The arithmetic here is simple enough to do by hand and easy enough to get wrong under time pressure — standing at a till, splitting a bill, or checking whether a "50% off, then 20% more" sign means what it appears to. Each tool shows the working so the answer is checkable.
Which one do you want?
| If you are… | Use |
|---|---|
| Percentages of, off, or change between numbers | Percentage Calculator — including reverse percentages, which are a division, never a subtraction. |
| Restaurant bill, split any number of ways | Tip Calculator. |
| Sale prices, including stacked discounts | Discount Calculator. |
The trap worth remembering: stacked discounts multiply, they do not add. 30% off then a further 20% off is 44% off, not 50%. Two 50% coupons make 75% off, never free.
Percentage traps worth knowing by heart
- Stacked discounts multiply, they do not add. 30% off then a further 20% off is 44% off, not 50%. Two 50% coupons come to 75% off, never free. The equivalent single discount is a + b − a×b.
- A percentage rise and the same percentage fall do not cancel. Up 20% then down 20% leaves you at 96% of where you started, not 100%. The second percentage is taken from a bigger number than the first was.
- Working backwards is division, not addition. If $63 is the price after 30% off, the original was 63 ÷ 0.70 = $90. Adding 30% to $63 gives $81.90 — wrong by nine dollars, and wrong in the direction that makes the deal look worse.
- Percentage and percentage point are different words. A rate moving from 4% to 5% has risen by one percentage point, or by 25%. Both are correct and they describe the same change; which one gets quoted usually depends on what the quoter wants you to conclude.
Frequently asked questions
If a price dropped 20% and then rose 20%, am I back where I started?
No — you are 4% below it. $100 falls to $80, and 20% of $80 is only $16, so you land at $96. Percentage changes are always measured against the value at the time.
What is the difference between 1 percentage point and 1 percent?
A rate moving from 4% to 5% has risen 1 percentage point, which is a 25% increase. Both descriptions are correct, which is exactly why headlines exploit the ambiguity.
How do I work backwards from a price that already includes tax?
Divide, do not subtract. A $108 total including 10% tax was $98.18 before tax ($108 ÷ 1.10), not $97.20.