# Percentage calculation

### Percent notations

$1%=\frac{1}{100}=0.01$

$100%=\frac{100}{100}=1$

$25%=\frac{25}{100}=\frac{1}{4}=0.25$

$x%=\frac{x}{100}=0.01x$

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

• G : Basic value
• p%: Percentage
• p : Percent number
• W : Percent value

$W=\frac{p\cdot G}{100}$

$p%=\frac{W}{G}$

$p=\frac{100\cdot W}{G}$

$G=\frac{100\cdot W}{p}$

### Slope in percent

The indication of a slope or a gradient in percent indicates how much the altitude changes in relation to the horizontal distance. For example, 10% slope are reached per 100m horizontally 10m vertical. This corresponds to an angle of 5.7 °. The angle is given by arctan (p / 100).

## Percentage

Percentage is the indication of ratios in a multiple of one hundredth of a whole. The hundredth is indicated by the % symbol and is equivalent to $\frac{1}{100}$. The specification of a number n% with the following percentage sign indicates that it is n hundredths of a whole. The statement makes sense only if the totality is indicated with. Background of the percentage calculation is to make size ratios comparable. For this purpose, the ratios are given with respect to the basis hundred.

### Definition

$p%=\frac{p}{100}$

In addition to the % operator and the spellings per cent and per cent are (of hundreds) used.

### Example of percentages

$5%=5\text{per cent}=\frac{5}{100}=0.05$

5 % are $5\cdot \frac{1}{100}$

### Terms of percentage calculation

$\text{Percent value}=\text{Basic value}\cdot \text{Percentage}$

The basic value, as the whole, multiplied by a percentage gives the percentage value of the whole. If for example a running track is 42km long, then 10% of the running track are given by $42\cdot \frac{10}{100}=4.2$

$\text{Basic value}=\frac{\text{Percent value}}{\text{Percentage}}$

The percent value and the corresponding percentage of the whole can be used to determine the basic value. How long, for example, is the total distance if 4.2km are 10% : $4.2\cdot \frac{100}{10}=42$

$\text{Percentage}=\frac{\text{Percent value}}{\text{Basic value}}$

The percentage is obtained by dividing the percent value and the total. Is the track portion 4.2km and the total distance is 42km then 4.2km match to $\frac{4.2}{42}=\frac{1}{10}=10%$