Percentages
Percentage Increase Formula
The percentage increase formula is (new − old) ÷ old × 100. Subtract the starting value from the new one, divide by the starting value, and multiply by 100. The same formula gives a percentage decrease when the result is negative. This guide explains why the starting value is the base, why equal rises and falls don’t cancel, and how to combine several changes.
How to calculate percentage increase
A percentage increase measures how much a value has grown, as a share of where it started. Three steps:
- Subtract the old value from the new value to get the increase.
- Divide the increase by the old value.
- Multiply by 100.
Example: a newsletter grows from 1,250 to 1,600 subscribers
The subscriber count rose 28%. To check, increase 1,250 by 28%: 1,250 × 1.28 = 1,600.
The percentage change calculator does this for any two values, shows the multiplier and the reverse change, and has a second mode for adding or removing a percentage. If you need the more basic “what is X% of Y” skills first, start with how to calculate percentage.
Why you divide by the starting value
A percentage always needs a reference point, and the question “how much did it increase?” is asked from the point of view of the start. An increase of 350 means something very different for a list of 1,250 than for a list of 100,000. Dividing by the old value scales the change to the size of the thing that changed.
Dividing by the new value answers a different question. 350 ÷ 1,600 ≈ 21.88% says what share of today’s list is new since the last count. That can be a useful figure, but it is not the percentage increase, and the two should not be mixed up.
When there is no start and finish, such as two shops’ prices on the same day, neither value is the natural base. In that case the symmetric percentage difference calculator, which divides by the average, is the fairer measure. The guide to percent difference vs percent change covers that choice.
How to calculate percentage decrease
A percentage decrease uses the same idea with the subtraction the other way round, so the answer comes out positive.
Example: a car’s value drops from $28,000 to $19,600
The car lost 30% of its value. Equivalently, it is now worth 70% of what it was: 19,600 ÷ 28,000 = 0.7.
A percentage decrease of a quantity that cannot go negative has a hard limit of 100%, which means the value has fallen to zero. There is no such limit on increases.
The percent change formula: one formula for both
Rather than keeping separate formulas for increase and decrease, most people use a single signed version:
For the car, (19,600 − 28,000) ÷ 28,000 × 100 = −30%. The minus sign carries the direction. When you say “decreased by 30%,” the word “decreased” takes over that job and you drop the sign. The negative number rules guide covers the sign arithmetic if the subtraction trips you up.
The multiplier form
Divide new by old instead of subtracting first and you get the growth multiplier. It holds the same information in a form that is much easier to calculate with.
| Multiplier | Percent change | In words |
|---|---|---|
| 0.25 | −75% | a quarter of the original |
| 0.5 | −50% | halved |
| 0.75 | −25% | three quarters remain |
| 0.9 | −10% | down a tenth |
| 1.05 | +5% | up a twentieth |
| 1.1 | +10% | up a tenth |
| 1.5 | +50% | half as much again |
| 2 | +100% | doubled |
| 3 | +200% | tripled |
| 10 | +900% | ten times the original |
Note the last three rows. Doubling is a 100% increase, not 200%, because the increase is only the part added on top of the original. “Three times as much” is a 200% increase. Mixing these up is one of the most common errors in reporting growth.
Why a rise and an equal fall don’t cancel
A share price goes from $200 up 50%, then down 50%. It does not end at $200.
The two changes are the same percentage but not the same amount, because each is measured from a different base. The rise is 50% of 200, which is 100. The fall is 50% of 300, which is 150. The bigger base makes the same percentage cost more.
The order makes no difference: down 50% then up 50% also ends at $150, since 0.5 × 1.5 is still 0.75. Any equal rise and fall, in either order, leaves you below the start.
The decrease that undoes an increase
To get back to the start after an increase of r, you need a decrease of r ÷ (1 + r). After a 30% rise, that is 0.3 ÷ 1.3 ≈ 23.08%. After a 50% rise it is about 33.33%, and after a 100% rise it is exactly 50%. The reverse percentage calculator applies this when you need the value before a change.
Example: finding the price before an increase
A monthly plan now costs $54 after an 8% price rise. What did it cost before?
The old price was $50. Taking 8% off $54 would give $49.68, which is wrong, because the 8% was a share of the old price, not the new one.
Combining several percentage changes
To combine changes, multiply their multipliers. Never add the percentages.
Two 10% increases give 21%, not 20%, because the second 10% applies to an amount that has already grown. That extra 1% is the same compounding effect that makes interest on interest grow over time. The simple interest formula guide shows the version where interest does not compound, for contrast. Stacked store discounts follow the same multiplying rule, which the guide on a percentage of a percentage works through.
Example: three years of changes
Sales rose 5% in the first year, 8% in the second, and fell 3% in the third. What is the total change, and the average yearly change?
