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# Addition and subtraction of fractions

If you want to add or substract fractions such as $$\displaystyle \frac{1}{\Tred{8}} + \frac{5}{\Tblue{6}},$$ you need to go through three steps.

• Re-write the fractions so that they have the same denominator. This can be done by cross multiplying by the denominators. $$\frac{1}{\Tred{8}} + \frac{5}{\Tblue{6}} = \frac{1\times\Tblue{6}}{\Tred{8}\times\Tblue{6}} + \frac{5\times\Tred{8}}{\Tblue{6}\times \Tred{8}} = \frac{6}{\Tviolet{48}} + \frac{40}{\Tviolet{48}}$$
• Add or subtract the numerators $\displaystyle \frac{\Tgreen{6}}{\Tviolet{48}} + \frac{\Tgreen{40}}{\Tviolet{48}} = \frac{\Tgreen{6} + \Tgreen{40}}{\Tviolet{48}}= \frac{\Tgreen{46}}{\Tviolet{48}}$
• Simplify the resulting fraction $\displaystyle \frac{46}{48} = \frac{23\times\Torange{2}}{24\times\Torange{2}} = \frac{23}{24}$

Here are some other examples

\begin{align*} &\frac{5}{6} + \frac{2}{3} = \frac{15 + 12}{18} = \frac{27}{18} = \frac{3}{2},\; &\frac{3}{5} + \frac{1}{3} = \frac{9 + 5}{15} = \frac{14}{15},\\ &\frac{5}{6} - \frac{2}{3} = \frac{15 - 12}{18} = \frac{3}{18} = \frac{1}{6}, \; &\frac{3}{5} - \frac{1}{3} = \frac{9 - 5}{15} = \frac{4}{15} \end{align*}
Fractions can be added when the denominators are the same.