Stacked Discount Calculator

Why two discounts do not add up

A stacked discount is two or more reductions applied one after another, each one charged on the price that is left. That single detail is why 20% plus 25% is not 45%.

Take a $200 item. The first 20% comes off $200, which is $40, and the price becomes $160. The second 25% comes off $160, which is $40, and the price becomes $120. The two discounts took $80 in total, and $80 out of $200 is 40%, not 45%.

The missing 5% is not a rounding error. It is the difference between 25% of $160 and 25% of $200, which is $40 against $50. The second discount is smaller in dollars because it is working on a smaller base, and that is the whole arithmetic of stacking.

The shorthand is to multiply what you keep, not add what you lose. Keeping 80% and then keeping 75% is 0.8 times 0.75, which is 0.6, so you pay 60% and save 40%. Once you see it as multiplication, three discounts are no harder than two.

Worked example: 20% then 25% on $200

Run the whole chain with a coupon and sales tax so the order of operations is visible.

1. Start: $200.00
2. 20% off: $200.00 times 0.80 is $160.00
3. 25% off: $160.00 times 0.75 is $120.00
4. $10 coupon: $120.00 minus $10.00 is $110.00
5. 8.25% tax: $110.00 times 0.0825 is $9.08
6. Total: $110.00 plus $9.08 is $119.08

Compare that with adding the percentages. 20% plus 25% is 45%, and 45% off $200 is $110.00, which is $10.00 below the real $120.00 after both discounts. Push the coupon through that chain and the price is $100.00 before tax, or $108.25 with 8.25% tax, against the $119.08 you pay. The wrong method understates the bill by $10.83.

The effective discount on this order is 45% of the original price, but only because the coupon adds the last 5 percentage points. The two percentages alone delivered 40%. Keep those two numbers apart or the next example will confuse you.

Stacking three or more discounts

Three discounts multiply the same way. Multiply the keep-factors, then subtract from 1 to get the discount.

The last row is the one to remember. Half off and then half off again leaves a quarter of the price, not zero. No sequence of percentage discounts ever reaches zero, because each one takes a share of what remains rather than a share of the original.

On a $200 item the 20/25/10 chain gives $200 times 0.54, which is $108.00. The same item under the naive 55% would be $90.00, an $18.00 gap that shows up at the register.

Where the coupon goes in the order

A dollar coupon and a percentage discount do not commute. The percentage should come first, because a percentage of a larger number takes off more money, and the coupon then comes off the smaller balance.

The gap between the first two rows is $4.00 before tax. The coupon-first route takes $10 off $200, leaving $190, and then the percentages take 40% of that, which is $76, so you pay $114. The percentage-first route takes 40% off $200 first, leaving $120, and then the coupon takes $10, so you pay $110.

In practice the store decides the order, and the checkout page applies it in the sequence its system was built for. If you have a choice, spend the percentage first. If the store insists on the coupon first, the difference is usually small enough that the code still saves money, but it is worth knowing why the total is a few dollars above your estimate.

Tax comes after the discount, not before

Sales tax is charged on the amount the customer pays. A discount that lowers the price lowers the tax with it, which is why the order matters for the total even though the discount itself is the same.

The second row is the mistake people make on paper: tax the full $200, then take 40% off the taxed total. It overstates the bill by $10.82, because the $9.08 of tax on the discounted price never should have been charged on the full price.

One wrinkle is worth knowing. A store's own discount reduces the taxable price, but a manufacturer coupon that the store gets reimbursed for often does not. The California Department of Tax and Fee Administration spells this out in Publication 113 on coupons, discounts and rebates: a retailer discount is not taxable, while a manufacturer coupon the retailer is paid back for is part of the taxable price. Most states follow the same outline, so if a big manufacturer coupon barely moves the tax line, that is why.

Rebates are a third case. A mail-in rebate does not reduce the price at the register, so you pay tax on the full amount and get the rebate back later. That makes a $20 rebate worth less than a $20 instant discount, and the gap is the tax on the $20.

The order that saves the most money

Sort the tools by how they behave, not by how big the number looks.

• Percentages first. A percentage takes a share of the largest base, so it does the most work at the start of the chain.
• Dollar coupons second. A fixed $10 is worth the same whenever it lands, and it is worth more as a share of a small balance.
• Tax last. Tax is a percentage of whatever is left, so it should never be part of the discount chain.
• Rewards and points last. Points are usually earned on the discounted price, so applying them earlier does not help.

The counterexample is a percentage with a cap. A 30% code capped at $25 stops helping once the price passes $83.33, because the cap takes over. On a $200 item that code is worth $25, which is 12.5%, and a $25 coupon is worth exactly the same. When a cap is in play, compare the capped dollar value against the coupon before you decide which to use first.

Free shipping belongs in the same comparison. Shipping is not a discount on the item, and in most states it is taxable when the item is taxable, so a $9 shipping charge at 8.25% costs $9.74. A code that drops the price by $10 but adds shipping can leave you paying more than the code you skipped.

Reading a store's stacked promo without the math

Retail promotions are written to sound additive. "Extra 25% off already reduced prices" is a stack, and the real number is always smaller than the sum of the signs.

Three phrases carry most of the weight. "Extra 25% off sale prices" means 25% comes off the marked-down price, which is a stack. "25% off your entire purchase" usually applies to the original price of full-price items only, which is not a stack at all. "Up to 70% off" is a ceiling across a range of items, not a rate you will see on the thing you want.

The Federal Trade Commission's guides against deceptive pricing cover comparative claims like "was $200, now $120", including the rule that a former price has to be one the item was genuinely sold at. That protects you from an invented original price, but it does not police the arithmetic of a stack, so the multiplication is still yours to do.

A quick habit: convert each percentage to a keep-factor and multiply in your head. 20% off keeps 0.8, 25% keeps 0.75, and 0.8 times 0.75 is 0.6. Two seconds of multiplication beats discovering a $10 gap at the register.

What this calculator does not cover

The math here assumes every discount applies to the whole order. Real carts are messier, and four things break the clean chain.

• Exclusions. A category code that skips gift cards, clearance or a named brand leaves a mixed cart where part of the price is discounted and part is not. The effective rate ends up between the advertised rate and zero.
• Minimum spend. A $25-off-$100 code does nothing on a $95 cart. Adding a $6 item to reach the threshold can be a good trade, and it can also cost more than the discount.
• Per-item caps. "$10 off each item" and "$10 off the order" look similar and behave nothing alike on a cart with five items.
• Membership prices. A loyalty price is usually applied before a coupon, and some coupons exclude items already sold at a member price.

Stacking also runs into store policy. Many retailers allow one percent-off code plus one dollar coupon, and block two percent-off codes. If a code is rejected at checkout, the reason is usually a rule rather than an error, and the order the system applies is fixed on its side.

Sales tax treatment of discounts, coupons and rebates varies by state and by the type of coupon. The figures here are an estimate for planning, so confirm the tax base with your state revenue department or a tax professional before you rely on them for a filing.