Percentage Change Calculator

change = new − old percentage change = (new − old) ÷ old × 100 multiplier = new ÷ old

Old value

The value you are starting from: 100 in the default case. It is the denominator of the whole calculation, which is why it cannot be zero — nothing is a percentage of zero. It may be negative; that is legal but reads oddly, and the page has a section on it below.

New value

The value you have ended up with: 125 in the default case. It may be smaller than the old value, in which case the change is negative, and it may be zero, which is a complete fall rather than an error.

Change

The plain difference, new minus old: 50 for 100 to 150. It carries the sign, so it says both how much and which way. It is the numerator of the percentage and it is shown on its own because a percentage without its absolute size is hard to act on — 25% of a small quantity and 25% of a large one are the same reading.

Percentage change

The change divided by the old value, times a hundred: +50% for 100 to 150 and -33.3333% for 150 to 100. Positive means the value rose, negative means it fell, and zero means it did not move. It is rounded to four decimal places, which is a display width; for changes that do not divide evenly the fourth place is rounded.

Multiplier

The new value divided by the old: 1.5 for 100 to 150, 1 for no change, 0.5 for a halving. It says what to multiply by rather than what to add, which is the reading that compounds correctly — applying two multipliers in sequence gives the right overall change, while adding two percentages does not.

The first use is a comparison where the direction is not known in advance: a metric that might have gone up or down, where you want one reading that covers both. The second is checking a claim such as a price or a traffic figure that has changed by a stated percentage — the multiplier line is often the quickest confirmation, since 1.5 either matches the numbers or does not. The third is a series of changes, where the multiplier is the reading to chain and the percentage is the reading to quote. The fourth is a fall to nothing, where the percentage is exactly -100% and the multiplier is 0, and both are useful for different reasons. The fifth is a negative starting point, such as a loss turning into a smaller loss, where the sign of the answer needs care and the page spells out why it can look wrong.

1. 100 → 150

   1. Find the change: 150 − 100 = 50
   2. Divide it by the old value: 50 ÷ 100 = 0.5
   3. Multiply by 100: 0.5 × 100 = 50, so the change is +50%
   4. The multiplier is 150 ÷ 100 = 1.5, which says the same thing: the value is half again as large

   The rise of the pair, and the one that gets put next to its own reversal below. The denominator is the old value, 100, which is why the answer is a round 50%. Everything here agrees with itself: the amount is 50, the percentage is 50% of 100, and the multiplier is 1.5. Use this row as the control when checking a fall, because the two directions are the pair that people expect to match and do not.

2. 150 → 100

   1. Find the change: 100 − 150 = -50
   2. Divide it by the old value, which is now 150: -50 ÷ 150 = -0.333333…
   3. Multiply by 100: -33.3333% after rounding to four places
   4. The multiplier is 100 ÷ 150 = 0.6667

   The same trip in reverse, and the reading that surprises people: going back down is -33.3333%, not -50%. The absolute amount moved is the same 50, and the multiplier is 0.6667 rather than 1.5 — but the denominator changed from 100 to 150, so the percentage cannot be the same. Put this example next to the one above: 100 to 150 and 150 to 100 are the same two numbers, and the percentages are not mirror images. This is why a rise and a fall of the same percentage do not return a value to where it started.

3. 100 → 125

   1. Find the change: 125 − 100 = 25
   2. Divide by the old value: 25 ÷ 100 = 0.25
   3. Multiply by 100: +25%
   4. The multiplier is 125 ÷ 100 = 1.25

   The default pair on the page, chosen because all three readings are plainly different from one another: the amount is 25, the percentage is 25%, and the multiplier is 1.25. Reading them as if they were interchangeable is the most common mistake with this calculation — a 25% rise is not an increase of 25 of anything in particular, it is a quarter of whatever the old value was.

4. 100 → 0

   1. Find the change: 0 − 100 = -100
   2. Divide by the old value: -100 ÷ 100 = -1
   3. Multiply by 100: -100%
   4. The multiplier is 0 ÷ 100 = 0

   A fall all the way to nothing, which the page accepts: the old value is what the division is measured against, and it is 100, so the calculation is perfectly defined. This is the one case where the percentage has a floor at -100% — nothing can fall by more than everything. The multiplier of 0 is the reading that compounds properly, since multiplying by 0 gives 0 whatever comes next.

