# Operations with Fractions

When adding and subtracting fractions , the first thing to check is if the denominators are the same.

### Adding and Subtracting Fractions with Like Denominators

If the denominators are the same, then it's pretty easy: just add or subtract the numerators, and write the result over the same denominator.

For example,

$\frac{2}{7}+\frac{4}{7}=\frac{6}{7}$

### Adding and Subtracting Fractions with Unlike Denominators

If the denominators are not the same, then you have to use equivalent fractions which do have a common denominator. To do this, you need to find the least common multiple (LCM) of the two denominators.

For example,

$\begin{array}{l}\frac{3}{4}+\frac{5}{3}=\frac{3\text{\hspace{0.17em}}×\text{\hspace{0.17em}}3}{4\text{\hspace{0.17em}}×\text{\hspace{0.17em}}3}+\frac{5\text{\hspace{0.17em}}×\text{\hspace{0.17em}}4}{3\text{\hspace{0.17em}}×\text{\hspace{0.17em}}4}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{9}{12}+\frac{20}{12}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{29}{12}\end{array}$

## Multiplying and Dividing with Fractions

### Multiplying a Fraction by a Fraction

To multiply two fractions, just multiply the numerators to get the numerator of the product, and multiply the denominators to get the denominator of the product.

$\frac{a}{b}\cdot \frac{c}{d}=\frac{a\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}c}{b\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}d}$

For example,

$\frac{4}{3}\cdot \frac{5}{7}=\frac{4\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}5}{3\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}7}=\frac{20}{21}$

### Multiplying a Fraction by an Integer

To multiply a fraction by a whole number, remember that any integer $n$ can be written as the fraction $\frac{n}{1}$ .

For example,

$6\cdot \frac{3}{13}=\frac{6}{1}\cdot \frac{3}{13}=\frac{6\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}3}{1\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}13}=\frac{18}{13}$

### Dividing by a Fraction

To divide by a fraction , multiply by the reciprocal of the fraction.

For example,

Divide $\frac{2}{3}$ by $\frac{4}{5}$ .

$\begin{array}{l}\frac{2}{3}÷\frac{4}{5}=\frac{2}{3}\cdot \frac{4}{5}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{10}{12}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{5}{6}\end{array}$