Percentage Increase Calculator
Result
Increase or decrease (%)
- Amount of change (new − original)
- 25.0000
- Multiplier (new ÷ original)
- 1.5000
A percentage increase calculator compares two values and reports how far the second has moved from the first, measured as a percentage of the first. Going from 50 to 75 is a percent increase of 50%, and going from 100 to 90 is a percent increase of -10%: it is the same formula both times, with the sign carrying the direction. The base is always the original value, so the percentage is measured against where the change started and not where it ended. That matters more than it sounds, because swapping the two values does not simply flip the sign: 50 to 75 is a rise of 50%, while 75 to 50 is a fall of 33.3333%. A percentage change that reports only a magnitude would hide that, which is why the direction is part of the answer here rather than something the reader has to supply. The panel reports the one change three ways. The percentage is the headline; the amount of change is the same movement in the original units, which is what a budget or a measurement actually moves by; and the multiplier is the new value divided by the original, so 1.5 means the new value is one and a half times the old one. The multiplier is the reading to use when the movement is going to be applied again, since applying a rise of 50% and applying a multiplier of 1.5 are not the same operation when they happen twice. Two things about the base are worth knowing before reading the panel, and neither one shows up as a broken page. A base of zero has no percentage change at all: nothing is defined when the divisor is zero, so the page reports an error instead of printing a figure. And a negative base is accepted, while reading oddly: the arithmetic is right, but a value that has moved upwards comes back with a minus sign in front of it, because the formula divides by a base that sits below zero.
Common increases and decreases
| Original value | New value | Percentage increase (%) |
|---|---|---|
| 100 | 50 | -50 |
| 100 | 80 | -20 |
| 100 | 90 | -10 |
| 100 | 100 | 0 |
| 100 | 110 | 10 |
| 100 | 125 | 25 |
| 100 | 150 | 50 |
| 100 | 200 | 100 |
| 100 | 300 | 200 |
| 20 | 25 | 25 |
| 50 | 100 | 100 |
| 200 | 100 | -50 |
The first nine rows all start from a hundred, which makes the percentage readable straight off the end value: an increase from 100 is the new value minus a hundred. Read down them and the ladder runs from -50% (halved) through 0 (unchanged) to 200% (tripled). The last three rows start somewhere else and exist to show that the base is what the percentage is measured against: 20 to 25 and 100 to 125 are both an increase of 25%, while 200 down to 100 is a decrease of 50% and 50 up to 100 is an increase of 100%. The same movement of 50 is worth a very different percentage depending on where it started, and no row here is a special case of the formula.
Formula
Percentage increase = (new − original) ÷ original × 100 (50 → 75) = 25 ÷ 50 × 100 = 50%
- Original value
- The value before the change. It is the base every reading on the panel is measured against, and it is the one input that cannot be zero.
- New value
- The value after the change. It may be smaller than the original, which is what makes the answer a decrease with a minus sign rather than an error.
- Amount of change
- The new value minus the original, in the same units as the two inputs: 25 for the default figures. It is the absolute half of the comparison, and the percentage is its size relative to the original.
- Multiplier
- The new value divided by the original: 1.5 for the default figures, meaning the new value is one and a half times the old one. Below 1 means the value shrank; exactly 1 means nothing moved.
- Percentage increase
- The amount of change divided by the original, then multiplied by a hundred. Positive when the value rose, negative when it fell, and zero when it did not move at all.
The increase from one value to another is what gets reported here, and the report is useful in three places. The first is a price, a salary or a bill that has changed, where the question is how big the change was relative to where it started: a rise from 50 to 75 is a 50% increase, and that figure is comparable across things of completely different sizes in a way the raw amount is not. The second is checking a claim: a listing that says something grew by half should come back as 50% here, and if it comes back as 33.3333% instead, the two numbers were put in the wrong boxes. The third is planning a further move, where the multiplier is the more useful of the three readings, because applying a 50% increase twice is not the same as applying a multiplier of 1.5 twice, and only one of those is what most people mean. A percentage change is also the natural way to compare a figure against a target or a previous period, which is the same calculation with the earlier figure as the original value.
Worked examples
50 to 75
- Amount of change: 75 − 50 = 25
- Divide by the original: 25 ÷ 50 = 0.5
- Multiply by a hundred: 0.5 × 100 = 50%
- Multiplier: 75 ÷ 50 = 1.5
The default case. All three readings describe one movement: a rise of half, an amount of 25, and a value that is now one and a half times the old one.
