# Per mille calculations

### Per mille notations

$1‰=\frac{1}{1000}=0.001$

$1000‰=\frac{1000}{1000}=1$

$250‰=\frac{250}{1000}=\frac{1}{4}=0.25$

$x‰=\frac{x}{1000}=0.001x$

With the following abbreviations the formulas for calculating per mille can be written clearly.

• G : Basic value
• p%: Per-mille
• p : Per-mille number
• W : Per-mille value

$W=\frac{p\cdot G}{1000}$

$p‰=\frac{W}{G}$

$p=\frac{1000\cdot W}{G}$

$G=\frac{1000\cdot W}{p}$

## Per mille calculations

Per mille statement is an indication of conditions in a multiple of one thousendth of a whole. The thousendth indicated by the per mille ‰ sign is the same as the factor $\frac{1}{1000}$. Specifying a number n‰ with the following per mille sign indicates that it is n-thousandth of a whole. Specifying this make sense only if the entirety is indicated with. Background of the per mille calculation is to make comparable proportions. For this purpose, the ratios based on the base thousand be specified.

### Definition

$p‰=\frac{p}{1000}$

In addition to the ‰ operator and the spellings per mille and ppm (parts per mille) used.

### Example

$5‰=5\text{ppm}=\frac{5}{1000}=0.005$

5 ‰ are $5\cdot \frac{1}{1000}$

### Notations

$\text{Per-mille value}=\text{Basic value}\cdot \text{Per-mille}$

The basic value, as the whole, multiplied by a per-mille gives the per-mille value of the whole. If for example a running track is 42km long, then 10‰ of the running track are given by $42\cdot \frac{10}{1000}=0.42$

$\text{Basic value}=\frac{\text{Per-mille value}}{\text{Per-mille}}$

The per-mille value and the corresponding per-mille of the whole can be used to determine the basic value. How long, for example, is the total distance if 4.2km are 10‰ : $4.2\cdot \frac{1000}{10}=420$

$\text{Per-mille}=\frac{\text{Per-mille value}}{\text{Basic value}}$

The per-mille is obtained by dividing the per-mille value and the total. Is the track portion 4.2km and the total distance is 42km then 4.2km match to $\frac{4.2}{42}=\frac{1}{10}=100‰$