Key facts
- A percentage is a number written out of 100: 20% is 20 ÷ 100, which is 0.2 as a decimal and 1/5 as a fraction.
- Increases and decreases are multipliers. 150 raised by 20% is 150 × 1.2 = 180; 150 reduced by 20% is 150 × 0.8 = 120.
- Percentage change always names an original value, so the answer depends on direction: 80 to 100 is +25%, but 100 to 80 is −20%.
- Reversing a change means dividing by its multiplier, not multiplying. A price of 120 after a 20% increase came from 120 ÷ 1.2 = 100, not from 120 × 0.8 = 96.
- Successive changes multiply rather than add: +10% then −10% leaves 99% of the starting value, a 1% loss.
What a percentage is
A percentage is a number written out of 100. The symbol % means "divided by 100", so 20% is 20 ÷ 100 = 0.2 as a decimal and 1/5 as a fraction. The three notations describe the same quantity, which is why 1/2, 0.5 and 50% are interchangeable.
Percentages are not capped at 100. 200% of 50 is 100, and an increase of 150% turns 100 into 250, because the multiplier is 1 + 1.5 = 2.5. A percentage above 100% simply means more than the whole you started with. A figure below 0% means a decrease: −20% is the same operation as a 20% decrease.
Because every percentage is a fraction out of 100, converting between the forms is exact division rather than an estimate. The NumberUtils percentage calculator handles the arithmetic behind all of the forms below.
The three "percentage of" questions
Almost every percentage problem is one of three questions. They look different but each is the same relationship read from a different direction, so the safest approach is to identify which of the three you have before choosing a formula.
What is 20% of 150?
Multiply the value by the percentage divided by 100: 20 ÷ 100 × 150 = 0.2 × 150 = 30.
30 is what percentage of 150?
Divide the part by the whole and multiply by 100: 30 ÷ 150 = 0.2, then 0.2 × 100 = 20%. The whole cannot be zero, because a part is always a share of something, so a value of zero as the whole has no percentage answer.
30 is 20% of what?
Divide the part by the percentage written as a decimal: 30 ÷ 0.2 = 150. This is the reversal of a percentage calculation, and it is the one most often attempted by multiplying instead of dividing.
The five everyday percentage calculations
Those three questions, together with increases, decreases and comparisons between two values, make up five everyday calculations. Knowing which one you have is most of the work; the rest is arithmetic.
| Calculation | Formula | Worked example |
|---|---|---|
| Percentage of a value | percent ÷ 100 × value | 20% of 150 = 0.2 × 150 = 30 |
| Value as a percentage of a whole | (part ÷ whole) × 100 | 30 as a percentage of 150 = 0.2 × 100 = 20% |
| Increase or decrease | value × (1 ± percent ÷ 100) | 150 up 20% = 150 × 1.2 = 180 |
| Percentage change | (new − old) ÷ old × 100 | 80 to 100 = 20 ÷ 80 × 100 = +25% |
| Percentage difference | |a − b| ÷ ((a + b) ÷ 2) × 100 | 8 and 12 = 4 ÷ 10 × 100 = 40% |
NumberUtils has a calculator for each of these. The percentage increase calculator and the percentage decrease calculator apply the multiplier form, the percentage change calculator gives the signed change between an original and a current value, and the percentage difference calculator gives the symmetric comparison. The conventions behind them are set out in the percentage methodology notes.
Percentage change and percentage difference
Percentage change needs an original value and a direction, because the original value is the denominator. The same two numbers therefore give different, equally correct answers depending on the order. Going from 80 to 100 is 20 ÷ 80 × 100 = +25%. Going from 100 to 80 is −20 ÷ 100 × 100 = −20%. Neither is a mistake; they answer different questions, which is why a fall of 20% and a rise of 25% are the same journey described in opposite directions.
Percentage difference has no original value. It divides the gap between two values by their average, so it is symmetric — swapping the values changes nothing — and it is never negative. For 8 and 12, the gap is 4 and the average is 10, giving 40%. Use the difference when comparing two quantities of the same kind, such as two prices or two measurements, and change when one value is genuinely a later version of the other.
Because a difference is measured against the average, it is a proportion of typical size rather than of a starting point. 8 and 12 differ by 40%, and 800 and 1,200 differ by exactly the same 40%, because scaling both values leaves the percentage unchanged.
Reversing a percentage change
To undo a percentage change, divide by the multiplier that produced the result. A price of 120 after a 20% increase came from 120 ÷ 1.2 = 100. Taking 20% off 120 instead gives 120 × 0.8 = 96, which is wrong: 0.8 is the multiplier for a 20% decrease, and a decrease is not the mirror image of an increase.
The same rule applies to reductions. A price of 80 after a 20% decrease came from 80 ÷ 0.8 = 100. A 100% decrease is the one case that cannot be reversed, because it maps every starting value to zero, so the NumberUtils calculators reject it rather than return a misleading result.
The asymmetry runs the other way too. Getting from 120 back to 100 is a decrease of 20 ÷ 120 × 100 = 16.67%, not 20%. Restoring a figure after an increase always needs a smaller percentage than the increase itself.
Successive changes multiply
Two percentage changes in a row multiply their multipliers, because the second is applied to the value the first produced. Starting from 100, a 10% increase gives 110, and a 10% decrease applied to 110 gives 99. The round trip leaves 99% of the original, so it is a 1% loss even though the two percentages are the same size.
Successive discounts work the same way, which is why they are often misread. Starting from 100, taking 20% off leaves 80, then taking 30% off 80 leaves 56. The total reduction is 44%, not the 50% that adding the two percentages would suggest.
Percentage points and percent
A percentage point is the plain difference between two percentages. A rate rising from 4% to 5% has risen by 1 percentage point, but measured against where it started, that is an increase of 1 ÷ 4 × 100 = 25%. The two figures describe different things: percentage points measure the change in the rate itself, while the relative percentage measures the change in terms of the original rate. Changes in interest rates, tax rates and poll results are usually reported in percentage points for this reason.
Choosing the base matters in business too: a markup and a margin describe the same profit as a percentage of different amounts. A 30% markup on a cost of 100 gives a selling price of 130, and the profit of 30 is 23.08% of that selling price. The markup and margin guide works through why the two figures coincide only when the profit is zero.
Where percentage calculations go wrong
- Adding percentages for changes in sequence. Multipliers compound: +10% then −10% is a 1% loss, not a wash.
- Reversing a change by multiplying instead of dividing, as in removing 20% from a price that already had 20% added.
- Using the new value as the denominator. Percentage change divides by the original, which is why 100 to 80 is −20% while 80 to 100 is +25%.
- Reading a rise of 1 percentage point as a rise of 1%. From 4% to 5% the relative increase is 25%.
- Taking the percentage off the wrong base. 20% of a selling price and 20% of a cost are different amounts, and only one of them is a margin.