100 down to 90
- Amount of change: 90 − 100 = -10
- Divide by the original: -10 ÷ 100 = -0.1
- Multiply by a hundred: -0.1 × 100 = -10%
- Multiplier: 90 ÷ 100 = 0.9
A fall is reported as a negative percentage rather than as a separate kind of answer, so one reading covers both directions. The multiplier stays positive and drops below 1.
3 to 4
- Amount of change: 4 − 3 = 1
- Divide by the original: 1 ÷ 3 = 0.333333...
- Multiply by a hundred: 33.3333% to four decimal places
A third of an increase never terminates, so this is where the four decimal places on the panel are visible. The result is rounded for display and the exact value is the repeating decimal.
A negative base, -50 to -25
- Amount of change: -25 − (-50) = 25
- Divide by the original: 25 ÷ (-50) = -0.5
- Multiply by a hundred: -50%
- Multiplier: -25 ÷ -50 = 0.5
The value has risen by 25 and the percentage is negative, because the base it is measured against is below zero. The arithmetic is right and the reading is the one to be careful with: a negative base is the case where a percentage change stops matching the instinct it was built on.
Limitations
The base is the original value and never the new one. That is what makes a rise and a fall asymmetric: 50 to 75 is an increase of 50%, while 75 to 50 is a decrease of 33.3333% and not -50%. A change and its reverse therefore do not cancel, and a quantity that rises by half and then falls by half does not end up where it started. A base of zero has no percentage change at all, and the page reports an error rather than a figure: dividing by zero is undefined, not infinite, and saying that something grew from nothing by some percentage would be answering a question that has no answer. A negative base is accepted and does have an answer, but the sign of that answer describes the division rather than the movement, so a value that rose from -50 to -25 comes back as -50%. The panel prints four decimal places, so a percentage that does not terminate is rounded there: a rise from 3 to 4 displays as 33.3333% and the exact figure is a repeating decimal that only a fraction can write down. The three readings share that limit, and the multiplier is rounded in the same way, so a multiplier and a percentage shown side by side can disagree in the last place. This page compares two plain numbers. It has no idea whether they are percentages already, and a move from 2% to 3% is a rise of one percentage point as well as a rise of 50%: the panel reports the second, which is the relative one, and the first is a different question about the same pair of figures.
Frequently asked questions
- How do I work out a percentage increase?
- Subtract the original value from the new one, divide that difference by the original value, and multiply by a hundred. Going from 50 to 75 is 25 ÷ 50 × 100, which is 50%.
- Why is the original value used as the base rather than the new one?
- Because the movement is being measured against where it started. Dividing by the new value instead would answer a different question about the same pair of figures: 50 to 75 would come back as 33.3333%, which is what fraction of the new value the change makes up, not how much the value grew.
- What happens if the value goes down?
- The answer is negative: 100 to 90 is a percentage increase of -10%. The direction is carried by the sign rather than by a separate reading, and the multiplier stays positive but drops below 1, here to 0.9.
- Can a percentage increase be more than 100%?
- Yes, and there is no cap on it. An increase of 100% is a doubling, so going from 100 to 300 is an increase of 200% with a multiplier of 3. Anything above a hundred percent simply means the value more than doubled.
- What does the multiplier column mean?
- It is the new value divided by the original. A multiplier of 1.5 means the new value is one and a half times the old one, and it is the reading to apply when the same movement happens again, since two successive increases of 50% are not two applications of 1.5.
- Why does the page show an error when the original value is zero?
- Because the percentage change from zero is undefined: the formula divides by the original value, and dividing by zero has no result. Nothing grew from zero by any finite percentage, so there is no figure to print.
- What if the original value is negative?
- It is accepted and the arithmetic is correct, but the sign reads backwards: going from -50 to -25 is a value that rose, and the panel reports -50%. The reason is that the formula divides by a base below zero, so the direction of the division and the direction of the movement point opposite ways.
References
- Percentage Change — the percentage change from a nonzero initial value to a final value, where a positive result is a percentage increase and a negative result is a percentage decrease; the entry states that the change is undefined for a zero initial value and that it is not symmetric, which are the two properties this page is built around — Wolfram MathWorld (United States)
- Percent — a ratio or fraction converted to a percentage by multiplying by a hundred, illustrated with an investment that grows: the new value is a percentage of the old one and the same movement is also stated as the amount it grew by — Wolfram MathWorld (United States)
- Relative Error — the relative form of a difference divides by the reference value rather than by the measured one, and the percentage form is a hundred times that relative figure, which is the shape both of this page's headline reading and of its multiplier — Wolfram MathWorld (United States)