How to Calculate Percentage Increase and Decrease

Percentage change answers “how much did this quantity move relative to where it started?” The start is the original value. The later figure is the new value. On Percent Folio the formula is Percentage Change = ((New − Original) / Original) × 100. The percentage change calculator requires a positive original and a nonnegative new value, because this version is built for quantities with a positive baseline.

A related but different question is “what number do I get if I apply a rate to a start?” That is the increase or decrease calculator: multiply by (1 + rate/100) or (1 − rate/100). Both tools appear in this article because people mix them, and because a 20% increase is not undone by a 20% decrease of the later number.

Original and new values

Name the baseline first. If a jar held 80 grams and now holds 100, original is 80 and new is 100. If you swap those labels, you are asking a different question. The calculator’s Swap Values button does that on purpose: it recomputes with the new original, and 100 down to 80 is a 20% decrease, not a 25% decrease.

Percentage increase

When new is larger than original, the change is positive. Example: original 80, new 100.

((100 − 80) / 80) × 100 = (20 / 80) × 100 = 25%

The page labels this INCREASE. Absolute change is 20, which is New − Original. Do not call 20 a “25% amount”; 20 is the gram difference, 25% is the relative change.

Percentage decrease

When new is smaller, the result is negative. Example: original 100, new 80.

((80 − 100) / 100) × 100 = (−20 / 100) × 100 = −20%

The page labels this DECREASE. Original 100 and new 0 is ((0 − 100) / 100) × 100 = −100%, a 100% decrease. That is a complete drop to zero from a positive start, not an undefined ratio.

The percentage change formula

Always divide by the original, not by the new value and not by the average of the two. Dividing by the average is percentage difference, which is symmetric and is documented on a separate calculator. Mixing those names is one of the eight mistakes listed on the homepage.

Why the original value is the denominator

Relative change needs a reference. “25% more than 80” means 0.25 × 80 = 20 more, landing on 100. “20% less than 100” means 0.20 × 100 = 20 less, landing on 80. The same two numbers produce two rates because each sentence chose a different reference. That is not a bug in the formula. It is the meaning of “relative to the start.”

Zero-baseline problems

If the original is 0, you cannot divide. Going from 0 to 10 is an infinite relative change in the ordinary formula; this site does not print Infinity. It says percentage change is undefined when the original value is zero. Going from 0 to 0 is also undefined as a ratio; the change calculator still refuses original zero rather than inventing 0%.

A negative original is refused on this particular tool, with a note that the convention is a positive baseline. Signed accounting problems need a different statement of “original.”

Percentage points versus percent change

A rate that moves from 20% to 25% increased by 5 percentage points. The relative percentage increase is (25 − 20) / 20 × 100 = 25%. Those are not both “a 5% increase.” Percentage points subtract the two percents as numbers. Percent change divides by the starting rate. The change calculator, if you enter 20 and 25 as original and new, reports 25% increase—the relative figure—not 5 percentage points. You still have to know which sentence you wanted.

Why increases and decreases are not always reversible by using the same rate

Start at 100. Increase 20%: 100 × 1.20 = 120. Decrease that 120 by 20%: 120 × 0.80 = 96, not 100. To reverse a 20% increase you divide by 1.20, which is the reverse calculator’s Mode B. The forward decrease of 20% is a different multiplier.

Forward example on the increase tool: 150 increased by 20% is 150 × 1.20 = 180. Decrease 150 by 20% is 150 × 0.80 = 120. Decrease 150 by 100% is 0. A decrease above 100% is rejected on that nonnegative calculator.

Related: how reverse percentages work and change versus difference.

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