The price after a percentage off — and, in the second mode, the original price when all you have is the sale tag.
Result
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A jacket is £170 after 15% off. What did it cost? Adding 15% to 170 gives £195.50, and it is wrong. The 15% was taken off the original price, not off 170. The right sum is 170 ÷ 0.85 = £200 — and the difference of £4.50 is the discount on the discount.
This is the same asymmetry that makes stacked discounts smaller than they look. "30% off, then a further 20% off" is not 50% off. It is 0.7 × 0.8 = 0.56, so 44% off. The second discount applies to the already-reduced price, which is why shops are happy to advertise them together.
A discount of more than 100% is arithmetically fine and commercially odd: the result goes negative, meaning the shop pays you. The calculator will show it rather than silently clamping to zero, because a tool that hides the answer is harder to trust than one that shows a strange one.
Divide the sale price by (100 − discount) ÷ 100. For £170 after 15% off: 170 ÷ 0.85 = £200. Do not add the percentage back — that gives a smaller, wrong answer, because the percentage was of the larger original number.
No, they multiply. 30% off followed by 20% off leaves 0.7 × 0.8 = 0.56 of the price, which is 44% off rather than 50%. Applying them in the other order gives exactly the same result, so it never matters which coupon goes first.
The saving is the original price minus the sale price. As a percentage of what you pay rather than of the original, it looks bigger: saving £30 on a £200 jacket is 15% of the original and 17.6% of the £170 you handed over. Retail always quotes the first.
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