A percent-off calculation answers a specific, practical question: if an item normally costs $X and is marked down by Y%, what's the final price? The formula is final price = original price × (1 − discount ÷ 100). A $50 item at 30% off becomes 50 × (1 − 0.30) = 50 × 0.70 = $35.
That single formula is really two steps compressed into one: first find the amount saved (30% of $50 = $15), then subtract it from the original price ($50 − $15 = $35). Both routes give the same answer; the compressed version is just faster once you trust it. Worked example one: a $120 jacket at 25% off. 120 × 0.75 = $90. Worked example two: a $19.99 book at 40% off. 19.99 × 0.60 = $11.994, which rounds to $11.99 in practice — a reminder that real-world prices round to the cent, even when the exact arithmetic produces a longer decimal.
Stacked discounts are where this calculation most often goes wrong. If a store advertises "30% off, plus an extra 20% off," it's tempting to add the percentages and treat it as 50% off — but that's not how sequential discounts work. Each discount applies to the already-discounted price, not the original: $100 at 30% off becomes $70, and then 20% off that $70 is $70 × 0.80 = $56, a total discount of 44%, not 50%. Stacked percentage discounts are always slightly less generous than their sum, because the second discount is calculated on a smaller base.
The reverse question — working out what an item cost before a discount, given the sale price and the percentage — uses the same formula solved for the original price: original = sale price ÷ (1 − discount ÷ 100). A $63 sale price after 30% off means the original was 63 ÷ 0.70 = $90. This is a different operation from simply adding 30% back onto $63 (which would incorrectly give $81.90), because the 30% was taken off a larger base than $63.
Worked example three, going the reverse direction with a stacked discount: an item is $80 after "25% off, then an extra 10% off." Reversing correctly requires undoing the steps in reverse order — the $80 was 90% of the post-first-discount price, so that price was 80 ÷ 0.90 = $88.89, and that $88.89 was 75% of the original, so the original was 88.89 ÷ 0.75 = $118.52. Reversing stacked discounts in one combined step (dividing $80 by 0.65, treating it as one 35% discount) gives a noticeably wrong answer, because — exactly as shown above — stacked discounts don't behave like a single combined percentage.
Sales tax adds another layer some shoppers get backwards: whether tax is applied before or after the discount usually doesn't change the final total mathematically (multiplication is commutative — discounting then taxing, or taxing then discounting, gives the same result), but it's worth checking a receipt if the numbers don't seem to match expectations, since retailers do occasionally apply tax to the pre-discount price by local rule. Percent-off calculations also generalize naturally to "percent more" situations — a markup rather than a markdown — by flipping the sign in the same formula: final price = original × (1 + markup ÷ 100). A wholesale item costing $40 with a 50% retail markup becomes 40 × 1.50 = $60. Retailers occasionally advertise a discount relative to an inflated "original" price rather than the price the item actually sold at previously, which is a business-practice issue rather than a math error, but it's worth knowing the formula is only as honest as the original price fed into it.
For the underlying percentage formula in more general form, see the Percentage Calculator and its companion guide.