Fraction Operations

To add (or subtract) two fractions :

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 $21$ .

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

To multiply two fractions:

1) Multiply the numerator by the numerator.

2) Multiply the denominator by the denominator.

For all real numbers $a, b, c, d (b \neq 0, d \neq 0)$

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

Example 2:

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

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

To divide by a fraction, multiply by its reciprocal .

For all real numbers $a, b, c, d (b \neq 0, c \neq, d \neq 0)$

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} = \frac{a d}{b c}$

Example 3:

Divide $\frac{3}{4} \div \frac{5}{7}$ .

$\begin{array}{l} \frac{3}{4} \div \frac{5}{7} = \frac{3}{4} \cdot \frac{7}{5} \\ = \frac{3 \cdot 7}{4 \cdot 5} \\ = \frac{21}{20} \end{array}$

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.

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

Example 5:

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

$\frac{11}{7} = 11 \div 7 = 1 R 4$

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 $15$ .

$\frac{45}{75} = \frac{45 \div 15}{75 \div 15} = \frac{3}{5}$

Fraction Operations

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