Recall how to add and subtract fractions.

When the denominators of two fractions are the same, we used the common denominator and add the numerators. For example:

$$\dfrac{3}{7}+\dfrac{2}{7}=\dfrac{3+2}{7}=\dfrac{5}{7}$$

When the denominators of the fractions to be added are different, we first find equivalent fractions with the lowest common denominator. For example:

 \begin{align*} \dfrac{3}{7}+\dfrac{2}{5}&=\dfrac{15}{35}+\dfrac{14}{35}\\\\ &=\dfrac{29}{35} \end{align*} or \begin{align*} \dfrac{3}{7}+\dfrac{2}{5}&=\dfrac{15+14}{35}\\\\ &=\dfrac{29}{35} \end{align*}

Subtraction is undertaken in a similar manner.

With pronumerals, the second notation is most useful.

Example 7

Express as a single fraction Solution
$$\dfrac{x}{7}+\dfrac{4x}{7}$$ \begin{align*} \dfrac{x}{7}+\dfrac{4x}{7} &=\dfrac{x+4x}{7} \\\\ &=\dfrac{5x}{7} \end{align*}
$$\dfrac{2z}{11}-\dfrac{z}{2}$$ \begin{align*} \dfrac{2z}{11}-\dfrac{z}{2} &=\dfrac{4z-11z}{22} \\\\ &=\dfrac{-7z}{22}\\\\ &=-\dfrac{7z}{22} \end{align*}
$$-\dfrac{m}{3}+\dfrac{2m}{5}$$ \begin{align*} -\dfrac{m}{3}+\dfrac{2m}{5} &=\dfrac{-5m+6m}{15} \\\\ &=\dfrac{m}{15} \end{align*}