# 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 $15$ .

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