How percentage change works
A percentage change tells you how much a value grew or shrank compared to where it started. You take the difference between the new value and the old value and express it as a share of the old value. If a monthly bill goes from 80 to 100, it rose by 20, and 20 is a quarter of 80, so the bill went up by 25%.
The sign tells you the direction. A positive result is a percent increase, a negative result is a percent decrease. The calculator shows the size of the change in large type and labels it clearly as an increase or a decrease, so you never have to read a minus sign to know which way it went.
Next to the percentage you get three more numbers. The difference is the plain gap, New minus Old. The multiplier is New divided by Old: a 25% increase is a multiplier of 1.25, a 20% decrease is 0.8. The change back to old is the percentage that would undo the change, which is rarely the same number with the opposite sign.
Why a rise and a fall of the same percent do not cancel out
Every percentage is measured against its own starting point. After a 25% increase, the starting point for the way back is the higher value, so a smaller percentage is enough to return. 80 plus 25% is 100, and 100 minus 20% is 80 again. The table shows the decrease that undoes a given increase.
| Increase | Decrease that undoes it | Example |
|---|---|---|
| 5% | 4.76% | 100 → 105 → 100 |
| 10% | 9.09% | 100 → 110 → 100 |
| 20% | 16.67% | 100 → 120 → 100 |
| 25% | 20% | 100 → 125 → 100 |
| 50% | 33.33% | 100 → 150 → 100 |
| 100% | 50% | 100 → 200 → 100 |
| 200% | 66.67% | 100 → 300 → 100 |
The same works the other way. A value that falls by 50% has to double, a 100% increase, to get back to where it was. Keep this in mind for prices, investments and anything else that moves up and down: a loss always needs a larger percentage gain to recover.
Finding the new value or the old value
Sometimes you know the percentage and need one of the values. To find the new value, multiply the old value by 1 plus the percentage as a decimal for an increase, or 1 minus it for a decrease. 80 with a 25% increase is 80 × 1.25 = 100. To find the old value, divide instead: if something costs 120 after a 20% increase, it cost 120 ÷ 1.20 = 100 before.
A common mistake is to work backward by applying the same percentage in the other direction. Taking 20% off 120 gives 96, not 100. The New value and Old value modes do the division for you and show the formula with your numbers.
Percent change vs. percentage points vs. percentage difference
Three ideas share similar names, and mixing them up leads to confusing headlines. When the two numbers you compare are already percentages, like an interest rate, a tax rate or a share of votes, the plain gap between them is measured in percentage points. A rate going from 4% to 5% is up 1 percentage point, and at the same time up 25% as a percentage change. Both statements are true, but the second sounds much larger, so say which one you mean.
| Measure | Formula | 4 and 5 | Use it for |
|---|---|---|---|
| Percent change | (B − A) ÷ |A| × 100 | 25% | A before and after, with a direction |
| Percentage points | B − A | 1 point | The gap between two percentages |
| Percentage difference | |A − B| ÷ ((A + B) ÷ 2) × 100 | 22.22% | Two values with no before and after |
Percentage difference compares two values on equal terms, like the price of the same item in two stores. It divides the gap by the average of both values, so the answer is the same whichever value you list first. The calculator shows it in Percent change mode whenever both values are positive.
Changes from zero and from negative numbers
A percentage change from 0 cannot be calculated. The formula divides by the old value, and nothing can be divided by 0. Going from 0 to 5 is not a 500% or an infinite increase; there simply is no percentage for it. In that case the calculator shows the plain difference and says why the percentage is missing. If you need a number for a report, write the change as an amount, like "up 5 from 0".
Negative old values need a decision. If a business goes from a loss of 50 to a loss of 25, the textbook formula (New − Old) ÷ Old gives −50%, which reads as a decrease even though the result improved. This calculator divides by the size of the old value, |Old|, instead. That way a move up is always an increase and a move down is always a decrease: from −50 to −25 is a 50% increase. When the old value is negative, the result says so, because readers can still take such a percentage the wrong way. When the sign flips, for example from −20 to 30, consider reporting the amount of the change instead of a percentage.
This page or the percentage calculator?
This page is built around one question: how much did something change, and what does that change mean in both directions. For other everyday percentage questions, like what 15% of 80 is or what share one number is of another, use the percentage calculator. For growth over several years, the CAGR calculator turns a total change into a yearly rate.
How to use the percentage change calculator
- 1Pick what you want to find: the percent change between two values, a new value, an old value, or the points between two percentages.
- 2Type your numbers. Decimals with a point or a comma both work, and the result updates as you type.
- 3Read the result, the difference, the multiplier and the change that would undo it. The formula line shows the math with your numbers.
- 4Copy the result as text, or copy a link that opens the calculator with the same numbers.
Frequently asked questions
How do I calculate the percentage change between two numbers?
Subtract the old value from the new value, divide the result by the old value and multiply by 100. Going from 80 to 100 is (100 − 80) ÷ 80 × 100 = 25%. A positive result is an increase, a negative result is a decrease.
Why does a 50% increase need a 33.33% decrease to get back?
Each percentage is taken from a different starting value. 100 plus 50% is 150. To get back to 100 you remove 50, and 50 is one third of 150, so the decrease is 33.33%. The bigger the increase, the wider the gap between the two percentages.
What is the difference between percentage change and percentage difference?
Percentage change has a direction: it compares a new value with an old one and divides by the old value. Percentage difference compares two values that have no before and after, like two prices in two stores, and divides the gap by their average. From 80 to 100 is a 25% change, but the percentage difference between 80 and 100 is 22.22%.
Can you calculate a percentage change from zero?
No. The formula divides by the old value, and dividing by 0 has no answer. Any rise from 0 would be infinitely large compared to 0. The calculator then shows the plain difference and says that the percentage is not defined.
How does the calculator handle a negative old value?
It divides by the size of the old value, ignoring its sign, so moving up is always an increase. From −50 to −25 is (−25 − (−50)) ÷ 50 × 100 = a 50% increase. Dividing by −50 instead would call that rise a 50% decrease, which is easy to misread.
What is a percentage point?
A percentage point is the plain difference between two percentages. A rate that moves from 4% to 5% rises by 1 percentage point. Measured as a percentage change, the same rate grew by 25%, because 1 is a quarter of 4. Use the Points mode to see both numbers.
How do I find the original value before a percentage increase?
Divide the new value by 1 plus the percentage as a decimal. If a price is 120 after a 20% increase, it was 120 ÷ 1.20 = 100 before. Taking 20% off 120 gives 96, not 100, because the 20% was based on the old value.