The cheat sheet: six questions
A shop marks a jacket 20% off and you have a coupon for another 20% off the sale price. That is 36% off, not 40%. Almost every percentage problem is one of six questions, and each has one formula. The percentage calculator answers any of them and writes out its working.
| Question | Formula | Worked example |
|---|---|---|
| What is X% of Y? | Y × X ÷ 100 | 15% of 80 = 80 × 15 ÷ 100 = 12 |
| A is what percent of B? | A ÷ B × 100 | 45 of 60 = 45 ÷ 60 × 100 = 75% |
| Increase Y by X% | Y × (1 + X ÷ 100) | 80 plus 15% = 80 × 1.15 = 92 |
| Decrease Y by X% | Y × (1 − X ÷ 100) | 80 minus 25% = 80 × 0.75 = 60 |
| Percentage change from old to new | (new − old) ÷ old × 100 | 80 to 92 = (92 − 80) ÷ 80 × 100 = 15% |
| The original before a change of X% | result ÷ (1 ± X ÷ 100): plus for a rise, minus for a fall | 60 after 25% off = 60 ÷ 0.75 = 80 |
Rows 3 and 6 are the same sum run in opposite directions: 80 × 1.15 = 92, and 92 ÷ 1.15 = 80. Most of the traps below come from mixing those two up.
Percentage CalculatorFind the box for your question, enter the numbers and read the answer with every step written out.Open itTrap 1: taking off what you added
A percentage is always a share of the number it is applied to. Take 20% off 100 and you have 80. Add 20% back and you get 80 × 1.2 = 96, because the 20% you added is a share of the smaller number. To get from 80 back to 100 you need 20 ÷ 80 = 25%.
Raising first gives the same 96: 100 × 1.2 = 120, then 120 × 0.8 = 96. To undo a 20% rise you need a fall of 20 ÷ 120, about 16.67%. The deeper the fall, the steeper the climb back:
| Fall | Rise needed to recover |
|---|---|
| 10% | 11.11% (10 ÷ 90) |
| 20% | 25% (20 ÷ 80) |
| 50% | 100% (50 ÷ 50) |
| 75% | 300% (75 ÷ 25) |
| 90% | 900% (90 ÷ 10) |
The rule is fall ÷ (100 − fall) × 100. The percentage decrease calculator shows the rise needed to win a fall back, and the percentage increase calculator shows the fall that undoes a rise.
Trap 2: stacking discounts and rises
Two percentages in a row multiply. They do not add. Take an 80 coat: 30% off gives 80 × 0.7 = 56, then a coupon for 20% off gives 56 × 0.8 = 44.80. You saved 35.20, which is 44% of 80, not 50%. Two discounts of 50% leave 0.5 × 0.5 = 0.25, so they are 75% off, not free.
Rises compound the same way: two 10% rises make 1.1 × 1.1 = 1.21, a 21% rise, which is the idea behind compound interest. With percentages the order does not matter, but with a fixed amount off it does, so check how a coupon is applied. The discount calculator shows the combined discount when you add a second one.
Trap 3: reverse percentages
A price of 60 after 25% off was not 60 plus 25%, which is 75. The 25% was taken from the original, so divide: 60 ÷ 0.75 = 80. Check it the other way: 25% off 75 is 75 × 0.75 = 56.25, not 60.
Tax is the same trap. If 120 includes tax at an example rate of 20%, the price without it is 120 ÷ 1.2 = 100 and the tax is 20. Taking 20% off 120 gives 96, which is wrong. The tax is 20 ÷ 120 = 16.67% of what you paid, which is the rate ÷ (100 + rate). The VAT calculator uses whatever rate you type, because rates differ by country and by kind of goods, and they change. Profit works the same way: a 25% markup on cost is only a 20% margin on price, as margin vs markup explained shows.
Trap 4: percent, percentage points and the base
If an interest rate goes from 4% to 5%, it rose by 1 percentage point, but by 1 ÷ 4 = 25% in relative terms. If it goes from 5% back to 4%, it fell 1 point, but by 1 ÷ 5 = 20%. Percentage points are a plain subtraction. Percentage change divides by the starting value, so the direction matters: 100 to 120 is +20%, while 120 to 100 is −20 ÷ 120, about −16.67%.
Two more edges. There is no percentage change from zero, because anything compared with nothing is an infinite multiple. And a small start exaggerates: 2 to 20 is +900% (18 ÷ 2 = 9), which is true and also misleading. Doubling is a +100% change and tripling is +200%. The percentage change calculator shows the sign, the plain change and the change that would reverse it.
Shortcuts for doing it in your head
- 10%: move the decimal point one place left. 10% of 250 is 25, and 1% is 2.5.
- 5% and 15%: 5% is half of 10% (12.5), and 15% is 10% plus 5% (37.5).
- Swap them: X% of Y equals Y% of X. 8% of 50 is awkward, but 50% of 8 is 4.
Use a calculator to check the mental answer, or when the numbers are not round.
Percentages in files and spreadsheets
Size savings are percentage decreases: a 5 MB file that shrinks to 1 MB is (5 − 1) ÷ 5 × 100 = 80% smaller. The guide to compressing a PDF has real before and after sizes to run through the formula. In a spreadsheet, with the old value in A2 and the new one in B2, =(B2-A2)/A2 formatted as a percentage is the percentage change. If your numbers sit in a CSV export, convert it to Excel first so long numbers and leading zeros survive.
Which calculator for which job
- Percentage calculator: all six questions on one page, with increase and decrease sharing one box.
- Percentage increase and percentage decrease: apply a rise or a fall, and see the way back.
- Percentage change: measure how far a number moved.
- Discount calculator: the sale price, the percent off between two prices, the original price, a coupon on top, and tax.
- VAT calculator: add VAT to a price or take it out, at the rate you enter.
Each one writes out its working, so you can check a sum or copy it into a report.
Discount CalculatorEnter a price and a discount, add a coupon, and see the combined saving with the working.Open it