Percentage Change, Increase, Decrease and Difference

The four percent formulas people confuse, when the base is zero, and why a 50% drop needs a 100% rise to recover.

Updated 2026-10-20

Four different questions get called “the percentage”. They have different formulas and different answers for the same numbers.

Percent of

What is 15% of 200? 0.15 × 200 = 30.

What percent

30 is what percent of 200? 30 ÷ 200 × 100 = 15%.

Percent change

From an old value to a new value:

change % = (new − old) ÷ |old| × 100

From 50 to 75: (75 − 50) ÷ 50 = 50% increase. From 75 to 50: (50 − 75) ÷ 75 = −33.3%, a decrease. The base is always the starting value, which is why the two directions give different magnitudes.

Percent increase and percent decrease are the same formula with the sign read off: a positive change is an increase, a negative one a decrease.

When old is zero

(10 − 0) ÷ 0 is undefined. There is no percentage increase from zero; our calculator reports “not defined” rather than infinity. Report the absolute change instead, or use the percentage of the new value.

Percent difference

When neither number is the “before”, use the symmetric form:

difference % = |a − b| ÷ ((a + b) ÷ 2) × 100

10 and 20: 10 ÷ 15 = 66.7%. This is the right figure when comparing two products’ prices or two measurements of the same thing, where neither is the reference.

Recovering from a loss

A 50% fall from 100 leaves 50. Getting back to 100 is a 100% rise, because the base is now 50. In general, recovering from a fall of p% needs a rise of p ÷ (100 − p) × 100 percent: a 20% fall needs 25%, a 90% fall needs 900%.

Percentage points

Going from 10% to 15% is a 5 percentage point increase but a 50% increase. Say which one you mean.

Applying a change

New = old × (1 + change ÷ 100). A 20% increase on 80 is 80 × 1.2 = 96. Reversing it: old = new ÷ (1 + change ÷ 100), so 96 ÷ 1.2 = 80. Subtracting 20% of 96 would give 76.8, which is wrong.

In Excel

Percent change: =(B1-A1)/ABS(A1), format as percent. Guard zero: =IF(A1=0,"n/a",(B1-A1)/ABS(A1)).

Worked examples

Rent. Rent goes from $1,450 to $1,595: (1,595 − 1,450) ÷ 1,450 = 10% increase.

Stock. A share falls from $80 to $60, then rises back to $80. The fall is −25%; the recovery is +33.3% (20 ÷ 60). Reporting the two as “down 25%, up 25%” would leave you at $75.

Population. Town A grows from 2,000 to 2,400 (20%); Town B from 50,000 to 55,000 (10%). B added more people, A grew faster. Percent change compares rates, not amounts.

Survey results. Approval moves from 40% to 44%. That is a 4 percentage point rise and a 10% increase. Headlines pick whichever sounds bigger.

Comparing two quotes. Plumber A quotes $380, plumber B $420. Percent difference: 40 ÷ 400 = 10%. Percent change from A to B: 10.5%; from B to A: −9.5%. Use percent difference when neither quote is the baseline.

Compound changes

Two successive changes multiply, not add. Up 10% then down 10% is 1.1 × 0.9 = 0.99, a net 1% loss. Up 50% then up 50% is 2.25x, a 125% increase. To find the average annual change over n years, take the nth root of the total ratio: a 60% rise over 5 years is 1.6^(1/5) = 1.0986, or 9.86% a year, not 12%.

Negative starting values

(New − old) ÷ |old| with the absolute value keeps the sign meaningful. From −50 to 50 is +200%; from −50 to −25 is +50% (an improvement). Some calculators divide by the signed old value and report nonsense; ours uses the absolute value.

Frequently asked

What is the percent change from 0? Not defined. Report the absolute change.

Is a 200% increase the same as tripling? Yes. 100% increase doubles, 200% triples.

How do I undo a 25% increase? Divide by 1.25 (equivalently, reduce by 20%), not subtract 25%.

What is the difference between percent and percentile? A percentile is a rank: the 90th percentile is the value below which 90% of observations fall. It has nothing to do with percent change.