Fraction Operations

To add (or subtract) two fractions

1) Find the least common denominator.

2) Write both original fractions as equivalent fractions with the least common denominator.

3) Add (or subtract) the numerators.

4) Write the result with the denominator.

Example 1:

Add $\frac{1}{3}+\frac{3}{7}$.

The least common denominator is $21$.

$\begin{array}{l}\frac{1}{3}+\frac{3}{7}=\frac{1\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}7}{3\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}7}+\frac{3\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}3}{7\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{7}{21}+\frac{9}{21}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{16}{21}\end{array}$

To multiply two fractions:

1) Multiply the numerator by the numerator.

2) Multiply the denominator by the denominator.

For all real numbers $a,b,c,d\left(b\ne 0,d\ne 0\right)$

$\frac{a}{b}\cdot \frac{c}{d}=\frac{ac}{bd}$

Example 2:

Multiply $\frac{1}{4}\cdot \frac{5}{6}$.

$\begin{array}{l}\frac{1}{4}\cdot \frac{5}{6}=\frac{1\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}5}{4\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}6}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{5}{24}\end{array}$

To divide by a fraction, multiply by its reciprocal.

For all real numbers $a,b,c,d\left(b\ne 0,c\ne ,d\ne 0\right)$

$\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\cdot \frac{d}{c}=\frac{ad}{bc}$

Example 3:

Divide $\frac{3}{4}÷\frac{5}{7}$.

$\begin{array}{l}\frac{3}{4}÷\frac{5}{7}=\frac{3}{4}\cdot \frac{7}{5}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{3\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}7}{4\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}5}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{21}{20}\end{array}$

Mixed numbers can be written as an improper fraction and an improper fraction can be written as a mixed number.

Example 4:

Write $7\frac{2}{5}$ as an improper fraction.

$\begin{array}{l}7\frac{2}{5}=\frac{7}{1}+\frac{2}{5}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{7\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}5}{1\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}5}+\frac{2}{5}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{35}{5}+\frac{2}{5}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{37}{5}\end{array}$

Example 5:

Write $\frac{11}{7}$ as a mixed number in simple form.

Therefore, $\frac{11}{7}=1\frac{4}{7}$.

A fraction is in lowest terms when the numerator and denominator have no common factor other than $1$.  To write a fraction in lowest terms, divide the numerator and denominator by the greatest common factor.

Example 6:

Write $\frac{45}{75}$ in lowest terms.

$45$ and $75$ have a common factor of $15$.

$\frac{45}{75}=\frac{45\text{\hspace{0.17em}}÷\text{\hspace{0.17em}}15}{75\text{\hspace{0.17em}}÷\text{\hspace{0.17em}}15}=\frac{3}{5}$