The formula: ((new − old) / old) × 100. The intuition: what fraction of the original value did the change represent? Positive is gain, negative is loss. Most "percentage math" errors in the wild are people getting the denominator wrong.
Worked examples
- $100 → $150: (50 / 100) × 100 = +50%
- $150 → $100: (−50 / 150) × 100 = −33.3%
- $10 → $50: (40 / 10) × 100 = +400%
- $80 → $0: (−80 / 80) × 100 = −100% (total loss)
Why up-then-down doesn't cancel
A 50% gain followed by a 50% loss doesn't get you back to start. $100 → $150 (+50%) → $75 (−50%). You lost money. That's because the same dollar amount is a different percentage of a different base. This is a common mental trap in investing discussions.
When the base is zero
Percentage change is undefined if the original value is zero — you'd divide by zero. In practice, report "infinite" or "new value" instead of a percentage. Going from 0 to anything positive is "gained $X", not "up X%".
Three modes: X% of Y, X is what % of Y, and % change between values. Shareable URL.