Addition and subtraction

Same-named fractions

You add or subtract fractions of the same name by adding or subtracting the numerators and maintaining the common denominator. In general:

$\frac{a}{b}+\frac{c}{b}=\frac{a+c}{b}$

$\frac{a}{b}-\frac{c}{b}=\frac{a-c}{b}$

Remember

Fractions have the same name if they have the same denominator.
Fractions are called dissimilar if they have different denominators.

Examples

$\frac{3}{8}+\frac{2}{8}=\frac{5}{8}$

$\frac{10}{13}-\frac{3}{13}=\frac{7}{13}$

$\frac{15x+4}{3y}-\frac{3x}{3y}=\frac{12x+4}{3y}$

Unlike fractions

In order to add or subtract unlike fractions, they are previously provided with the same name. To do this, you expand the fractions so that all fractions have the same denominator. .

Hint

The easiest way to provide fractions with the same name is by naming the fractions with the other denominators.
For example: $\frac{1}{\color{blue}{3}}+\frac{3}{\color{green}{4}}=\frac{1\cdot\color{green}{4}}{\color{blue}{3}\cdot\color{green}{4}}+\frac{3\cdot\color{blue}{3}}{\color{green}{4}\cdot\color{blue}{3}}$ $=\frac{4}{\color{red}{12}}+\frac{9}{\color{red}{12}}$

Actually, you just have to think: "What do I have to multiply with the denominator so that all fractions have the same denominator?"

Sometimes it is enough to just expand a fraction.
For example: $\frac{1}{4}+\frac{1}{\color{red}{8}}=\frac{1\cdot\color{green}{2}}{4\cdot\color{green}{2}}+\frac{1}{\color{red}{8}}$ $=\frac{2}{\color{red}{8}}+\frac{1}{\color{red}{8}}$

Examples

Calculate and shorten if necessary

$\frac{3}{5}+\frac{2}{15}=\frac{3\cdot3}{5\cdot3}+\frac{2}{15}$ $=\frac{9}{15}+\frac{2}{15}$ $=\frac{11}{15}$

$\frac{2}{4}-\frac{1}{7}=\frac{2\cdot7}{4\cdot7}-\frac{1\cdot4}{7\cdot4}$ $=\frac{14}{28}-\frac{4}{28}$ $=\frac{10}{28}$ $=\frac{5}{14}$

$\frac{x}{ab}+\frac{y}{ac}=\frac{x\cdot c}{ab\cdot c}+\frac{y\cdot b}{ac\cdot b}$ $=\frac{cx}{abc}+\frac{by}{abc}$ $=\frac{cx+by}{abc}$