Percentage Calculator: How It Works
Percentages cause more everyday arithmetic errors than any other operation, and almost all of them come from the same source: forgetting that a percentage is always a percentage of something, and that the something can change between the two halves of a calculation.
The three calculations
| Question | Formula | Example |
|---|---|---|
| What is X% of Y? | Y × X ÷ 100 | 15% of 240 = 36 |
| X is what % of Y? | X ÷ Y × 100 | 36 of 240 = 15% |
| % change from A to B | (B − A) ÷ A × 100 | 200 → 250 = +25% |
Why an increase and decrease do not cancel
A price rises 20% then falls 20%. It does not return to where it started.
100 → 120 → 96. The 20% rise was of 100; the 20% fall was of 120. Different bases, different amounts.
This is why a stock that drops 50% needs a 100% gain to recover, and why 'we cut costs 30% then they went up 30%' leaves you 9% better off than you started, not level.
| Fall | Rise needed to recover |
|---|---|
| 10% | 11.1% |
| 25% | 33.3% |
| 50% | 100% |
| 80% | 400% |
| 90% | 900% |
Stacked discounts
'30% off, then an extra 20% off' is not 50% off. The second discount applies to the already-reduced price: 100 → 70 → 56, which is 44% off. The shortcut is to multiply the remaining fractions: 0.7 × 0.8 = 0.56.
Usefully, the order does not matter — 0.8 × 0.7 gives the same result. What does matter is whether tax is applied before or after the discounts, which varies by jurisdiction and can change the total.
Percentage points versus percent
This distinction is routinely lost in reporting and it changes the meaning entirely. If an interest rate moves from 4% to 6%:
- It rose by 2 percentage points.
- It rose by 50 percent.
Both are true. A headline saying 'rates up 50%' and one saying 'rates up 2 points' describe the same event and land very differently. When you see a percentage change applied to something already measured in percent, check which is meant.
Reverse percentages
To find the original price from a discounted one, divide rather than adding the percentage back. An item costing 68 after a 15% discount was originally 68 ÷ 0.85 = 80, not 68 × 1.15 = 78.20. The same logic applies to removing tax from an inclusive price: divide by (1 + rate).
Percentages of percentages
Increasing a 5% conversion rate 'by 20%' could mean raising it to 6% (a relative increase) or to 25% (adding 20 points). Almost always the first is meant, but the ambiguity is real and worth resolving explicitly in any report — write '5% → 6%' rather than '+20%' and the question disappears.