Percentage Change from A to B

Percentage change answers a before-and-after question: compared with where we started, how large is the move? It is not “what percent is A of B,” and it is not the subtraction of two rates that are already percents. Those are neighboring tools. Mixing them up is how a 25% increase and a 10 percentage-point increase get reported as if they were synonyms.

The formula

Percent change = (new − old) / |old| × 100

From 80 to 100. (20 / 80) × 100 = +25%.

From 100 to 80. (−20 / 100) × 100 = −20%.

From 250 to 250. 0%. Nothing happened, and the formula agrees.

From 40 to 10. (−30 / 40) × 100 = −75%. A quarter of the start remains. People sometimes say “down by a factor of four” and sometimes “down 75%.” Both can be right: 10 is one fourth of 40, and the decline is three fourths of 40. Say which comparison you mean if the audience is tired.

The absolute value around the old number matters when the old number is negative, which is rare for prices and common for profit. Going from a loss of 20 to a loss of 10 is an improvement. The signed story gets political quickly (“percent change in profit”). For ordinary positive quantities — prices, headcount, kilograms — |old| is just old, and the sign of the result matches the direction of the move.

A table of everyday moves

Doubling is +100%, not +200%. Tripling is +200%, because you added twice the original. “Percent of” and “percent change” split on this point: 200% of 2 is 4, while a 200% increase of 2 is 6. If a sentence is ambiguous, force it to pick “of” or “increase.”

What the formula is not

It is not the difference of two percentages. Unemployment moving from 5% to 7% is 2 percentage points. The relative change in that rate is (7 − 5) / 5 = 40%. Both numbers are computable. Only the second one is “percentage change” in the sense of this page. The vocabulary is sorted in percentage increase versus percentage points.

It is not a speed. “Up 10%” does not say per year unless you say per year. A 10% rise over five years is a different path from 10% a year for five years (that second path compounds to about 61%). If time is part of the claim, put time in the sentence, then decide whether you need this formula once or the compound interest formula repeatedly.

It is undefined when the start is zero. From 0 to 15 there is no percent change to report. Report the absolute change. Dashboards that show thousands of percent because last month was almost zero are dividing by a crumb. Read the counts.

Successive changes

A price goes 100 → 120 → 90.

• First leg: +20%
• Second leg: (90 − 120) / 120 = −25%
• Overall: (90 − 100) / 100 = −10%

You cannot add +20 and −25 to get the overall −5. The factors are 1.20 × 0.75 = 0.90, which is −10%, matching the overall formula. When you only have the leg percents and not the endpoints, multiply the factors. When you have the endpoints, ignore the legs and use the endpoints. Doing both and averaging them is not a method.

When to use the percentage change calculator

Use the percentage change calculator when both numbers are measurements of the same thing at two times or in two versions: a fare, a weight, a follower count, a bill. Put the earlier one in the “old” slot even if it is larger.

Use the increase or decrease calculators instead when you are given a rate and one value and you want the other value. Use percentage points, said out loud, when both inputs are already percents and you only subtracted.

Worked comparisons that are not changes

A class goes from 28 students to 35. That is a change: (35 − 28) / 28 ≈ 25%. “35 is what percent of 28” is 125%, which is the new size as a share of the old size, equal to 100% plus the 25% increase. Both numbers describe the same pair. Reporting 125% as “attendance rose 125%” adds an extra hundred points of excitement the formula did not earn. If the sentence contains “rose,” “fell,” “up,” or “down,” you want the change, and the answer should be able to go negative. If the sentence contains “what percent is,” you want the ratio, and it is usually positive.

A price index level of 110 versus 100 is a 10% change and also “110% of the old index.” News copy that says the index “is at 110%” often means the level, not the change. Ask whether 100 is still in the number.

Negative bases, briefly

Profit goes from −€20 to +€10. The gap is +€30. Dividing by the absolute value of the old profit gives 30/20 = 150%, and the sign of the gap says things improved. Some analysts refuse percent change across a sign flip because “percent of a loss” is a story readers mishandle. If you publish it, publish the euro gap beside it. The calculator will emit a number whenever the start is not zero. You decide whether the number deserves a sentence.

For prices across years, a raw percent change still mixes inflation into the story. If you need “more expensive in today’s money,” pair this formula with a price index, as sketched in inflation and percent change. Use the percentage change calculator once the two endpoints are the same kind of unit. Do not feed it a rate and a single price; that is the increase tool. If the two numbers are already rates — 5% and 7% — decide whether you wanted this relative change or a subtraction in percentage points before you copy the result into a sentence someone will act on.