OneTaskBy Tim King · Updated
Percentage Change Calculator
Average Percentage Change Per Period and the Rise Needed to Get Back
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How the Percentage Change Calculator works
This percentage change calculator measures how far a value has moved from an old figure to a new one, as a percentage of where it started. It writes out the working, shows what change would undo it, and, if you enter a number of years or months, the average rate per period.
Percentage change compares the size of a move with the starting value, so a rise of 20 means a lot more on 80 than on 8,000. It is the figure behind price rises, pay increases, population growth and test score improvements, and the one most often worked out wrongly by dividing by the new value instead of the old.
How the percentage change calculator works
The formula is (new − old) ÷ old × 100. A positive answer is an increase and a negative one a decrease. The return figure uses the same formula with the values swapped. The average per period is the compound rate, (new ÷ old)^(1 ÷ periods) − 1, which is what grew each period on top of the last.
First example. A price goes from 80 to 100. (100 − 80) ÷ 80 × 100 = +25%. Going back from 100 to 80 is only −20%, because the base is now bigger.
Second example. A salary rises from 50,000 to 72,000 over 5 years. That is +44% in total, but the compound average is about +7.57% a year, not 44 ÷ 5 = 8.8%.
How many percent did a share fall if it went from 250 to 190? (190 − 250) ÷ 250 × 100 = −24%.
The same two numbers measured different ways, from 50,000 to 72,000
| Measure | Formula | Result |
|---|---|---|
| Percentage change | (new − old) ÷ abs(old) × 100 | +44% |
| Multiplier | new ÷ old | 1.44× |
| Percentage difference | abs(new − old) ÷ average × 100 | 36.07% |
| Change back to the old value | (old − new) ÷ abs(new) × 100 | −30.56% |
| Simple average per period | change ÷ periods | +8.8% |
| Compound average per period | (new ÷ old)^(1/periods) − 1 | +7.57% |
Starting from 100: the percentage change to each new value, and the change needed to get back to 100
| New value | Change | Percentage change | Change to get back |
|---|---|---|---|
| 25 | −75 | −75% | +300% |
| 50 | −50 | −50% | +100% |
| 60 | −40 | −40% | +66.67% |
| 75 | −25 | −25% | +33.33% |
| 80 | −20 | −20% | +25% |
| 90 | −10 | −10% | +11.11% |
| 95 | −5 | −5% | +5.26% |
| 99 | −1 | −1% | +1.01% |
| 101 | +1 | +1% | −0.99% |
| 105 | +5 | +5% | −4.76% |
| 110 | +10 | +10% | −9.09% |
| 120 | +20 | +20% | −16.67% |
| 125 | +25 | +25% | −20% |
| 150 | +50 | +50% | −33.33% |
| 175 | +75 | +75% | −42.86% |
| 200 | +100 | +100% | −50% |
| 250 | +150 | +150% | −60% |
| 300 | +200 | +200% | −66.67% |
| 400 | +300 | +300% | −75% |
| 500 | +400 | +400% | −80% |
| 1000 | +900 | +900% | −90% |
Tips
- Always divide by the old value. Dividing by the new one gives the change back, a different number.
- Percent is not percentage points. An interest rate going from 4% to 5% is +1 point but +25%.
- Rises and falls do not cancel. Up 50% then down 50% leaves you at 75% of where you began.
FAQ
What is the formula for percentage change?
Take the new value, subtract the old value, divide by the old value and multiply by 100. From 40 to 52 that is (52 − 40) ÷ 40 × 100 = 30%, an increase. A negative result, such as from 52 to 40, gives −23.08%, a decrease, measured against the starting figure of 52.
What is the difference between percentage change and percentage difference?
Percentage change has a direction: it runs from an old value to a new one and divides by the old. Percentage difference compares two values with no before or after, so it divides by their average and is never negative. Use difference for two shops' prices, change for one price over time.
Why does a 25% rise need only a 20% fall to undo it?
Each percentage is taken from a different starting point. After a 25% rise, 80 has become 100, and 20% of 100 is the same 20 that was added. The bigger the rise, the wider the gap: a 100% rise is undone by a 50% fall, and a 50% fall needs a 100% rise to recover.
How do you work out the average percentage change per year?
Do not divide the total change by the number of years, because each year's growth builds on the last. Divide the new value by the old, take the root for the number of years, subtract 1 and multiply by 100. Enter the number of years in the periods box and the calculator does this.
See also: all Arithmetic tools, Number Theory, Algebra and Geometry, Statistics, Trigonometry and Calculus.
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