Reverse Discount Calculator

The two things this calculator works out

Most discount math runs in one direction: you start with a price and take a percentage off it. Reverse discount math runs the other way. You already hold the receipt, and you want the number that sat behind the discount. This calculator handles both versions of the question.

The first version is the common one. You know the sale price and the discount that came off, and you want the price before the sale. The formula is short:

• Original price = sale price / (1 - discount / 100)
• A sale price of $72.00 at 20% off is $72.00 / 0.80 = $90.00
• What you saved = original price - sale price, which is $18.00 here

The second version turns the question around. You know the original price and the sale price, and you want the discount. The formula is:

• Discount = (original price - sale price) / original price x 100
• A mark down from $90.00 to $72.00 is (18.00 / 90.00) x 100 = 20%

The habit to break is dividing by the wrong base. Take 20% off $90.00 and you get $72.00, but adding 20% back onto $72.00 gives $86.40 rather than $90.00. Percentages are not symmetrical, and that one slip causes most of the wrong answers people get at the kitchen table.

Worked example: $72 paid with 20% off

Here is the number trail for the example the calculator opens with, worked through step by step with sales tax added at the end.

• Sale price you paid before tax: $72.00
• Discount taken off: 20%, so you paid 80% of the original
• Original price: $72.00 / 0.80 = $90.00
• Saved on the item: $90.00 - $72.00 = $18.00
• Sales tax at 7.25%, charged on the $72.00 selling price: $5.22
• Total you paid: $72.00 + $5.22 = $77.22

Buy two of them and the same numbers scale up. The subtotal before tax is $144.00, the tax on it is $10.44, and the total is $154.44. What you saved on the pair is $36.00.

Notice what the tax does not touch. It never applies to the $90.00. Sales tax is charged on the actual selling price, which is the $72.00 you were asked to pay, so the discount itself is never taxed. If a receipt shows tax worked out on the pre-sale price, the store has made a mistake and it is worth asking about.

Why the discount comes off before the tax is added

People say "take the discount off first, then add the tax", and that order is the right one to use. There are two reasons it matters even when the final total looks identical either way.

With a straight percentage discount the arithmetic happens to commute. Twenty percent off $90.00 followed by 7.25% tax lands on $77.22, and 7.25% tax on $90.00 followed by 20% off lands on the same $77.22. The totals agree but the taxable amount does not, and the taxable amount is what a receipt, a return and an auditor care about. The tax attaches to the selling price, and the selling price is the discounted one.

With a fixed dollar discount the order starts to change the answer. Take an $80.00 item with a $20.00 store coupon at 7.25% tax. The store discounts first, so the tax applies to $60.00, which is $4.35, and you pay $64.35.

A manufacturer coupon is treated differently. The store hands you the discount and then gets reimbursed, so in most states the tax still applies to the full $80.00. The tax is $5.80 and your total is $80.00 x 1.0725 - $20.00 = $65.80. Now reverse it. To recover the original price from the $65.80 you paid, add the coupon back first and remove the tax second: ($65.80 + $20.00) / 1.0725 = $80.00. Take the other order, dividing out the tax first, and you land on $61.35 + $20.00 = $81.35, which is $1.35 off. Once a fixed dollar amount is in the mix, the sequence is not cosmetic.

Stacked discounts: 20% off plus 10% off is 28% off

Stores lean on a second discount on top of the first: 20% off, plus an extra 10% at checkout. The two do not add. They compound, because the second one is taken off an already reduced price, and the result is always less than the sum of the two percentages.

Those stacked pairs are worth 28%, 37%, 36.25%, 60% and 27.1% off respectively. The gap grows as the discounts get deeper, which is why clearance racks that say an extra 50% off the lowest marked price can still disappoint.

To undo a stack you divide by each multiplier in turn. A $57.60 sale price at 20% off then 10% off is $57.60 / 0.80 = $72.00, then $72.00 / 0.90 = $80.00. Doing it in one step with a single 28% divisor works too, because 0.72 is the combined multiplier, but the two-step version is easier to check against a receipt.

Percent off versus dollars off

A percent-off sale and a dollar-off coupon are two different animals, and they reverse in different ways. To undo a percent you divide. To undo a fixed dollar amount you add. Mixing the two up is the second most common mistake after adding the percentage back.

The same coupon is worth more on a cheaper item and less on an expensive one. A $20.00 coupon on a $90.00 item beats 20% off, because $20.00 is 22.2% of $90.00. Put the same coupon against a $150.00 item and it is worth 13.3%, which is worse than a flat 20% off.

Reversing a dollar amount is the easy case. If a tag shows a sale price of $70.00 after $20.00 off, the original was $90.00, and no division is involved. When a receipt shows both a percent and a dollar coupon, undo the dollar one first, then the percent, then the tax.

The divide-by table for common discounts

Once you know the shortcut you rarely need a calculator, because each discount percent has a fixed number to divide by. The table covers the percentages that show up most often in US stores, using a $100.00 sale price as the worked figure.

The pattern to notice is how fast the multiplier runs away at the deep end. Undoing 70% off multiplies the sale price by 3.333, so a $100.00 clearance price came from a $333.33 original. That is also a warning about trusting a big discount claim, because a price that low often started somewhere nearer the sale price than the tag admits.

Checking a sale claim you saw in an ad

A reverse discount calculation doubles as a fact check. When a tag reads was $90.00, now $72.00, the discount is 20%. When a store lifts the was price in the weeks before a sale, the percentage it advertises is larger than the one you collect.

The check takes one division. Divide the now price by the was price and subtract the result from 1. A jacket tagged was $90.00, now $79.00 is 12.2% off. If that jacket sat at $82.00 for the six weeks before the sale, the real discount is $3.00, or 3.7%, and the was price on the tag was never a price anyone paid.

The Federal Trade Commission requires advertising to be truthful and not misleading, and its advertising guidance for small business sets out how former price claims are treated. A was price that no one ever paid is not a bargain, and a comparison price made up for a sale weekend is the thing the rule is aimed at.

The same check works on a coupon code at checkout. If the code promises 20% off and the price only falls by $4.00 on a $90.00 item, the code is worth 4.4%, and something else in the cart, such as a shipping charge that rose at the same moment, has quietly soaked up the saving.

Where people get this wrong

The mistakes cluster into a handful of habits, and each one shows up on a receipt in a predictable way.

• Adding the percentage back instead of dividing. Twenty percent added to $72.00 is $86.40, not $90.00. The base moved when the discount was taken.
• Adding stacked discounts. Twenty percent plus 10% is 28%, not 30%, and the gap widens as the percentages grow.
• Taxing the pre-sale price. The tax belongs on the selling price, so a $72.00 sale price carries $5.22 at 7.25% rather than the $6.53 that $90.00 would carry.
• Using the sale price as the base for the percentage. The $18.00 you saved is 25% of the $72.00 you paid, but the discount off the original is 20%.
• Stripping the tax last. When you are working backwards from a total that includes tax, remove the tax first, then undo the discount, then add any fixed coupon back.

On the tax point, if you itemize on your federal return, the IRS guidance on deductible taxes explains how state and local sales tax can be claimed instead of state income tax, which is worth a look before you write off a year of large purchases as unrecoverable.

This is a planning estimate, not tax advice. Rates and coupon rules differ by state, and a receipt from a real store always beats a calculator. Confirm the tax treatment for a real purchase with your state revenue department or a tax preparer.