Why use this calculator
Percentage change is the most quoted and most misread number in business and news reporting. A price that rises 25% and then falls 25% does not return to where it started, a discount of 50% followed by a further 20% is not 70% off, and a fall from 8% to 6% can honestly be described as either a two point drop or a 25% drop depending on who is selling the story.
This calculator gives you the change amount, the percentage, the multiplier and the reverse percentage side by side, so the asymmetry is visible rather than assumed. It also refuses to invent a percentage when the starting value is zero, which is the single most common way a spreadsheet produces a nonsense growth figure.
How to use it
- Enter the original figure in the old value field.
- This is whatever came first in time or whatever you are treating as the baseline, such as last quarter's revenue, the price before a rise, or a starting weight.
- Enter the current or final figure in the new value field.
- Both fields accept negative numbers, which matters when you are comparing a loss to a profit, and the calculator will tell you when the result is not meaningful rather than printing a misleading zero.
- Read the percentage change as the headline, then check the multiplier if you want to apply the same move to another number, and the reverse change if you need to know what it would take to undo it.
What each result means
- Change amount
- The raw difference between the two figures in whatever units you entered, before any percentage is applied.
- Percentage change
- The change expressed as a proportion of the original value. Positive is a rise, negative is a fall.
- Direction
- Whether the move is an increase, a decrease or no change, stated in words so a negative sign cannot be missed.
- New value as a share of old
- The new figure as a percentage of the old one. A hundred and twenty-five per cent here is the same as a twenty-five per cent increase.
- Multiplier
- The factor you would multiply the old value by to get the new one, which is the form to use when compounding several changes.
- Reverse change
- The percentage move required to get from the new value back to the old one. It is always smaller in magnitude after a rise and larger after a fall.
The formula
Percentage change = ((new value minus old value) divided by the old value) times 100
The change amount is simply new value minus old value. The multiplier is the new value divided by the old value, so a 25% increase is a multiplier of 1.25. The reverse change is ((old value minus new value) divided by the new value) times 100, which is why a 25% rise needs only a 20% fall to undo it. When the old value is zero the division is undefined and no percentage exists.
How this number is used in practice
The asymmetry between a rise and a fall is where money is lost. An investment that drops 50% needs a 100% gain to break even, not another 50%, because the second percentage is measured against a smaller base. The same arithmetic explains why a supplier who raises prices 10% and then offers a 10% discount has still left you paying 99% of the original rather than 100%.
Reading the multiplier rather than the percentage makes this obvious: 0.5 times 2 returns you to 1, while 1.1 times 0.9 is 0.99.
Statistical agencies are careful to distinguish a percentage change from a percentage point change, and it is worth copying their discipline. When an interest rate moves from 4% to 5%, that is a one percentage point rise and a 25% relative rise, and both statements are true.
Headlines choose whichever is more dramatic. When you are comparing figures over time, also check that the base has not been restated or rebased, because a change measured against a revised baseline is not comparable with one measured against the original.
For everyday budgeting the useful habit is to convert every quoted percentage into a multiplier before you combine it with anything else. Chained discounts, stacked price rises, compounding pay increases and sales tax all behave like multipliers, so 1.05 times 1.03 tells you immediately that two consecutive rises of 5% and 3% amount to 8.15%, not 8%.
Worked examples
A price rises from 100 to 125
A change of 25.00, a 25.00% increase and a multiplier of 1.2500x. Reversing it takes a 20.00% decrease.
Revenue falls from 50,000 to 42,500
A change of -7,500.00, a 15.00% decrease and a multiplier of 0.8500x. Recovering the lost ground needs a 17.65% increase.
An investment halves from 20,000 to 10,000
A 50.00% decrease, but the reverse change shows a 100.00% gain is needed just to break even.
A business turns a loss of -5,000 into a profit of 15,000
A change amount of 20,000.00, which the percentage cannot describe sensibly because the base is negative.
