Reverse percentage: finding the price before a discount

A sale tag says “15% off,” you pay $63.75 at checkout, and you want to know what the item cost before the discount. This is a reverse percentage problem — you have the result and the rate, and you need the starting number. It trips people up because the instinct is to run the percentage forward again, which gives the wrong answer. This guide covers the correct formula, why the obvious shortcut fails, and how the same logic applies to tax, tips, and raises.

The formula

A discount takes a percentage off the original price, so:

Sale price = Original × (1 − r)

where r is the discount as a decimal. Solve for the original by dividing instead of multiplying:

Original = Sale price ÷ (1 − r)

For a 15% discount, r = 0.15, so you divide by 0.85. A $63.75 sale price gives 63.75 ÷ 0.85 = 75 — the original price was $75. You can verify: 15% of $75 is $11.25, and $75 − $11.25 = $63.75. It checks out.

Why the shortcut of adding the percent back fails

The mistake is treating the sale price as the base for the add-back. If you take 15% of $63.75 (the sale price) and add it, you get $63.75 + $9.56 = $73.31 — off by $1.69 from the correct $75. The reason: 15% was subtracted from the original price, a larger number, so 15% of the original is a larger dollar amount than 15% of the smaller sale price. Percentages are always relative to their base, and here the base is the number you're trying to find, not the number you already have. This is the same base-confusion that shows up in percentage point vs percentage differences — the direction of the calculation changes which number is the denominator.

The same logic in reverse for increases

The general form works for increases too, just with addition instead of subtraction:

Original = Final ÷ (1 + r)

A common real-world version: a receipt total of $107 includes 7% sales tax. The pre-tax price is 107 ÷ 1.07 = $100, not 107 − (107 × 0.07) (which gives $99.51, the wrong number for the same reason as above). A raise works the same way: if your new salary is $54,000 after a 8% raise, your old salary was 54,000 ÷ 1.08 = $50,000, not 54,000 minus 8% of 54,000.

Chained and repeated discounts

Retail sometimes stacks two percentages — say, 20% off, then an extra 10% off at checkout. These do not add to 30%; each percentage applies to the already-reduced price:

Final = Original × (1 − 0.20) × (1 − 0.10) = Original × 0.72

A stacked 20% and 10% discount is equivalent to a single 28% discount, not 30%, because the second percentage is computed on a smaller base after the first cut. To reverse a chained discount, divide by each factor in turn: Original = Final ÷ 0.90 ÷ 0.80. Order doesn't matter for the division since multiplication is commutative, but skipping a step and dividing by 0.70 instead of 0.72 will overstate the original price.

Quick reference by scenario

Whenever you're reversing a percentage, ask which direction the original percentage moved the number, then apply the opposite operation to the same base:

• Discount taken off → divide the result by (1 − rate) to get the original.
• Tax or markup added on → divide the result by (1 + rate) to get the pre-tax or pre-markup price.
• Multiple sequential percentages → divide by each factor separately, in any order.

The percentage calculator on this site handles the forward direction — percent of a number, one number as a percent of another, and signed percent change — with the formula shown for every result, so you can check a reverse calculation by running the forward version and confirming it lands back on your known figure.