Percentage Change Calculator

Why 100 to 80 is a 20% decrease but 80 to 100 is a 25% increase

Percentage change always divides by the value you started from, never the value you ended at. That single rule is the whole measure, and it is also why the two directions of the same journey do not match. Falling from 100 to 80 moves 20 units against a base of 100, so it is a 20% decrease. Climbing back from 80 to 100 moves the identical 20 units, but now against a base of 80, so it is a 25% increase. The numerator is the same in both directions; the denominator is not, and the denominator is what percentage change is really about.

Read the sign as the direction — positive is an increase, negative is a decrease, zero is no movement — and read the growth multiplier as the same fact in a form you can carry forward. A 25% increase is 1.25×; a 20% decrease is 0.80×. The multiplier is also what you reverse with: 125 ÷ 1.25 = 100. Subtracting 25% from 125 gives 93.75, which is not where you began, and that substitution is the most common way a reverse-percentage answer ends up quietly wrong. Reverse mode divides; it never subtracts the same percentage back off.

Percentage change is also not percentage difference. Change is directional and needs a starting point. Difference is symmetric and compares two independent readings against their average, which is what you want when neither number came first — two labs, two suppliers, two quotes. The percentage calculator has a dedicated difference mode for that case. On this page, enter percentages as whole numbers (25 means 25%, not 0.25); results are rounded only for display, to the number of decimals you select, while the arithmetic underneath is carried at full precision.

A 50% loss needs a 100% gain, and the gap widens as the loss deepens

Halve 100 and you are left with 50. Add 50% back and you reach 75, not 100. The loss was measured against 100 and the recovery has to be measured against 50, so equal percentages moving in opposite directions never cancel out. To get from 50 back to 100 you have to double, which is a 100% gain: Gain% = 50 ÷ (100 − 50) × 100 = 100%.

What the formula shows once you run it across several losses is that recovery becomes disproportionately harder, not proportionally harder. A 20% loss leaves 80, and 20 ÷ 80 × 100 = 25% is what it takes to return. A 25% loss leaves 75, and 25 ÷ 75 × 100 = 33.33%. A 50% loss leaves 50, and 50 ÷ 50 × 100 = 100%. Each step down the ladder shrinks the base you have left to grow from while the target stays exactly where it was, so the required gain accelerates away from the loss that caused it.

This is the arithmetic behind a sentence that sounds reasonable and is not: “we are down 25% this quarter, so 25% growth next quarter puts us back.” It does not. It puts you at 93.75% of where you started, because the second percentage is taken from the smaller number the first one left behind. Loss Recovery mode performs the division for you, but the reasoning is worth holding on to — any time an equal percentage rise is offered as the answer to a percentage fall, the two figures are sitting on different bases and the account will not close.

Two percentage changes multiply — they never add

Percentage changes over consecutive periods do not sum, because each one applies to whatever the previous one left behind. Two consecutive 25% increases are not a 50% total: 100 × 1.25 = 125, then 125 × 1.25 = 156.25, a cumulative change of 56.25%. Those extra 6.25 points are the second increase acting on the first increase’s output rather than on the original starting value.

The same multiplication explains a pairing that looks like a coincidence. A 25% increase followed by a 20% decrease lands you exactly back at the start, because 1.25 × 0.80 = 1.00. Reverse the order and the answer is identical, since multiplication does not care about sequence. But an increase and a decrease of the same size do not cancel at all: 100 raised by 20% is 120, and 120 cut by 20% is 96 — a 4% net fall out of two moves that sound like they should undo each other.

This is why Multi-Period mode reports per-period and cumulative change as two separate columns instead of one running total. The per-period figures tell you what happened inside each interval; the cumulative figure is the product of every multiplier in the chain, never their sum. Averaging the per-period percentages produces a third number again, and it is usually the least defensible of the three, because a straight average treats a large swing on a small base as equal in weight to a small swing on a large one. When several chains need comparing, put them beside each other in Scenario Comparison rather than collapsing each one into a single average.

A rate moving from 3% to 5% rose 2 points and 66.67%

When the quantity you are measuring is itself a percentage — an interest rate, a conversion rate, a tax rate, a margin — there are two correct answers to “how much did it change?”, and they are nowhere near each other. The arithmetic gap is 2 percentage points: 5 minus 3. The relative change is 2 ÷ 3 × 100 = 66.67%, because percentage change divides by where the rate started, and the rate started at 3.

Both figures are true and they answer different questions. What is never true is describing the move as “a 2% increase”, which silently merges the two and misleads every reader who takes it at face value. Say “two percentage points” when you mean the arithmetic gap, say “66.67%” when you mean the relative move, and name which one you are using before the number rather than after it.

