A fraction is one way of writing a rational number. In

$$\frac ab,$$

$a$ is the numerator and $b\ne0$ is the denominator. Different fractions can represent the same number; multiplying numerator and denominator by the same nonzero value does not change the fraction.

Arithmetic with fractions

Fractions can be added or subtracted after expressing them with a common denominator. For example,

$$\frac12+\frac13=\frac36+\frac26=\frac56.$$

Multiplication operates directly on numerators and denominators:

$$\frac ab\cdot\frac cd=\frac{ac}{bd}.$$

Division by a nonzero fraction is multiplication by its reciprocal.

Ratios

A ratio compares two quantities through a quotient. If a mixture contains $2$ parts of one ingredient for every $3$ parts of another, their ratio is

$$\frac23.$$

Ratios compare relative size rather than the absolute amount present.

Proportions

When two ratios are equal, they form a proportion. For example,

$$\frac23=\frac46.$$

This expresses the same relative relationship at two different scales.

If $3$ notebooks cost €6 at a constant price per notebook, then $6$ notebooks cost €12: doubling one quantity doubles the other while preserving the ratio.

Percentages

A percentage is a ratio expressed relative to one hundred. Thus

$$25%=\frac{25}{100}=\frac14.$$

Fractions, ratios, proportions and percentages provide several representations of the same underlying idea of relative quantity.