What Is a Percentage, Really?
A percentage is simply a fraction expressed as parts per hundred. The word comes from the Latin per centum — "by the hundred." When you see 25%, it means 25 out of every 100, which is the same as the fraction 1/4, or the decimal 0.25.
This might sound elementary, but understanding that equivalence is the key to all percentage arithmetic. Every formula below is just a rearrangement of the same core relationship:
From this single equation you can derive every other percentage formula by isolating a different variable. Let's go through each one.
Formula 1 — What Is X% of Y?
This is the bread-and-butter percentage question. You know the percentage and the whole; you want the part.
Formula 2 — X is What Percent of Y?
Here you know both numbers but want to express their relationship as a percentage.
Formula 3 — Percentage Change
Percentage change tells you how much a value has grown or shrunk relative to where it started. It is arguably the most important percentage formula in business and finance.
Formula 4 — Reverse Percentage
A reverse percentage works backwards: you know a part and the percentage it represents, and you want to find the original whole. This is invaluable for finding original prices before a discount was applied.
Note the common mistake: people subtract 30% from $63 and guess $44.10, which is wrong because $63 is already the 70% price, not the 100% price.
Formula 5 — Add or Subtract a Percentage
When you need to apply a percentage directly to a number — adding VAT, a markup, a commission, or subtracting a discount — use these two formulas:
Bonus — Compound Percentage Change
When a percentage change is applied repeatedly — annual interest, year-over-year growth, recurring discounts — the changes compound, meaning each period's change is applied to the new value, not the original.
This is why compound interest is so powerful — and why compound inflation is so damaging. A 7% annual return doubles your money in roughly 10 years (the Rule of 72: divide 72 by the interest rate to estimate the doubling time).
Quick Reference Cheat Sheet
| Question | Formula | Example |
|---|---|---|
| What is X% of Y? | (X ÷ 100) × Y | 20% of 150 = 30 |
| X is what % of Y? | (X ÷ Y) × 100 | 30 ÷ 150 × 100 = 20% |
| % change from A to B? | ((B−A) ÷ |A|) × 100 | 80→100 = +25% |
| X is Y% — find whole | X ÷ (Y ÷ 100) | 30 ÷ 0.20 = 150 |
| Y increased by X% | Y × (1 + X/100) | 200 × 1.15 = 230 |
| Y decreased by X% | Y × (1 − X/100) | 200 × 0.85 = 170 |
| Compound growth (n periods) | Start × (1 + r/100)^n | 1000 × 1.07^10 = 1967 |
3 Percentage Mistakes People Make Every Day
- Adding percentages directly. If a price rises 20% then falls 20%, people assume you're back to the start. You're not — you end up at 96%. (100 × 1.2 × 0.8 = 96.) Percentage changes are multiplicative, not additive.
- Confusing percentage points and percent change. A tax rate rising from 20% to 25% is 5 percentage points, but a 25% relative increase. The distinction matters enormously in news and financial reporting.
- Reverse percentage errors. If something is $70 after a 30% discount, the original was NOT $70 + 30% = $91. The correct calculation is $70 ÷ 0.70 = $100. The discount was taken off the original, not added back to the discounted price.
Stop Guessing, Start Calculating
Percentages are everywhere — in every price tag, every business report, every news headline about economic growth. The good news is that all percentage arithmetic comes from a single formula rearranged five different ways. Once you recognise which type of problem you're facing, the calculation is straightforward.
Keep this guide bookmarked as a reference, and use the Percentage Calculator above whenever you need a quick answer without the mental gymnastics.