Percentage change explained
Percentage change measures how far a final value moved relative to the magnitude of its initial value. This page uses the signed result: a positive percentage is an increase, a negative percentage is a decrease, and zero means no change.
Percentage change formula
Percentage change equals final value minus initial value, divided by the absolute value of the initial value, multiplied by one hundred percent.
Initial is the old or baseline value. Final is the new value. Vertical bars in |Initial| mean absolute value, or distance from zero.
(Final − Initial) ÷ |Initial| × 100%, used when Final > Initial.
(Initial − Final) ÷ |Initial| × 100%, reported as a positive decrease amount when Final < Initial.
How to calculate percentage change
- Subtract the initial value from the final value.
- Divide that numeric change by the magnitude of the initial value.
- Multiply by 100 to express the result as a percentage.
- Interpret a positive sign as an increase and a negative sign as a decrease.
For example, a price rising from £40 to £46 changes by £6; £6 ÷ £40 × 100 = 15%. The price increased by 15%. The same method can describe salary from £30,000 to £31,500 (5% increase), revenue from 200,000 to 180,000 (10% decrease), population from 8,000 to 8,400 (5% increase), a test score from 72 to 81 (12.5% increase), or a measurement from 2.4 cm to 2.1 cm (12.5% decrease).
Percentage increase versus percentage decrease
The signed general formula handles both. If the final value is larger, the numerator is positive and the answer is an increase. If the final value is smaller, the numerator is negative and the answer is a decrease. When writing a decrease without a sign, use its magnitude: −20% means a 20% decrease.
Changes are directional, not reversible. A fall from 100 to 50 is a 50% decrease, but recovering from 50 to 100 requires a 100% increase because the second calculation uses the smaller value, 50, as its baseline.
Percentage change versus percentage difference
Percentage change is directional and uses the initial value as the denominator. Use it when one value clearly comes before the other. Percentage difference treats two values symmetrically and usually divides their absolute difference by their average. Use percentage difference when neither value is the baseline, such as comparing two independent measurements.
Percentage change versus percentage-point change
Percentage points compare two percentages by subtraction. If a test pass rate rises from 40% to 50%, it rises by 10 percentage points, but the relative percentage increase is (50 − 40) ÷ 40 × 100 = 25%. These statements describe different quantities.
Why the initial value is the denominator
The question is “how large was the movement compared with where it started?” The initial value supplies that reference. A change of 10 is large relative to an initial value of 20 (50%) but small relative to an initial value of 1,000 (1%). Swapping the values changes the reference and therefore changes the answer.
Worked examples
Percentage increase: 120 to 150
- Substitute: (150 − 120) ÷ |120| × 100.
- Subtract: 150 − 120 = +30.
- Divide by the baseline magnitude: 30 ÷ 120 = 0.25.
- Multiply by 100: 0.25 × 100 = +25%.
Interpretation: 150 is 25% higher than 120.
Percentage decrease: 80 to 50
- Substitute: (50 − 80) ÷ |80| × 100.
- Subtract: 50 − 80 = −30.
- Divide: −30 ÷ 80 = −0.375.
- Multiply: −0.375 × 100 = −37.5%.
Interpretation: 50 is 37.5% lower than 80.
Negative to negative: −40 to −20
- Substitute: (−20 − (−40)) ÷ |−40| × 100.
- Subtract: −20 + 40 = +20.
- Divide: 20 ÷ 40 = 0.5.
- Multiply: 0.5 × 100 = +50%.
Interpretation: the number increased by 20, moving closer to zero; under the absolute-baseline convention this is a 50% increase.
Crossing negative to positive: −20 to 10
- Substitute: (10 − (−20)) ÷ |−20| × 100.
- Subtract: 10 + 20 = +30.
- Divide: 30 ÷ 20 = 1.5.
- Multiply: 1.5 × 100 = +150%.
Interpretation: the value increased by 30 and crossed zero, a 150% increase relative to the starting magnitude.
No change: 75 to 75
- Substitute: (75 − 75) ÷ |75| × 100.
- Subtract: 75 − 75 = 0.
- Divide: 0 ÷ 75 = 0.
- Multiply: 0 × 100 = 0%.
Interpretation: the values are equal, so there is no change.
Zero initial value: 0 to 10
- Substitute: (10 − 0) ÷ |0| × 100.
- Subtract: 10 − 0 = +10.
- The next step would require 10 ÷ 0.
- Division by zero is undefined, so no percentage can be calculated.
Interpretation: the numeric change is +10, but the percentage change is undefined. It should not be labeled infinite.
Edge cases and interpretation
- Initial value of zero: percentage change is undefined because the formula requires division by zero. Report the numeric difference instead.
- Negative initial values: this calculator divides by |Initial|, so the result sign follows the numeric direction. Moving from −40 to −20 is positive; moving from −20 to −40 is negative. Some fields use different conventions, so state the convention when reporting.
- Changes beyond 100%: these can be valid. From 50 to −10 is −120% because the value fell by more than the entire starting magnitude.
- Rounding: calculate with unrounded numbers and round only the displayed answer.
Editorial review: Formula, reverse-solving branches, examples, zero-baseline handling, and accessibility behavior were checked by the Starlight Tools Editorial Team using independent substitution and boundary tests. Last reviewed: July 14, 2026. Last updated: July 14, 2026. Calculations run locally in your browser; entered values are not uploaded.