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Addition and subtraction of fractions (using LCM)

To add fractions, we need to re-write them with the same denominators. One way is to multiply the denominators.

$$ \frac{1}{8} + \frac{5}{6} = \frac{(1\times 6)+ (8\times 5)}{8\times 6} = \frac{46}{48} = \frac{23}{24}$$

There is a smarter way which uses smaller numbers.

  • Find the lowest common multiple of the denominators $$ \LCM(8,6) = \Tviolet{24}.$$
  • Re-write the fractions so that their denominators are equal to the lowest common multiple. $$ \frac{1}{\Tred{8}} + \frac{5}{\Tred{6}} = \frac{1\times \Tblue{3}}{\Tred{8} \times \Tblue{3}} + \frac{5\times \Tblue{4}}{\Tred{6} \times \Tblue{4}} = \frac{3}{\Tviolet{24}} + \frac{20}{\Tviolet{24}} $$
  • Add the numerators and simplify if necessary. $$\displaystyle \frac{3}{\Tviolet{24}} + \frac{20}{\Tviolet{24}} = \frac{23}{\Tviolet{24}}.$$

Here are some other examples:

\begin{align*} \frac{5}{6} + \frac{2}{3} = \frac{5 + 4}{6} = \frac{9}{6} = \frac{3}{2}, &\quad \frac{2}{15} + \frac{1}{6} = \frac{4 + 5}{30} = \frac{9}{30} = \frac{3}{10},\\ \frac{5}{6} - \frac{2}{3} = \frac{5 - 4}{6} = \frac{1}{6}, &\quad \frac{2}{15} - \frac{1}{6} = \frac{4 - 5}{30} = \frac{-1}{30} \end{align*}