How do you calculate percentage increase?
Short answer. Subtract the old value from the new one, divide by the old value, then multiply by 100. Rent going from 1,450 to 1,595 is (1,595 − 1,450) ÷ 1,450 × 100 = a 10% increase. To add a percentage instead, multiply: 1,450 × 1.10 = 1,595.
Two different questions hide behind the same phrase. Sometimes you have two numbers and want to know how much bigger the second one is. Sometimes you have one number and a percentage and want the result of adding it on. The arithmetic runs in opposite directions, so it is worth knowing which one you are in.
Finding the increase between two numbers
percentage increase = (new − old) / old × 100
The old value goes on the bottom. That is the whole trick. An increase is measured against the starting point, so the starting point sets the scale.
Rent rises from 1,450 to 1,595 a month.
rise 1,595 − 1,450 = 145 divide 145 / 1,450 = 0.1 × 100 10% increase
The one pass version divides first: 1,595 ÷ 1,450 = 1.1, and 1.1 − 1 = 0.1. Fewer keystrokes, same answer, and it lands you directly on the multiplier that the next section needs.
Increasing a number by a percentage
Going the other way, do not work out the extra and then add it. Multiply once by 1 + p/100 and the addition is already inside.
| Increase | Multiplier | Applied to 60 |
|---|---|---|
| Up 5% | × 1.05 | 63 |
| Up 8% | × 1.08 | 64.8 |
| Up 12% | × 1.12 | 67.2 |
| Up 20% | × 1.2 | 72 |
| Up 50% | × 1.5 | 90 |
| Up 100% | × 2 | 120 |
| Up 150% | × 2.5 | 150 |
This is also the form that survives being done twice. Two 10% rises in a row are not a 20% rise, they are 1.1 × 1.1 = 1.21, which is 21%. The extra one percent is the second rise landing on the first one.
Increases above 100%
Past 100% the wording starts misleading people, so read the multiplier instead of the percentage. A number goes from 25 to 80.
rise 80 − 25 = 55 divide 55 / 25 = 2.2 × 100 220% increase multiplier 80 / 25 = 3.2 times as much
A 220% increase and 3.2 times the original are the same event described twice. Percentage increase counts only the part that was added; the multiplier counts the whole result. So a 100% increase is a doubling, a 200% increase is a tripling, and anything advertised as "300% bigger" is four times the size, assuming whoever wrote it meant that. Often they did not.
Doing it in your head
Find 10% by moving the decimal point one place left, then build the rest out of that.
increase 240 by 15% 10% of 240 = 24 half of that = 12 add both = 36 new total 240 + 36 = 276 check 240 × 1.15 = 276
Same scaffolding as working out a restaurant tip, which is the 10% then adjust method under a different name. For 5% take half of the 10%. For 20% double it. For 8%, take 10% and subtract a fifth of it.
Several increases in a row do not average
Three years of raises: 5%, then 8%, then 12%. The temptation is to call that an average of 8.3333% a year. It is not.
total 1.05 × 1.08 × 1.12 = 1.27008
27.008% over three years
true average per year
cube root of 1.27008 ≈ 1.0829549
about 8.2955% a year
plain average of 8.3333% instead
(13/12)³ = 2197/1728 = 1.2714120...
27.1412%, which is too high
Averaging the percentages overstates the result, because each year's rise is computed on a different base. The honest average is the one that, repeated three times, reproduces the total. That is the geometric mean: multiply the multipliers, then take the nth root.
Working backwards from the new value
If you know the result and the percentage, divide by the multiplier. Do not subtract the percentage from the new number.
a bill is 1,595 after a 10% rise original = 1,595 / 1.1 = 1,450 wrong: 1,595 − 10% of 1,595 = 1,435.50
The 10% belonged to 1,450, not to 1,595, so taking 10% off the new figure removes too much. The general rule sits in the percentage change post, along with what happens when the move goes the other way and why a fall of 50% needs a rise of 100% to undo it.
Where the decimals bite
Percentage work generates long decimals fast. A price moving from 37 to 42 is a rise of 5, and 5 ÷ 37 = 0.135135135..., repeating without end. Rounded, a 13.51% increase. That is fine as a final answer and risky as an input to the next step.
The rounding you did not ask for is worse. Most calculators and spreadsheets store numbers in binary floating point, which cannot hold 1.1 exactly, so two chained 10% rises come out as 1.1 × 1.1 = 1.2100000000000002 rather than 1.21. Invisible once, and visible by the time it has run down a column of a hundred rows. Prism's keypad computes in exact decimal arithmetic instead, which is why 0.1 + 0.2 returns exactly 0.3 there and 1.1 × 1.1 returns exactly 1.21.
Quick reference
| You want | Do this | Example |
|---|---|---|
| Increase between two numbers | (new − old) / old × 100 | 1,450 to 1,595 is +10% |
| Add a percentage to a number | old × (1 + p/100) | 60 up 12% is 67.2 |
| Undo an increase | new / (1 + p/100) | 1,595 after +10% came from 1,450 |
| Two increases in a row | multiply the multipliers | 1.1 × 1.1 = 1.21, so +21% |
| Average of yearly increases | nth root of the product | 1.27008 over 3 years is +8.2955% a year |
If the increase you are pricing is the gap between what something cost you and what you sell it for, the same arithmetic goes by another name and gets confused with its cousin daily. That is markup versus margin.
Or let the percentages tool keep track of the base
Prism has a percentages calculator with From and To fields: type 1,450 and 1,595, read 10% off the Change row. Percentages are free forever, with no ads and no subscription. See the app.