Percentage and per mille are ways of expressing a part of a whole. A percentage means a hundredth part, while a per mille means a thousandth. We come across them when calculating discounts, taxes, concentrations or the alcohol content of blood. In this article we explain the formulas, show the difference between these quantities, and give examples of the calculations.
What is a percentage?
A percentage is one hundredth part of a whole. The notation 1% means the fraction 1/100, that is 0.01. To calculate a percentage of a number, we use the formula:
- value = (percentage / 100) × number
For example, 20% is simply 0.2 of the whole.
What is a per mille?
A per mille is one thousandth part of a whole. The notation 1‰ means the fraction 1/1000, that is 0.001. The formula is analogous:
- value = (per mille / 1000) × number
Per mille is used where percentages would give very small numbers, for example for the alcohol content of blood.
The difference between percentage and per mille
The most important relationship is simple:
- 1% = 10‰ — one percent is ten per mille
- 1‰ = 0.1% — one per mille is one tenth of a percent
To convert percentages into per mille, we multiply by 10; to convert per mille into percentages, we divide by 10.
Example: percentage of a number
A shop offers a 15% discount on goods costing 200 zł. We calculate the amount of the discount.
- We apply the formula: value = (percentage / 100) × number.
- We substitute: value = (15 / 100) × 200.
- We calculate: value = 0.15 × 200 = 30 zł.
- Price after the discount: 200 - 30 = 170 zł.
The discount is 30 zł, and the customer will pay 170 zł.
Example: per mille of a number
We want to calculate 5‰ of an amount of 4000 zł.
- We apply the formula: value = (per mille / 1000) × number.
- We substitute: value = (5 / 1000) × 4000.
- We calculate: value = 0.005 × 4000 = 20 zł.
Five per mille of 4000 zł is 20 zł. If we wrote it as a percentage, it would be 0.5%.
What percentage of the whole is it?
Sometimes you have to calculate what percentage of one number another one is. We then use the formula: percentage = (part / whole) × 100. For example, 30 zł out of 200 zł is: (30 / 200) × 100 = 15%.
Calculating a value after a percentage change
We often calculate the price after an increase or a decrease. To add 23% VAT to a net amount of 100 zł: 100 × 1.23 = 123 zł gross. To subtract a 10% discount from 250 zł: 250 × 0.9 = 225 zł. The multipliers 1.23 and 0.9 simplify the calculations by combining the percentage with the whole in a single step.
Percentage points versus percentages
In the media, percentages are often confused with percentage points, but these are two different things. If an interest rate has risen from 4% to 6%, it has risen by 2 percentage points, but at the same time by 50% in relative terms (because 2 is half of 4). A percentage point describes the absolute difference between two percentage values, while the percentage change relates this difference to the starting value. Distinguishing between these concepts is crucial when interpreting data on inflation, interest rates or poll results.
Per mille in practice
We come across per mille not only in chemistry. The alcohol content of blood is given precisely in per mille. A result of 0.5‰ means 0.5 grams of alcohol per 1000 grams of blood. Per mille is also used to describe the gradient of terrain or small admixtures, where percentages would be too large a number.
Most common mistakes
- Confusing percentage with per mille — they differ tenfold, which is easy to overlook.
- Dividing by 100 instead of by 1000 — a per mille is a thousandth, not a hundredth part.
- Calculating a discount from the wrong base — a percentage is always calculated from the full starting amount.
- Adding up percentages from different bases — two 10% reductions must not be added as 20%.
Frequently asked questions
- How much is 100%? One hundred percent means the whole, that is the entire set or amount under consideration.
- Can you have more than 100%? Yes, for example a price increase of 150% means that the new price is 2.5 times higher than the original.
- How do I convert a fraction into a percentage? You multiply the fraction by 100. For example, 0.25 is 25%.
Summary
A percentage is a hundredth, while a per mille is a thousandth part of a whole, with 1% = 10‰. To calculate a percentage or per mille of a number, we divide it by 100 or 1000 respectively and multiply by the value. These are basic operations useful in finance and physics.
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