# Fractions and simplification of fractions

A fraction represents a rational number as the ratio of two integers, the numerator and the denominator. The denominator cannot be zero.

$$ \textrm{Fraction} = \frac{\Tblue{\textrm{Numerator}}}{\Tred{\textrm{Denominator}}}$$The fraction $$\Tblue{17}/\Tred{26}$$ has numerator $$\Tblue{17}$$ and denominator $$\Tred{26}$$.

A fraction is in its simplest form, or irreducible, if the numerator and denominator have **no common factor**. A fraction has a **unique** simplest form. To simplify a fraction, find the **highest common factor** of the numerator and the denominator and divide both by it.

$$\Tblue{7}/\Tred{9}$$ is irreducible. $$\Tblue{18}/\Tred{27}$$ is not. The HCF of $$\Tblue{18}$$ and $$\Tred{27}$$ is $$9$$. The fraction can be simplified to $$\Tblue{2}/\Tred{3}$$. $$\frac{\Tblue{18}}{\Tred{27}} = \frac{9\times \Tblue{2}}{9\times \Tred{3}} = \frac{\Tblue{2}}{\Tred{3}}. $$

Two fractions are equivalent if they have the same value. They have the same simplest form.

$$\Tblue{3}/\Tred{9}$$ and $$\Tblue{2}/\Tred{6}$$ are equivalent; their simplest form is $$\Tblue{1}/\Tred{3}$$.