The percentage-change formula
For a positive original value:
Percentage change = (new value − original value) ÷ original value × 100%
A positive result is an increase and a negative result is a decrease. The denominator is the original value, not the new value and not the average of the two.
If monthly sales rise from $800 to $1,000, the change is ($1,000 − $800) ÷ $800 = 25%. If they then fall from $1,000 to $800, the change is −$200 ÷ $1,000 = −20%. The dollar movement is the same size, but the reference amount differs.
Work backward from a percentage change
To find a new value after a percentage change, multiply the original by one plus the decimal rate. For a decrease, use a negative rate.
New value = original value × (1 + rate)
To recover the original value, divide the new value by 1 + rate. For example, a price of $120 after a 20% increase started at $120 ÷ 1.20 = $100. Subtracting 20% from $120 would give $96 and does not reverse the original increase.
| Situation | Calculation | Result |
|---|---|---|
| Increase $100 by 20% | $100 × 1.20 | $120 |
| Decrease $100 by 20% | $100 × 0.80 | $80 |
| Original price before a 20% increase to $120 | $120 ÷ 1.20 | $100 |
| Original price before a 20% decrease to $80 | $80 ÷ 0.80 | $100 |
Percentage points describe differences between rates
A conversion rate that rises from 4% to 5% increases by 1 percentage point. Relative to its original rate, it increases by (5 − 4) ÷ 4 = 25%.
Both descriptions are correct but communicate different quantities. A statement that the rate “increased by 1%” is ambiguous if the writer actually means one percentage point.
Use percentage points for the arithmetic difference between percentages. Use percentage change for relative movement, and name the starting value when the change could otherwise seem misleading.
Consecutive changes compound
A 10% increase followed by a 10% decrease changes $100 to $100 × 1.10 × 0.90 = $99. Equal percentage movements do not cancel because the second movement applies to a different base.
For multiple periods, multiply the growth factors. A 5% increase followed by an 8% increase produces (1.05 × 1.08 − 1) × 100 = 13.4% growth, not 13%.
When comparing growth over several years, distinguish total percentage change from an annualized rate. A total gain of 50% over five years is not the same as earning 10% each year. The compound annual rate is (ending ÷ beginning)^(1/years) − 1, when the values and assumptions make that calculation meaningful.
Zero and negative starting values need care
Percentage change from zero is undefined because the formula divides by zero. Going from zero sales to $500 is a $500 increase; it is not a finite percentage increase. Show the absolute change and explain the starting point.
The signed formula can also be confusing with a negative starting value. Moving from a loss of $100 to a loss of $50 is an improvement, but the signed formula gives −50%. Crossing from a loss to a profit is even harder to summarize with one intuitive percentage.
For a straightforward increase-or-decrease interpretation, use a positive original value. For zero or negative bases, report the absolute movement and describe the direction in context rather than silently changing the formula.
Round for communication, not during calculation
Keep full precision in the calculation and round the displayed result to a useful number of decimals. Small differences near a reporting threshold can disappear when the input rates are already rounded.
For comparisons built from counts, calculate from the underlying counts where possible. A displayed rate of 4.0% could represent several different unrounded values, which can materially change the relative percentage movement in a small effect.
Frequently asked questions
Why does a 50% loss require a 100% gain to recover?
After a value falls from 100 to 50, it must gain 50 on a base of 50. That is a 100% increase.
Can percentage change be calculated from zero?
Not as a finite percentage using the standard formula. Report the absolute change and state that the original value was zero.
What is the difference between 1% and 1 percentage point?
One percentage point is an absolute difference between rates. A 1% relative increase multiplies the original rate by 1.01.
Sources & calculation notes
The formulas and worked examples above show the calculation method. All example inputs are illustrative; a mathematical result does not validate the assumptions or replace a project-specific assessment.
Use this guide thoughtfully. Examples illustrate a calculation method, not a guaranteed outcome. The usefulness of any result depends on the definitions, measurements, and assumptions used.