When people use it
- Measuring how much a supplier price, rent or utility bill has risen
- Reporting sales, traffic or revenue growth between two periods
- Checking the real size of a discount or a stacked promotion
- Tracking investment gains and the recovery needed after a loss
- Comparing a forecast against the actual result as a variance percentage
Tips
- Always name the base. A 25% rise from 100 and a 25% rise to 100 are different numbers, and ambiguity here is how sales copy misleads.
- Convert percentages to multipliers before combining them. Two 10% rises are 21%, not 20%, because 1.1 times 1.1 is 1.21.
- A percentage point is not a percentage. A rate moving from 4% to 5% rose one percentage point and 25% in relative terms.
- After a fall of x%, the gain needed to recover is always larger than x%, because it is measured against the smaller number.
- Never report a percentage change from a zero base. Quote the absolute change instead, which is what this calculator does.
- When the old value is negative, percentage change stops behaving intuitively. Use the change amount and describe the move in words.
What it assumes, and where it stops being accurate
- Both figures are measured in the same units and on the same basis, so the comparison is like for like.
- The old value is the baseline. Swapping the two fields produces a different percentage, which is correct rather than a fault.
- No adjustment is made for inflation, seasonality or a change in the number of trading days between the two periods.
- Percentage change is undefined when the old value is zero, and the calculator reports that rather than substituting zero.
- Results are rounded to two decimal places for display, so recombining rounded figures may differ in the last cent.
Terms used on this page
- Base value
- The starting figure a percentage change is measured against, and the denominator in the formula.
- Multiplier
- The factor the old value is multiplied by to reach the new value. A 20% rise is a multiplier of 1.2.
- Percentage point
- The arithmetic difference between two percentages, as distinct from the relative change between them.
- Variance
- In budgeting, the difference between a planned figure and the actual result, usually reported as a percentage.
Questions people ask
- What is the formula for percentage change?
- Subtract the old value from the new value, divide the result by the old value, then multiply by one hundred. In symbols that is ((new minus old) divided by old) times 100. The old value is always the denominator, which is why swapping the two figures gives a different answer.
- Why does a 50% loss need a 100% gain to recover?
- Because each percentage is measured against a different base. Falling from 20,000 to 10,000 is half of 20,000, but climbing back is 10,000 on a base of only 10,000, which is the whole of it. The reverse change row shows this recovery figure directly.
- What if the old value is zero?
- No percentage exists. Division by zero is undefined, and any move away from zero is effectively an infinite increase. The calculator says so rather than printing a zero or an error. Report the absolute change instead, for example from zero to twelve sales.
- What is the difference between percentage change and percentage difference?
- Percentage change measures a move relative to a starting value and has a direction. Percentage difference compares two values relative to their average and has no direction, which makes it the right choice when neither figure is naturally the baseline, such as comparing two laboratory measurements.
- Can I use negative numbers?
- You can enter them, and the change amount stays correct and meaningful. The percentage becomes difficult to interpret once the base is negative, because the sign of the denominator flips the sign of the result. In that case quote the absolute change and describe the direction in words.
- How do I combine two percentage changes?
- Multiply their multipliers rather than adding the percentages. A 5% rise followed by a 3% rise is 1.05 times 1.03, which equals 1.0815, so the combined change is 8.15%. Adding them would understate it, and the error grows quickly with larger percentages.
Use this with
Percentage Change Calculator answers one part of the question. These pick up where it stops: Percentage Calculator, Discount Calculator, Markup Calculator, Inflation Calculator and Profit Margin Calculator.
Where these figures come from
This calculation is jurisdiction-neutral arithmetic. No country-specific rules are applied.
- US Bureau of Labor Statistics: Consumer Price Index. Worked examples of percent-change calculations on index numbers
- Australian Bureau of Statistics: Consumer Price Index. Quarterly and annual percentage change methodology
- UK Office for National Statistics: Inflation and price indices. Published rates of change and the index-number arithmetic behind them
Percentage Change Calculator produces an estimate for general information. It is not financial, tax, medical, legal or engineering advice. read the full disclaimer. Spotted a wrong number? Report it and it gets fixed. The form picks up this calculator and the inputs you used.