Maths
Four percentage mistakes almost everyone makes
Percentage change, stacked discounts, working back to the original price and percentage points: four calculations that trip people up, explained with worked examples.
1. Up 20 % then down 20 % is not back to the start
A share price goes from 100 up 20 % to 120, then falls 20 %. It ends at 96, not 100, because the second 20 % is taken from 120: 120 × 0.8 = 96.
The same thing happens with salaries, rents and prices. To undo a 20 % rise you need a fall of 1 − 1/1.2 ≈ 16.7 %. To undo a 50 % fall you need a 100 % rise. In Percentage Calculator, use *Percentage increase or decrease* with the old and new values to see the real change.
2. Two discounts don’t add up
A shop offers 30 % off, and a coupon gives another 20 % off. That is not 50 % off. The coupon applies to the already reduced price:
- 100 × 0.7 = 70 after the first discount
- 70 × 0.8 = 56 after the coupon
So the total discount is 44 %. The order of the two discounts doesn’t matter (0.7 × 0.8 = 0.8 × 0.7), but adding them does. Use *Add or subtract a percentage* twice to check a deal.
3. Working back to the original price
You paid 80 after a 20 % discount. What was the original price? The tempting answer is 80 + 20 % = 96. It is wrong: 20 % of 96 is 19.2, and 96 − 19.2 = 76.8.
The right method is to divide by what is left: 80 ÷ (1 − 0.20) = 100. The same logic gives prices before tax: a price of 120 that includes 20 % VAT is 120 ÷ 1.2 = 100 before tax, with 20 of tax, not 24. The calculator’s *Original price before a discount* mode does this for you.
4. Percentage points versus percent
An interest rate rises from 2 % to 3 %. Has it gone up 1 % or 50 %? Both statements are made, and only one is clear.
- It rose by 1 percentage point (3 − 2).
- It rose by 50 percent in relative terms ((3 − 2) ÷ 2).
News reports about interest rates, unemployment or election results should use percentage points for the first meaning. When you see a headline claim, ask which one is meant. The *Percentage points vs percent* mode shows both side by side.
A habit that prevents all four
Before trusting a percentage, ask “percent of what?”. Every mistake above comes from taking the percentage of the wrong base: the original instead of the new value, the full price instead of the reduced one, the final price instead of the original. Percentage Calculator shows the formula next to each answer, so you can see which base it used.