5. -50 → -25

   1. Find the change: -25 − (-50) = 25, so the value has moved up
   2. Divide by the old value: 25 ÷ (-50) = -0.5
   3. Multiply by 100: -50% — a fall, according to the formula, even though the number got larger
   4. The multiplier is -25 ÷ -50 = 0.5

   The case that looks like a bug and is not. The value moved from -50 to -25, which is up, and the amount of change is +25, which is positive; the percentage is -50% because the numerator is positive and the denominator is negative. Reading a percentage change on a quantity that crosses zero is genuinely awkward, and the honest thing is to show the arithmetic rather than pretend the answer is intuitive. The multiplier points the same way and is equally odd: it is below 1 for a value that got larger. Both are correct applications of the definitions above, and both are reasons to read the amount of change as well as the percentage.

The old value cannot be zero. A percentage of zero has no value — going from nothing to fifty is a rise of infinity times and a rise of 100%, and neither of those is a useful answer, so the page refuses the input rather than inventing one. The new value may be zero: that is a fall of exactly -100%, which is a real result and the floor for this calculation. Both values may be negative, and a change between two negative values is where the percentage is most likely to read contrary to intuition: -50 to -25 produces -50% for a value that got larger, because the denominator is negative. The page reports it and explains it rather than flagging it as an error. The percentage is rounded to four decimal places, so a change that does not divide evenly is approximate in the last place — 150 back to 100 is -33.3333%, which is not exactly a third. Nothing here is a judgement about size: the page does not say whether a 3% change is large or small, because that depends entirely on what is being measured, and a 3% move in body weight and a 3% move in a share price are not the same event. The page also handles exactly two values and no more: it is not a compound growth calculator and it will not chain a series of changes into a total, though the multiplier line is the reading to use if you want to do that by hand.

How do you calculate percentage change?

The formula: subtract the old value from the new one, divide by the old value, and multiply by 100. From 100 to 150 the change is 50, and 50 ÷ 100 = 0.5, which is +50%. The sign tells you the direction: positive for a rise, negative for a fall.

Why is a 50% rise followed by a 50% fall not back where it started?

Because the two percentages are measured against different bases. Rising from 100 to 150 is +50%, but falling from 150 back to 100 is -33.3333% — the return journey is measured against 150. Falling 50% from 150 would land at 75, which is below the starting point. This is the asymmetry of percentage change, and it is why the multiplier reading is worth using when changes are chained.

Can the percentage be negative?

Yes, and negative simply means the value fell. A fall from 150 to 100 is -33.3333%, and a fall to nothing is exactly -100%, which is as far as it goes. A negative answer is not an error unless it contradicts the amount of change, which happens only when one of the two values is itself negative.

What is the multiplier line for?

It is the new value divided by the old, so 1.5 for 100 to 150 and 0.6667 for 150 to 100. It says what to multiply by rather than what to add, and that makes it the reading that combines correctly: two multipliers in sequence give the right overall change, while two percentages added together do not.

What if the old value is zero?

Then there is no percentage to give, and the page says so instead of returning a number. A change from zero is undefined in percentage terms — you could call it an infinite rise or a 100% rise and neither would be wrong. A new value of zero is a different matter and is accepted: that is a fall of -100%.

Can I use this with negative values, such as a loss becoming a smaller loss?

Yes, and the answer will sometimes look backwards. From -50 to -25 the value has got larger and the amount of change is +25, but the percentage is -50%, because the denominator is negative. The page shows the arithmetic in that case rather than hiding it; the multiplier is below 1 for the same reason.

• Percentage Change — the relative change between an initial and a final value, including the fact that it is not symmetric between the two directions and is undefined when the initial value is zero — Wolfram MathWorld (United States)
• Percentage Change — the method on this page, with a worked example showing that a percentage rise is not reversed by the same percentage fall — Maths Is Fun (Rod Pierce)
• Percentage — a number expressed as a fraction of one hundred, which is the form the main reading on this page is printed in — Wolfram MathWorld (United States)
• Ratio — the quotient of two numbers, which is what the multiplier line on this page is, and which is the reading that combines correctly when changes are chained — Wolfram MathWorld (United States)