Stacked Discounts: Why 20% Off + 10% Off Is 28%, and How to Back Out Sales Tax

The sign says “20% off everything, plus an extra 10% at checkout.” Your brain says 30%. The register says 28%. Later you look at a $108 receipt, want to know what the item cost before 8% tax, take 8% of $108 and get $8.64 of tax. The real figure is $8.00.

Both mistakes have the same cause: a percentage is always a percentage of whatever the price is at that moment, and the price keeps moving under you. The second discount comes off a smaller number than the first. The tax on your receipt was charged on a smaller number than the total.

Below are two short recipes, one for stacking discounts and one for backing tax out of a total, plus a table of the mistakes people make and what they cost. The arithmetic matches what our discount calculator and sales tax calculator do, so you can check any number here in either one.

Recipe 1: stacked discounts multiply

Turn each discount into the share of the price you still pay, then multiply those shares together.

20% off  → you pay 0.80
10% off  → you pay 0.90
together → 0.80 × 0.90 = 0.72 → you pay 72%, which is 28% off

On an $80 item: $80 × 0.80 = $64, then $64 × 0.90 = $57.60. The second 10% only takes $6.40, not $8, because it’s 10% of $64. That missing $1.60 is the 2 points between “30%” and “28%”.

If you want the combined discount in one line, it’s a + b − a × b with both as fractions. For 20% and 10%: 0.2 + 0.1 − 0.02 = 0.28. The a × b term is the part you lose by taking the second discount from an already reduced price. It is tiny for small discounts and huge for big ones:

Discounts	What it sounds like	You actually pay	Real discount	Gap
10% + 10%	20% off	81%	19% off	1 point
20% + 10%	30% off	72%	28% off	2 points
30% + 20%	50% off	56%	44% off	6 points
40% + 40%	80% off	36%	64% off	16 points
50% + 50%	100% off (free)	25%	75% off	25 points

The last row is the one that makes it obvious. Half off, then half off again, can’t be free. Half of the half is still a quarter of the original price. A $100 coat at “50% off plus an extra 50%” costs $25.

Two more things fall out of the multiplication:

The order of two percent discounts doesn’t matter. 0.8 × 0.9 and 0.9 × 0.8 are the same number, so “10% then 20%” and “20% then 10%” both land on 28% off.
The order of a dollar coupon does matter. Subtraction and multiplication don’t commute. On an $80 item at 25% off with a $10 coupon, taking the coupon after the percent gives $80 × 0.75 − $10 = $50.00. Taking it first gives ($80 − $10) × 0.75 = $52.50. Coupon last is better for you, because the percent then applies to a bigger number.

The discount calculator’s “Sale price” mode handles exactly this: an original price, a percent off, an optional “Extra % off” that it applies to the already reduced price, an optional fixed-dollar coupon with a “before % off” or “after % off” switch, and optional sales tax charged on the discounted price. When you fill in both percents, it prints the combined figure and spells out that it’s not the sum. It shows each step (first discount, extra discount, coupon) with the dollars taken off and the running price.

Recipe 2: back sales tax out of a total by dividing

A total that includes tax is the pre-tax price times (1 + rate). So you get back to the pre-tax price by dividing by the same thing:

pre-tax = total ÷ (1 + rate)       $108 ÷ 1.08 = $100.00
tax     = total − pre-tax          $108 − $100 = $8.00
check   = pre-tax × (1 + rate)     $100 × 1.08 = $108.00

That’s the whole recipe. Always do the third line. If multiplying your answer by (1 + rate) doesn’t give back the total, you went the wrong way.

The tempting shortcut is to take 8% of the total. That gives $8.64 of “tax” and a “pre-tax price” of $99.36. Run the check: $99.36 × 1.08 = $107.31, not $108. The shortcut is wrong because the 8% was charged on the $100, not on the $108. Taking 8% of $108 also taxes the tax.

The sales tax calculator has this as its own mode. Switch from “Add tax to a price” to “Remove tax from a total,” enter the total and the rate, and it computes total ÷ (1 + rate), rounds to the cent, and shows the pre-tax price, the tax and the total side by side. If you don’t know your combined rate, tick “Build a combined rate (state + county + city)” and add the parts. Picking a state fills in its base rate from a built-in reference table, which you can edit if it’s out of date.

When the cents don’t come back exactly

Real receipts round tax to the cent, so dividing a total won’t always land on a clean price. Try $10.00 at 8.875%. Dividing gives $9.1848…, which rounds to $9.18. But $9.18 plus its tax (rounded to the cent) is $9.99, and $9.19 plus its tax is $10.01. No pre-tax price produces exactly $10.00 at that rate. If your answer is a cent off when you check it, that’s rounding, not a mistake. For a multi-item receipt, the store may also have rounded per line rather than on the total, which can move things another cent.

The mistakes table

What you did	Example	What you got	Correct answer
Added two discounts	20% + 10% on $80	$56.00 (30% off)	$57.60 (28% off)
Added two big discounts	40% + 40% on $100	$20.00 (80% off)	$36.00 (64% off)
Took the rate from the total	8% of $108	$8.64 tax	$8.00 tax
Multiplied the total by (1 − rate)	$108 × 0.92	$99.36 pre-tax	$100.00 pre-tax
Added the discount back to undo it	$60 sale at 25% off, $60 × 1.25	$75.00 original	$80.00 original ($60 ÷ 0.75)
Used the coupon first when the store uses it last	$80, $10 coupon, 25% off	$52.50	$50.00

The fifth row is the discount version of the tax mistake. A sale price is the original times (1 − discount), so the original is the sale price divided by (1 − discount). The discount calculator has a “Find original price” mode for this, and it shows the division it did (for example $60.00 ÷ 0.75). Its “Find % off” mode goes the other way: give it the original and what you paid, and it returns the percent off.

Discounts first, then tax (usually)

When a store discount and sales tax both apply, the tax is normally charged on the discounted price. The discount calculator works that way: tax is on the reduced price, not the original. California’s tax agency spells out one exception that catches people. Its Publication 113 on coupons, discounts and rebates says a retailer’s own coupons “are excluded from your total taxable sales,” but the amounts manufacturers pay “to reimburse you for the value of the manufacturer’s coupons” are “included in your total taxable sales.” So a manufacturer coupon can lower what you pay without lowering the tax. Other states set their own rules, so if the cents on a receipt matter, check the receipt.

The same “which base?” question comes up at restaurants. If you want to tip on the food and not on the tax, back the tax out first with the sales tax calculator, then use the tip calculator. Its “Tip on the pre-tax amount” option takes the tax in dollars and tips on the bill minus that amount.

If this kind of trap is familiar, it’s the same family as “down 20%, then up 20%, isn’t back where you started,” which we covered, along with undoing a discount and stripping VAT, in percentage points vs percent change. If you’re the one setting the sale price, margin vs markup shows how fast a discount eats into your profit.

The short version

Stacking: turn each discount into what you still pay (1 − d) and multiply. Never add.
Undoing a discount: divide the sale price by (1 − d).
Backing out tax: divide the total by (1 + rate). Never take the rate from the total.
Always check: run your answer forward and see if you get the number you started with.

For a quick answer, put the deal into the discount calculator, with the extra percent and any coupon, and it will show every step. For a receipt, use “Remove tax from a total” in the sales tax calculator. For any other percent question, the percentage calculator has ”% of a number,” “X is what % of Y” and ”% change.” All three run in your browser.