per cent

The per cent may be in general interpreted to mean a ‘hundredth’. So e.g. 5 per cent is ‘5 hundredths’, i.e. $\\frac{5}{100}$ .

In practice, giving some number of per cents, one means so many hundredths of a quantity given in the same or being clear from the context; for example, we can say that the illiteracy in the world is about 20 per cent – meaning that 20/100 of the adults of the world cannot read. If we say that the interest (rate) of a loan is 8 per cent, it means that one must pay interest for the loan 8/100 of the amount of the loan in a year.

If a percentage of a quantity has changed e.g. from 12% to 15%, we must not say that it has grown 3% but that it has grown 3 percentage points.

Determination of percentage

How many percent a number $a$ is of a second number $b$ ? The answer, the per cent number $p$ , is obtained from

\\displaystyle p=\\frac{a}{b}\\cdot 100.

(1)

The number $b$ here is called the base value and $a$ the per cent value(?). Essentially, the procedure in (1) may be replaced by converting the ratio $\\frac{a}{b}$ to hundredths, which can be done formally by multiplying this ratio by $1=\\frac{100}{100}=100\\%$ :

\\frac{a}{b}=\\frac{a}{b}\\cdot 100\\,\\%.