Percentage Increase Calculator – Percent Change

What this calculator does, and what it cannot decide

This page compares two numbers. You give it a starting value and an ending value, and it returns the gap between them expressed three ways: as a signed percentage, as an absolute difference, and as the multiplier that carries one to the other. That is the whole of its job, and it is a job it can do exactly.

What it cannot decide is whether percent change is the right statistic for your two numbers in the first place. A percentage is a ratio, and a ratio taken against a very small baseline can be arithmetically correct and practically meaningless. Two orders rising to four is a 100% increase, and so is two million rising to four million. The arithmetic cannot tell those apart, and it does not try to. Deciding whether the relative change or the absolute change is the figure worth quoting is your step, not the calculator's.

The formula, and why the denominator is the old value

Percent change is the absolute change measured as a share of where you began:

((new − old) / |old|) × 100

The numerator is uncontroversial. The denominator is where the meaning lives. Percent change asks how large a movement was relative to the thing that moved, so the baseline has to be the value you started from. Dividing by the new value instead would answer a different question, and dividing by the average of the two would answer a third one, which is percent difference rather than percent change.

The absolute value bars around the denominator matter only when the baseline is negative, and they are there so the sign of the answer still reads the way a reader expects. Multiplying by 100 is the last step and the least interesting one: it converts a ratio into the units of percent, where the symbol stands for the number 0.01. The NIST reference at the foot of this page is the authority for that definition.

One calculation, many names

Percent increase, percent decrease, percent change, percentage change and percent gain are five labels for a single operation. Nothing in the arithmetic changes when the direction of travel changes. Subtract, divide by the baseline, multiply by 100, then read the sign.

The labels persist because the contexts differ. A retailer says percent off, an investor says percent gain, an analyst says percent change, and a student is asked for percentage increase. This page prints the direction alongside the number, so a negative result is labelled a decrease rather than left for you to interpret. There is no separate mode to switch into and no second tool to open when the value went down.

Working it out by hand

The calculator opens on 80 and 100. Here is that case in full, followed by two that are less obvious.

The same pair, reversed

Now start at 100 and end at 80. The absolute change is 80 − 100 = −20, and the baseline is now 100 rather than 80, so the division is −20 ÷ 100 = −0.20 and the result is a 20% decrease. The multiplier is 80 ÷ 100 = 0.80. Note that reversing the pair did not simply flip the sign of the 25%. The denominator moved with the baseline, which is why a 25% rise is undone by a 20% fall rather than by a 25% one.

A negative baseline

Take −50 rising to −20, the shape of an account moving back toward zero. The absolute change is −20 − (−50) = 30, a genuine increase. Dividing by the absolute value of the baseline gives 30 ÷ 50 = 0.60, so the page reports a 60% increase, which is the reading a person expects. The multiplier for the same pair is −20 ÷ −50 = 0.40, which is below 1 and looks like a shrink. Both figures are correct. They disagree because dividing one negative by another discards the direction, and when the baseline is negative the signed percentage is the figure to trust.

Percent off, and which direction you are going

Percent off is percent decrease with a price tag on it, and the arithmetic is identical. An item marked down from $80 to $60 has fallen by 20 ÷ 80, which is 25% off, and the multiplier is 0.75. If you already have the two prices, this page is the tool.

Most percent-off questions run the other way, though, and the distinction is worth being exact about. If you know a price and a discount and you want the amount you will pay, that is the inverse operation: multiply rather than divide. This page does not do that, and saying otherwise would claim a capability the tool does not have. The discount calculator takes a price and a percentage and returns what you pay.

Why an increase and an equal decrease do not cancel

This is the mistake that costs people the most, and it follows directly from the denominator. Start at 100 and add 50%: the 50% is taken against 100, so you gain 50 and land on 150. Now subtract 50%: this time the 50% is taken against 150, so you lose 75 and land on 75. The two moves were the same percentage and different amounts, because the base underneath them changed.

Multipliers make the outcome obvious in one line. A 50% increase is ×1.5 and a 50% decrease is ×0.5, and 1.5 × 0.5 = 0.75, a 25% net decrease. Any sequence of percentage moves collapses this way: convert each to its multiplier, multiply them together, and read the product. That is the practical reason the multiplier sits in the result panel next to the percentage rather than being left out as a duplicate.

Common percent changes at a glance

Every row is exact, and each can be checked in the tool above by entering the two values from the example column:

Percentage points are not percent

When the two numbers you are comparing are themselves percentages, the word percent turns ambiguous and the error it hides is large. A rate moving from 4% to 6% has risen by two percentage points, and it has also risen by 50% in relative terms, because 2 ÷ 4 is 0.5. Both statements describe the same movement and they are not interchangeable.

The rule of thumb is short. If your inputs are rates, shares or proportions and you subtract one from the other, the answer is in percentage points. If you divide by the baseline, as this page does, the answer is a relative percent change. Entering 4 and 6 here returns 50%, which is correct and is not the two-point figure a reader may be expecting, so name the units when you quote it. This convention is editorial rather than metrological, and the NIST reference below does not cover it.

Edge cases

• A baseline of zero has no answer. Percent change from 0 is undefined, because there is no quantity to take a share of. The page says so rather than printing a number, and it points you to the absolute change instead. This is a property of the arithmetic, not a limitation of the tool.
• An empty field is not a zero. Clearing an input leaves the result waiting rather than treating the blank as 0 and computing a confident percentage from it. Type an explicit 0 if you mean zero.
• The percentage and the multiplier can disagree in sign. This happens only when the baseline is negative, as in the −50 to −20 case above. The signed percentage is the figure that reads correctly.
• Order is not symmetric. 80 to 100 is +25% and 100 to 80 is −20%. Swapping the inputs changes the denominator, so the two results differ in magnitude rather than being the same number with opposite signs.
• Decreases stop at 100%, increases do not. From a positive baseline, a 100% decrease lands exactly on zero and nothing below is reachable by a larger percentage. Increases have no ceiling: 40 to 120 is +200%.
• Identical values are reported as no change. The result is 0% with a multiplier of 1, and the page labels the direction as no change rather than calling zero an increase.

Your numbers stay in the page

The values you enter are processed by this page in your browser. They are not sent to a server, not stored after you close the tab, and not visible to anyone else. There is no account and nothing to sign up for.

Related tools

• Percent decrease calculator when the drop is the framing you want stated up front.
• Percentage calculator for a plain percentage of a single number rather than a change between two.
• Discount calculator for the inverse: a price and a percentage off, returning what you pay.
• Percent error calculator when one of your two values is a known true value and the other is a measurement.
• Rate of change calculator when the change is measured against time rather than against the starting value.

Related guides

• Percent Change Formula How to calculate percent change between two numbers, handling increases and decreases.
• How to Calculate Percentage Increase Step by step with three worked examples (price, salary, and traffic).
• Percentage Increase Formula What each piece of the formula means and why the old value is the base.

Sources

Percent change is definitional arithmetic rather than a standard any body sets, so there is one citation on this page and it covers one thing. NIST Special Publication 811 § 7.10.2 records that the symbol % represents the number 0.01, which is what the final multiplication in the formula relies on.

It does not cover the percentage-point convention described above, and it is not cited for it. That distinction is editorial usage, widely followed and not standardised, and this page states it as its own rather than attributing it to a reference that is silent on the matter.