The reverse discount formula
Original price = sale price / (1 - discount as a decimal). The logic is simple: the sale price is the original price with the discount taken out, so dividing by what remains puts the discount back. A 25% discount leaves 75%, or 0.75, of the price, so you divide the sale price by 0.75.
Worked example: $60 at 25% off
The tag says $60, the sign says 25% off. Compute 1 - 0.25 = 0.75. Divide: $60 / 0.75 = $80. The original price was $80. Check it forward: $80 x 0.75 = $60. It matches. Another: a $42.50 sale price at 15% off. Compute 1 - 0.15 = 0.85. Divide: $42.50 / 0.85 = $50. The item was $50 before the discount.
Reversing stacked discounts
When two discounts applied one after another, reverse them one step at a time, starting with the last discount. Example: an item rings up at $57.60 after 20% off plus an extra 10% off. Reverse the 10% first: $57.60 / 0.90 = $64. Then reverse the 20%: $64 / 0.80 = $80. The original price was $80. Order matters only in that you undo the last discount first; the final answer is the same either way because multiplication is commutative.
Why bother finding the original price?
Two reasons. First, verification: compare your computed original price with the "was" price on the tag. If the tag claims $100 but your reverse math says $80, the reference price is inflated and the real discount is smaller than advertised. Second, comparison shopping: two stores may show different sale prices with different discount percents, and the original price is the common baseline that tells you which shelf price is genuinely lower.
Finding the discount percent from two prices
Sometimes the tag shows the original and sale prices but no percent. Compute it: (original - sale) / original x 100. An $80 item selling for $60: ($80 - $60) / $80 = 0.25, a 25% discount. A $45 item at $36: ($45 - $36) / $45 = 0.20, or 20% off. This is the fastest way to compare two competing sales: the store advertising the bigger dollar saving is not always the bigger percent saving.
When sales tax is baked into the price
A few tags and receipts show the tax-inclusive total, which you must strip out before reversing the discount. Remove tax first: total / (1 + tax rate). A $64.80 receipt at 8% tax means the pre-tax sale price was $64.80 / 1.08 = $60. Then reverse the discount: at 25% off, $60 / 0.75 = $80 original. Skipping the tax step inflates your answer, because you would be treating tax dollars as part of the discounted price.
The mistakes to avoid
The classic error is multiplying instead of dividing: $60 x 1.25 = $75, which is wrong. Adding the percent back does not reverse a discount, because the percent applies to a different base. Another error is reversing stacked discounts as a single sum: $57.60 / 0.70 = $82.29, which is wrong, because the discounts multiplied (0.80 x 0.90 = 0.72), so the correct reversal is $57.60 / 0.72 = $80. When in doubt, check your answer by running the discounts forward.