Math walkthrough · reverse-discount arithmetic
How to find the original price before a discount
You know the sale price and the percent it was marked down — now you need the number it started from: to check a "was" anchor, reconcile a receipt, or fill in an expense report. The formula is one line, and the intuitive shortcut is wrong in a way worth seeing once.
The reverse formula
original price = sale price ÷ (1 − discount rate)
Sale price of $80 at 20% off: $80 ÷ (1 − 0.20) = $80 ÷ 0.8 = $100. It's the forward formula run backwards — the sale price is 80% of the original, so the original is the sale price divided by 0.8. Division undoes the multiplication exactly.
Why "add the percent back" is wrong
The intuitive move — take the sale price and add 20% back — gives $80 × 1.2 = $96, not $100. Four dollars short, and the error is structural, not rounding: adding 20% computes 20% of $80, but the original discount was 20% of $100. The base changed. Percents are only meaningful against their base, and reversing a discount means dividing, never adding back.
The bigger the discount, the bigger the miss: reverse $59 at 40% off correctly — $59 ÷ 0.6 = $98.33 — while adding 40% back gives $82.60, a $15.73 error on a question as simple as "what did this cost before?"
Worked examples
| Sale price | Discount | Original (÷ route) | Add-back error |
|---|---|---|---|
| $80 | 20% | $100 | $96 |
| $59 | 40% | $98.33 | $82.60 |
| $228 | 25% | $304 | $285 |
For a price that carried multiple successive discounts, invert each layer in reverse order — divide by (1 − d) for the last discount first — or use the stacked formula from the successive discount guide and divide once by its combined factor.
When you'd actually use it
- Checking a "was" anchor. If the tag says "was $125, now $87.50 at 30% off," the reverse formula says the original was $125 — consistent. If it doesn't reconcile, the anchor is inflated, and the discount percentage is marketing. (Keep the receipt for the comparison — receipt ledger apps keep the numbers searchable if your inbox is where receipts disappear.)
- Reconciling a receipt. Line-item discounts and the final total should tell the same story; the reverse formula is the check.
- Expense reports and claims. Reimbursement forms often want the pre-discount figure — the ÷ route gives it in one step.
Reverse the discount
Enter the sale price and the discount it carried — see the correct original price, and what the "add the percent back" mistake would have given you.
$100
original price — 80 ÷ (1 − 20/100)
The "add it back" mistake: 80 × (1 + 20/100) = $96 — off by $4, because the percent would apply to a different base.
Useful for checking a "was" price anchor, reconciling a receipt, or expense reports that need the pre-discount figure.
The forward version of this arithmetic — what you pay after a discount — is the main discount calculator; the single-layer mental math is in how to calculate 20 percent off.
Sources & verification
Sources last verified 2026-10-21. Retailer policies and sale calendars change without notice — treat the retailer's own help pages as the authoritative version before you buy.
Arithmetic only — tax handling, gift-card interplay, and per-line rounding at the register can shift real receipts slightly from the pure figures.