Sales are up about 10% over the three years. The average yearly rate is about 3.23%, not the 3.33% you get from averaging 5, 8 and −3. The correct average is the single rate that, applied three times, produces the same total; that requires the cube root, not division by 3.
Percentage increase of a percentage: points vs percent
When the values being compared are themselves percentages, such as interest rates, unemployment rates or test pass rates, there are two ways to describe a change, and both are correct.
“The rate went up 1 percentage point” and “the rate went up 25%” describe the same event. Problems start when someone says “up 1%” and means the points, or quotes the 25% to make a small move sound dramatic. Use “percentage points” for the simple subtraction and “percent” for the relative change, and the ambiguity disappears.
Percentage increase in pricing: markup and margin
A markup is a percentage increase from cost to selling price. An item that costs $40 and sells for $52 has a markup of (52 − 40) ÷ 40 = 30%. The markup calculator solves for any of cost, markup, price or profit.
Margin uses the same $12 profit but divides by the selling price: 12 ÷ 52 ≈ 23.08%. That is the decrease from price back to cost, which is why a 30% markup and a 23.08% margin describe the same item. It is the reverse-change relationship from earlier, in business language. The guide on margin vs markup covers converting between them.
Common mistakes
- Dividing by the new value. From 1,250 to 1,600 is +28%. Dividing 350 by 1,600 gives about 21.88%, which is not the increase.
- Calling a doubling a 200% increase. Doubling is +100%. Tripling is +200%.
- Adding successive changes. +10% twice is +21%. Multiply the multipliers.
- Undoing a rise by subtracting the same percent. To reverse +8% you divide by 1.08; taking 8% off overshoots.
- Averaging yearly rates with a plain mean. Over several periods, use the root of the total multiplier.
- Starting from zero. If the old value is 0, the percentage increase is undefined. Report the absolute change instead.
- Mixing points and percent. State which one you mean when the values are rates.
For one-off “X% of Y” arithmetic with no before and after, the plain percentage calculator is the quicker tool.
Frequently asked questions
What is the formula for percentage increase?
The percentage increase formula is (new − old) ÷ old × 100. Subtract the starting value from the final value, divide by the starting value, and multiply by 100. From 1,250 to 1,600: 350 ÷ 1,250 × 100 = 28%. If the result is negative, the value fell, and the figure is a percentage decrease.
How do you calculate percentage decrease?
Subtract the new value from the old value, divide by the old value, and multiply by 100: (old − new) ÷ old × 100. A car that drops from $28,000 to $19,600 has lost 8,400 ÷ 28,000 × 100 = 30% of its value. The base is always the original amount, which is why a decrease can never exceed 100% for a positive quantity.
What is the percent increase from 20 to 25?
It is a 25% increase. The change is 25 − 20 = 5, and 5 ÷ 20 = 0.25, which is 25%. Going back from 25 to 20 is a 20% decrease, because the same change of 5 is now measured against the larger number, 25.
What is the percentage increase from 50 to 120?
It is a 140% increase. The change is 70, and 70 ÷ 50 = 1.4, or 140%. The multiplier is 120 ÷ 50 = 2.4, so the new value is 2.4 times the old one. The increase is always the multiplier minus 1, which is why 2.4 times corresponds to +140%, not +240%.
Is percent change the same as percent increase?
Percent change is the general term and can be positive or negative. A percent increase is a positive percent change, and a percent decrease is a negative one. All three use the same formula, (new − old) ÷ old × 100; the sign of the answer tells you which you have. Use “change” when you do not know the direction in advance.
How do I calculate percentage increase in Excel?
With the old value in A2 and the new value in B2, enter =(B2-A2)/A2 and format the cell as a percentage. Excel then shows 0.28 as 28%. Do not multiply by 100 as well if you use percentage formatting, or the result will display a hundred times too large. Negative results are decreases.
Why doesn’t a 50% drop followed by a 50% rise get back to the start?
Because the rise is calculated on a smaller number. Halving 200 gives 100, and 50% of 100 is only 50, so you end at 150. The combined multiplier is 0.5 × 1.5 = 0.75, a 25% loss. To recover from a 50% drop you need a 100% increase, since the value has to double.
How do I calculate the average percentage increase over several years?
Multiply the yearly multipliers to get the total, then take the root for the number of years and subtract 1. For +5%, +8% and −3%: 1.05 × 1.08 × 0.97 = 1.09998, and the cube root is about 1.0323, so the average is about 3.23% a year. A plain average of the three percentages, 3.33%, overstates it.
Calculators for this
Every one of these shows the working, so you can check the method as well as the answer.
More guides
https://fourfunctioncalculator.com/percentage-increase-formula/