What is a fraction?

Fractions are the following numbers.

\[ \frac{1}{2},\frac{3}{10},\frac{1}{100} \]

The number above the line is called the numerator and the number below the line is called the denominator.

Division and fractions have the following relationship.

\[ a\div b=\frac{a}{b} \]

Example

Example 1

Calculate \(\displaystyle \frac{1}{2}+\frac{1}{3} \).

Solution

\begin{align} \frac{1}{2}+\frac{1}{3}&=\frac{1\times3}{2\times3}+\frac{1\times2}{3\times2}\notag\\[+10pt] &=\frac{3}{6}+\frac{2}{6}\notag\\[+10pt] &=\frac{5}{6}\notag \end{align}

EXample 2

Calculate \(\displaystyle \frac{1}{2}-\frac{1}{3} \).

Solution

\begin{align} \frac{1}{2}-\frac{1}{3}&=\frac{1\times3}{2\times3}-\frac{1\times2}{3\times2}\notag\\[+10pt] &=\frac{3}{6}-\frac{2}{6}\notag\\[+10pt] &=\frac{1}{6}\notag \end{align}

Example 3

Calculate \(\displaystyle \frac{1}{2}\times\frac{1}{3} \).

Solution

\[ \frac{1}{2}\times\frac{1}{3}=\frac{1\times1}{2\times3}=\frac{1}{6} \]

Example 4

Calculate \(\displaystyle \frac{1}{2}\div\frac{1}{3} \).

Solution

\[ \frac{1}{2}\div\frac{1}{3}=\frac{1}{2}\times\frac{3}{1}=\frac{1\times3}{2\times1}=\frac{3}{2} \]