How to Find the Original Price Before a Discount

Divide the price you paid by one minus the discount. An item costing £60 after 25% off was £80, because 60 ÷ 0.75 = 80. The instinct to add 25% back to £60 gives £75, which is wrong, and it is wrong in the same direction every single time.

Why adding it back does not work

A percentage is always a percentage of something, and the discount was taken from the original price, not from the sale price. When you add 25% back, you are taking 25% of the smaller number, so you recover less than was removed.

original £80, 25% off
  discount taken  = 25% of 80 = £20   → pay £60

adding it back
  25% of 60       = £15               → £75   wrong

dividing
  60 / (1 - 0.25) = 60 / 0.75         → £80   correct

The error grows with the size of the discount. At 10% off it is small enough to miss; at 50% off, adding the percentage back lands 25% below the true original, which is a difference nobody would accept if they noticed it.

The formula

For any discount expressed as a percentage, the original price is the price paid divided by one minus that percentage as a decimal.

original = paid / (1 - discount / 100)

£45 after 40% off   →  45 / 0.60  = £75
£90 after 10% off   →  90 / 0.90  = £100
£12 after 70% off   →  12 / 0.30  = £40

The same shape works for anything that was reduced by a proportion, not only money: a figure after a 15% cut, a weight after a 5% loss, a headcount after a 20% reduction.

Going the other way, for tax

The mirror problem is removing a percentage that was added rather than taken off, which is what you want when a price includes sales tax or VAT and you need the amount before it. The structure is the same, with a plus instead of a minus: divide by one plus the rate. A price of £120 including 20% VAT is £100 before tax, because 120 ÷ 1.20 = 100. Subtracting 20% of £120 gives £96, and is the same mistake in a different coat.

Checking whether a sale is what it says

This calculation is most useful in reverse. If a listing claims "was £200, now £150", that is a 25% discount, and you can check the claim against the arithmetic rather than the sticker.

discount % = (was - now) / was × 100

(200 - 150) / 200 × 100 = 25%

The number worth being suspicious of is the "was" price itself, not the maths. A discount is only meaningful against a price the item genuinely sold at, which is why advertising rules in many countries require the higher price to have been charged for a defined period. The arithmetic can confirm the percentage is honest and can tell you nothing about whether the reference price was.

Two discounts do not add up

A further 20% off an item already reduced by 30% is not 50% off. Each reduction applies to what is left after the previous one, so the two multiply rather than add.

£100 → 30% off → £70 → 20% off → £56

total discount = 44%, not 50%
0.70 × 0.80 = 0.56

To reverse a stacked discount, divide by each factor in turn, or divide once by their product. Stacked reductions always come out less generous than the sum of their parts, which is exactly why they are advertised the way they are.