What is the percentage change formula?

Percentage change = (new value − original value) ÷ |original value| × 100. The absolute value in the denominator ensures the formula produces a sensible result even when the original value is negative — the sign of the percentage still correctly reflects whether the value increased or decreased.

Why does a 50% drop require more than a 50% gain to recover?

Because percentage change is always calculated against a different base after the value moves. A $100 value that drops 50% becomes $50. Recovering to $100 from $50 requires a 100% gain, not 50%, because the gain is now calculated against the smaller $50 base. This asymmetry is a direct consequence of the denominator changing between the two calculations.

Is percentage change the same as percentage points?

No, and mixing them up is a common error. If a rate moves from 10% to 15%, that's a 5 percentage-point increase, but a 50% percentage increase relative to the original 10% ((15−10)/10 × 100). Percentage points measure an absolute difference between two percentages; percentage change measures a relative change.

Can percentage change be negative?

Yes — a negative result means the new value is lower than the original. The sign is meaningful: it's not just magnitude, it tells you the direction of the change.

What if the original value is zero?

Percentage change is undefined when the original value is zero, because you'd be dividing by zero. Any calculator, including this one, will show an undefined or not-a-number result in that case rather than a misleading percentage.

Everyday maths guide · Reviewed 28 August 2026

Why a 50% loss needs a 100% gain to undo.

Percentage change looks like simple arithmetic, but the shifting denominator behind it produces results that trip people up constantly — from investment losses to price comparisons.

The formula, and why the denominator matters

Percentage change = (new value − original value) ÷ |original value| × 100. The subtraction on top measures the absolute change; dividing by the original value expresses that change relative to where you started. That denominator is the entire reason percentage change behaves asymmetrically: it's always measured against the starting point of that specific calculation, not against some fixed reference.

The recovery asymmetry, worked through

Take $100 that falls 50% to $50 — a straightforward application of the formula: (50 − 100) ÷ 100 × 100 = −50%. To get back to $100 from $50, the formula runs again with a new original value of $50: (100 − 50) ÷ 50 × 100 = +100%. The dollar amount of the move is the same ($50) in both directions, but the percentage isn't, purely because the base of the second calculation is smaller than the base of the first. This is why percentage losses and gains are never simple mirror images of each other.

Percentage change versus percentage points

These are frequently confused. If an interest rate moves from 4% to 6%, that's a 2 percentage-point increase — a straightforward subtraction. Run it through the percentage change formula instead and you get (6 − 4) ÷ 4 × 100 = 50%, a relative percentage increase. Both statements are true and describe the same move, but they answer different questions: percentage points measure the absolute gap, percentage change measures the relative size of that gap against the starting value.

What the percentage change calculator computes

Enter an original value and a new value. The calculator returns the percentage change along with the absolute difference, so you can see both the relative and dollar (or unit) size of the move side by side.

Calculate a percentage change ↗ See how percentage change compounds over time ↗

Frequently asked questions

What is the percentage change formula?: Percentage change = (new value − original value) ÷ |original value| × 100. The absolute value in the denominator ensures the formula produces a sensible result even when the original value is negative — the sign of the percentage still correctly reflects whether the value increased or decreased.
Why does a 50% drop require more than a 50% gain to recover?: Because percentage change is always calculated against a different base after the value moves. A $100 value that drops 50% becomes $50. Recovering to $100 from $50 requires a 100% gain, not 50%, because the gain is now calculated against the smaller $50 base. This asymmetry is a direct consequence of the denominator changing between the two calculations.
Is percentage change the same as percentage points?: No, and mixing them up is a common error. If a rate moves from 10% to 15%, that's a 5 percentage-point increase, but a 50% percentage increase relative to the original 10% ((15−10)/10 × 100). Percentage points measure an absolute difference between two percentages; percentage change measures a relative change.
Can percentage change be negative?: Yes — a negative result means the new value is lower than the original. The sign is meaningful: it's not just magnitude, it tells you the direction of the change.
What if the original value is zero?: Percentage change is undefined when the original value is zero, because you'd be dividing by zero. Any calculator, including this one, will show an undefined or not-a-number result in that case rather than a misleading percentage.

Source note: This guide describes the standard percentage change formula as a general illustration. All calculations happen in your browser — nothing you enter is sent to a server or stored. This is educational information, not financial advice.