Percentages in everyday life

In short: Per cent means “out of a hundred”: 1 % is one hundredth of a whole. With three quantities – base value, percentage rate and percentage amount – almost every percentage problem can be solved.

In more detail: The basic formula is percentage amount = base value · rate / 100. If you rearrange it for the quantity you are looking for, you can reliably work out discounts, taxes, pay rises or survey figures.

In detail

• The three quantities: The base value G is the whole (100 %), the rate p gives the share in per cent, and the percentage amount W is the share as a number. Example: 20 % of €80 is 80 · 20 / 100 = €16.
• Rearranging: Looking for the rate: p = W / G · 100 (12 out of 48 is 25 %). Looking for the base value: G = W · 100 / p (€15 is 30 % of €50).
• Using factors: An increase of p % equals multiplying by 1 + p/100, a decrease multiplying by 1 − p/100. €80 minus 25 % is 80 · 0.75 = €60.
• Up and down is not the same: If a price rises by 20 % and then falls by 20 %, you do not end up at the starting value: 100 · 1.2 · 0.8 = 96. The second percentage refers to a different base value.
• Working back from VAT: A gross price of €119 at 19 % VAT contains a net price of 119 / 1.19 = €100. Subtracting 19 % of €119 instead gives the wrong result of €96.39.
• Per cent and percentage points: If an interest rate rises from 4 % to 5 %, that is an increase of 1 percentage point, but a rise of 25 % compared with the old value. News reports often mix the two up.
• Per mille: One per mille (‰) is one thousandth, i.e. 0.1 %. It is used, for example, for blood alcohol levels or for very small shares.
• Quick mental maths: You get 10 % by moving the decimal point one place to the left; 5 % is half of that and 1 % a tenth of it. So 15 % of €60 is €6 + €3 = €9.
• Context: Percentages are fractions with the denominator 100, and repeated percentage growth leads to compound interest (bruchrechnung-grundlagen, zinseszins).