Discount Calculator

Discount calculations look simple — take a percentage off a price — but the reverse direction, working out what a discounted price was *before* the discount, trips up more people than the forward calculation does, largely because the intuitive-but-wrong approach is to just add the percentage back rather than accounting for how percentages compound. The forward calculation is straightforward: discount amount equals the original price multiplied by the discount percentage, divided by 100, and the final price is the original price minus that discount amount. A £50 item with a 30% discount has a discount amount of £50 × 0.30 = £15, making the final price £50 − £15 = £35. The reverse calculation — given a final sale price and the discount percentage that was applied, find the original price — is where the common mistake happens. The naive (and incorrect) approach is to assume that if 30% was taken off, you can find the original price by adding 30% back to the sale price: £35 × 1.30 = £45.50. But this is wrong, and the error grows larger as the discount percentage increases. The correct approach recognizes that the sale price already represents 70% of the original price (100% − 30% discount = 70% remaining), so the original price is the sale price divided by 0.70, not multiplied by 1.30: £35 ÷ 0.70 = £50 — which correctly reverses the original £50 → £35 calculation. The gap between the wrong method (£45.50) and the correct one (£50) is £4.50 on this single example, and it grows substantially larger at higher discount percentages. This asymmetry exists because percentage decreases and percentage increases aren't mirror images of each other around the same base. Taking 30% off £50 moves you down by £15 to £35, but to get back from £35 to £50 you need an increase of £15 on a base of £35, which is £15 ÷ £35 ≈ 42.9% — not 30%. The percentage needed to reverse a discount is always larger than the discount itself, and the gap widens dramatically as the discount grows: a 50% discount needs a 100% increase to reverse (since half of something needs to be doubled to get back to the whole), and a 75% discount needs a 300% increase to reverse. This is exactly why the reverse formula — dividing by (1 − discount%/100) rather than multiplying by (1 + discount%/100) — matters, and why this calculator provides a dedicated reverse mode rather than expecting users to intuit the correct adjustment themselves. A worked comparison across discount sizes makes the pattern clear: a 10% discount needs an 11.1% increase to reverse; a 25% discount needs a 33.3% increase; a 50% discount needs a 100% increase; and a 90% discount needs a 900% increase. As the discount percentage approaches 100%, the percentage needed to reverse it grows toward infinity — which makes sense, because a "100% off" final price is £0, and no percentage increase on £0 can ever recover a positive original price. This is precisely the edge case the reverse mode guards against, showing a dash rather than attempting a division by zero when the discount is exactly 100%.