Scope and assumptions
Single percentage discount; rate as decimal; sale price already reflects the discount.
Discounts run forwards easily: list price in, sale price out. The reverse is where people stall, because the obvious move — adding the percentage back onto the sale price — is wrong, and it's wrong in a way that looks almost right.
Recovering the original needs one division, and the divisor is the share you paid, not the rate you saved.
The reverse division
Original Price = Sale Price ÷ (1 − rate)
| Sale price | Off | Divide by | Original |
|---|---|---|---|
| $80 | 20% | 0.80 | $100.00 |
| $140 | 30% | 0.70 | $200.00 |
| $56 | 30% | 0.70 | $80.00 |
| $35 | 15% | 0.85 | $41.18 |
Why the divisor is (1 − rate)
A 20% discount means you handed over 80% of the original. So the relationship is:
sale = 0.80 × original
Undo it by dividing both sides by 0.80. Dividing by 0.20 would instead return the discount amount — $16 in this case — which is a different quantity entirely, and a common way to produce an answer that's wildly too large.
The reverse formula is just the forward one solved for a different variable. Nothing new is happening; the base is simply being recovered.
The add-back error
Adding 20% to $80 gives $96, not $100. The reason is subtle and worth holding onto: the 20% discount was 20% of the original, which is larger than the sale price. Once the price falls, the same 20% is a smaller dollar amount, so adding it back under-recovers.
Percentages don't reverse by applying the same rate in the opposite direction. They reverse by dividing.
What the recovered price tells you
Reversing a tag is also a way to audit it. If a jacket is $80 at "20% off," the implied original is $100. If you know the jacket never sold above $90, the $100 reference price was manufactured for the sale — and the "discount" is smaller than advertised.
When the tag gives dollars, not percent
"$20 off, now $80" doesn't require the reverse formula at all: the original is simply $100. If you know the rate but not the original, divide the dollar discount by the rate — $20 ÷ 0.20 = $100. What you can't do is recover the original from the dollar discount alone, with no rate and no reference price.
The Common Traps in Reverse Discounting
Dividing by the rate instead of its complement; adding the rate back to the sale price; forgetting to convert 20% into 0.20; rounding the recovered original before checking it; attempting a reversal from a dollar discount with no rate; and treating the recovered figure as fact when the "was" price may have been inflated.
Related discount guides
The reverse direction (sale price → original). For the forward price-to-pay method, see Sale Price After Percent Off, and for the full formula index start at the hub: Discount Formula.
FAQ
How do I reverse two stacked discounts to find the original? Divide successively. After 20% then 10% off, the survivor factor is 0.80 × 0.90 = 0.72, so original = sale ÷ 0.72.
Can I recover the original from the dollar discount and the rate? Yes. If you know the rate, the original is (dollar discount ÷ rate). $20 off at 20% implies an original of $100.
Why doesn't adding the percentage back to the sale price work? Because the rate was measured against the larger original. Adding 20% to $80 gives $96, but the original was $100 — the base changed when the price fell.
Must a retailer's "was" price be genuine? In the US, yes — the FTC's guides against deceptive pricing treat an inflated "was" price as a fake discount, and the UK's Consumer Protection from Unfair Trading Regulations 2008 enforce the same idea. A reference price never actually charged — price inflation, or a "fake anchor" — can breach these consumer protection laws; rules vary, so check local guidance.
Sources: Khan Academy, Percent word problems; Maths is Fun, Percentages; FTC, Guides Against Deceptive Pricing.
Disclaimer: Illustrative. Real tags may show manufactured "was" prices; verify legitimacy. Informational only.
Sources and references
- Khan Academy — Percent word problemshttps://www.khanacademy.org/math
- Maths is Fun — Percentageshttps://www.mathsisfun.com/percentage.html
- FTC — Guides Against Deceptive Pricinghttps://www.ftc.gov/business-guidance/resources/guides-against-deceptive-pricing-practices
Maintenance and corrections
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