### Calculating with numbers: Calculating with fractions

### Addition and subtraction of fractions

Addition and subtraction of fractions with the same denominator

Adding and subtracting two fractions with the same denominator is easy: the numerators are added or subtracted while the denominator is left unchanged. if possible, the answer is simplified.

**Example**

\[\begin{aligned}\tfrac{5}{14} +\tfrac{3}{14}&=\tfrac{5+3}{14}=\tfrac{8}{14}=\tfrac{4}{7} \\[0.2cm] \tfrac{5}{14} -\tfrac{3}{14}&=\tfrac{5-3}{14}=\tfrac{2}{14}=\tfrac{1}{7}\end{aligned}\]

Addition and subtraction of fractions with the different denominator

Adding and subtracting two fractions with the different denominators can be done via the following algorithm:

- rewrite the rational numbers as fractions with the same denominator;
- add or subtract these fractions;
- if possible, simplify the answer.

**Example**

\[\begin{aligned}\tfrac{3}{10} +\tfrac{2}{15}&=\tfrac{\phantom{1}3\times 15}{10\times 15}+\tfrac{10\times 2\phantom{1}}{10\times 15}\\&=\tfrac{45}{150}+\tfrac{20}{150}=\tfrac{65}{150}=\tfrac{13}{30} \\[0.2cm] \tfrac{3}{10} -\tfrac{2}{15}&=\tfrac{\phantom{1}3\times 3}{10\times 3}-\tfrac{\phantom{1}2\times 2}{15\times 2}\\&=\tfrac{9}{30}-\tfrac{4}{30}=\tfrac{5}{30}=\tfrac{1}{6}\end{aligned}\]

The examples below with two or three fractions illustrate the computational work.

We use it to rewrite the fractions as fractions with a common denominator. \[\frac{9}{10}=\frac{9\times 13}{10\times 13}=\frac{117}{130}\quad\text{and}\quad\frac{4}{13}=\frac{4\times 10}{13\times 10}=\frac{40}{130}\] Then the computation is easy: \[\begin{aligned}\frac{9}{10}+\frac{4}{13} &=\frac{117}{130}+\frac{40}{130} &\blue{\text{construction of fractions with a common denominator}} \\ \\ &=\frac{117+40}{130} &\blue{\text{collection of numerators}} \\ \\ &={{157}\over{130}}&\blue{\text{calculation and sometimes simplification}} \end{aligned}\]

Mathcentre video clips

Addition and Subtraction (22:32)