculator plus the markup calculator cover value and margin. Everything else is on the free online tools hub.

Frequently Asked Questions

This page performs arithmetic on figures you enter. It is not financial or legal advice, it cannot know a retailer's terms, and whether particular offers may be combined at all is governed by those terms rather than by the calculation.