Orpheus

Percentage Increase and Decrease

The change between two numbers, and why it does not reverse.

Inputs
Increase
25%
Absolute change
20
Multiplied by
1.25×
Change going back
20%
Percentage points
20
change = (new − old) ÷ old × 100

The reverse change is a different percentage, because it is measured against a different base.

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Subtract the old value from the new, divide by the old, multiply by 100. Going from 80 to 100 is a 25 percent increase; going back from 100 to 80 is a 20 percent decrease, because the base changed.

How to use Percentage Change Calculator

  1. Enter the original value. The change is always measured against this.
  2. Enter the new value. Lower gives a decrease, higher an increase.
  3. Read both directions. The reverse figure is shown, since it is rarely the same number.

About calculating percentage change

Percentage change is the most misused piece of everyday arithmetic, and nearly all of the misuse comes from forgetting that a percentage is always a percentage of something specific. Change the base and the same absolute movement becomes a different percentage, which is why increases and their reversals never match: a rise from 80 to 100 is 25 percent, and the fall back is 20. At larger movements the asymmetry becomes dramatic. A fifty percent loss requires a hundred percent gain to recover; a ninety percent loss requires nine hundred. This is not a quirk of notation but the whole reason volatility damages compounded returns. The second confusion, between percent and percentage points, is exploited routinely. A rate moving from four percent to five has risen by one percentage point and by twenty-five percent, and a headline may honestly choose whichever sounds better. Interest rates, tax rates, unemployment and polling numbers are all rates, so almost every claim about them is ambiguous unless it says which it means. Finally, successive changes multiply rather than add, which is why a sequence of ten percent gains and ten percent losses drifts steadily downward however many times it repeats.

Frequently asked questions

Why is a 25% increase reversed by a 20% decrease?
Because a percentage is always of something, and the something changed. 25 percent of 80 is 20; 20 percent of 100 is also 20. Same absolute change, different base, therefore different percentage. This is why a stock that falls 50 percent needs to rise 100 to recover.
What is the difference between percent and percentage points?
If a rate goes from 4 percent to 5 percent, that is a rise of one percentage point and of 25 percent. Both are correct and they mean very different things — which is why the distinction is exploited constantly in reporting about interest rates and polling.
How do I calculate percentage change from zero?
You cannot. The formula divides by the original value, and dividing by zero is undefined — there is nothing for the change to be a percentage of. Report it as an absolute change, or as "from zero to n".
Do successive percentage changes add up?
No, they multiply. Up 10 percent then down 10 percent leaves you at 99, not 100, because the decrease applies to the larger number. Sequences of gains and losses always end below what simple addition suggests.

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