The asymmetry from the first section governs this case too. A rate falling from 5% back to 3% is still a 2 percentage-point move, but the relative change is now 2 ÷ 5 × 100 = 40%, not 66.67%, because the denominator changed with the direction. Percentage Points mode reports both readings side by side for exactly this reason: seeing the point gap and the relative change together is what stops one being quoted in place of the other.

100% growth is not a growth rate until you say over how long

A move from 100 to 200 is a 100% change, and it stays a 100% change whether it took twelve months or a decade. Total percentage change carries no time dimension at all, which makes it the wrong figure to quote in any conversation where the speed of the move is the point being argued.

Convert it to a per-year rate and the difference stops being subtle. Doubling inside a single year is a 100% annual rate. The same doubling spread over five years is a compound annual growth rate of roughly 14.87% per year — the rate which, applied five times over, multiplies the starting value by two. The 100% figure and the 14.87% figure describe the same journey, and swapping one for the other would justify opposite decisions.

Year-over-year and month-over-month comparisons are simply percentage change applied to two time-separated readings: this period against the same period last year, or against last month. They are exact and they are easy to misread, because a striking YoY figure can be an artefact of an unusually weak base period rather than evidence of an unusually strong current one — always look at the base before you celebrate the percentage. Use Multi-Period mode when you have several readings and want per-period and cumulative change together, and the CAGR companion when you need one annualised rate. CAGR carries a limitation worth stating plainly: it assumes positive start and end values, and it describes a perfectly smooth path that real growth almost never takes, so it summarises a trajectory rather than reporting one.

When the original value is zero there is no percentage to report

The formula divides by the original value, so a starting point of zero has no answer — not infinity, not an enormous number, simply undefined. A move from 0 to 125 is not a percentage increase of any size, because 125 is not a multiple of nothing. The honest report is the absolute change: it rose from 0 to 125. This calculator says the result is undefined rather than printing a figure that would look authoritative and mean nothing at all.

Negative values and pairs that straddle zero are the subtler version of the same problem. The arithmetic still runs, but the language stops matching it: a deficit narrowing is an improvement that the formula can report carrying a minus sign, and a value crossing from below zero to above it can produce a percentage that reads like a collapse. The calculator still computes those figures, because suppressing them would hide information you may need, but in those cases the absolute change is the number to quote and the percentage is the one that needs a sentence of explanation beside it.

A very small original value causes a related distortion without ever being invalid. The same 25-unit move is a 25% increase on a base of 100 and a 100% increase on a base of 25 — identical movement, percentages four times apart. That is a limitation of the measure itself rather than a defect in the calculation, and the defence is always the same: publish the percentage and the absolute change together, never the percentage on its own. This tool does not know which of the two your audience needs, and it cannot tell you whether the baseline you chose is a fair one to divide by.

Fold change, weight change, and body fat: three lookalikes with different answers

Fold change is percentage change wearing different clothes. Fold change is New ÷ Original, so a doubling is a 2-fold change, while percentage change is (Fold change − 1) × 100, so the same doubling is a 100% increase. A 0.5-fold change is a 50% decrease. This calculator reports the percentage form directly; convert with that relationship whenever a journal, a lab protocol, or a supplier spec asks for the fold figure instead.

Weight change is ordinary percentage change and needs no adaptation: enter your earlier measurement as the original value and your current one as the new value, in whatever unit you weighed yourself in. The answer is how far you moved relative to where you started, which is the figure most programmes actually track.

Body fat percentage is a different kind of quantity altogether. It is a part-of-whole estimate derived from body measurements, not a comparison of two readings, and this calculator does not compute it. What it will do is give you the change between two body fat figures you already have — and because those figures are themselves percentages, that is a percentage-points-versus-relative-change question, so read the section above before you quote a number from it.

Some percentage questions are not change questions at all. For a slice of a number, a reverse percentage, or a symmetric percentage difference, use the percentage calculator. For pricing built up from cost, the markup and profit margin calculators keep two measures apart that are constantly confused. For inclusive-versus-exclusive tax arithmetic, the VAT calculator. And the formula on this page cites no external authority because none exists: percentage change is elementary arithmetic, written out above in full. Where an authority does exist for an application of it — a statutory rate, an index methodology — cite that authority for the inputs rather than citing this page for the sum.

Related calculators

These tools take specific percentage problems further:

More in Math, or browse all calculators.

Read the guide

There is no dedicated percentage-change guide yet. For compound growth over time — the idea behind this calculator's CAGR companion — see How Compound Interest Works With Regular Contributions.