Lesson 1.3: Fractions

New Concept

This lesson reviews fractions, which represent parts of a whole. You will learn to simplify, add, subtract, multiply, and divide fractions, including complex fractions and those with variables. Mastering these operations is foundational for solving algebraic equations.

What’s next

Get ready to master these skills. You’ll start with interactive examples on simplifying fractions, followed by practice cards covering all four basic operations.

Property

A fraction is written $\frac{a}{b}$, where $b \neq 0$ and $a$ is the numerator and $b$ is the denominator. A fraction represents parts of a whole. The denominator $b$ is the number of equal parts the whole has been divided into, and the numerator $a$ indicates how many parts are included.

Equivalent Fractions Property
If $a$, $b$, and $c$ are numbers where $b \neq 0$, $c \neq 0$, then $\frac{a}{b} = \frac{a \cdot c}{b \cdot c}$ and $\frac{a \cdot c}{b \cdot c} = \frac{a}{b}$.
A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator. To simplify, factor the numerator and denominator into prime numbers, then divide out common factors.

Examples

• To simplify $-\frac{108}{63}$, we find the prime factors: $-\frac{2 \cdot 2 \cdot 3 \cdot 3 \cdot 3}{3 \cdot 3 \cdot 7}$. We cancel the common factors $3 \cdot 3$ to get $-\frac{2 \cdot 2 \cdot 3}{7} = -\frac{12}{7}$.

Property

To multiply fractions, multiply the numerators and multiply the denominators. An LCD is NOT needed. For $b \neq 0$ and $d \neq 0$, $\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$.

To divide fractions, multiply the first fraction by the reciprocal of the second. An LCD is NOT needed. For $b \neq 0$, $c \neq 0$, and $d \neq 0$, $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$.

Examples

• To multiply $(-\frac{14}{15})(\frac{9}{20})$, multiply numerators and denominators: $-\frac{14 \cdot 9}{15 \cdot 20} = -\frac{(2 \cdot 7) \cdot (3 \cdot 3)}{(3 \cdot 5) \cdot (4 \cdot 5)}$. After canceling common factors, we get $-\frac{21}{50}$.

Property

To add or subtract fractions, an LCD (Least Common Denominator) is needed.

How to Add or Subtract Fractions

1. Determine if the fractions have a common denominator. If not, rewrite each fraction as an equivalent fraction with the LCD.
2. Add or subtract the numerators and place the result over the common denominator.
3. Simplify the resulting fraction, if possible.

Examples

• To add $\frac{7}{12} + \frac{5}{18}$, the LCD of 12 and 18 is 36. We rewrite the fractions as $\frac{7 \cdot 3}{12 \cdot 3} + \frac{5 \cdot 2}{18 \cdot 2} = \frac{21}{36} + \frac{10}{36}$, which equals $\frac{31}{36}$.

Property

A complex fraction is a fraction in which the numerator and/or the denominator contains a fraction.

How to Simplify a Complex Fraction

1. Simplify the numerator.
2. Simplify the denominator.
3. Divide the numerator by the denominator. Rewrite the complex fraction as a division problem and simplify if possible.

Examples

• To simplify $\frac{\frac{x}{2}}{\frac{xy}{6}}$, rewrite as division: $\frac{x}{2} \div \frac{xy}{6}$. Then multiply by the reciprocal: $\frac{x}{2} \cdot \frac{6}{xy} = \frac{6x}{2xy}$. After canceling common factors, the result is $\frac{3}{y}$.

Property

For any positive numbers $a$ and $b$,

Examples

• The fraction $-\frac{5}{9}$ can be written as $\frac{-5}{9}$ or $\frac{5}{-9}$. All three represent the same negative value.

• When simplifying $\frac{12}{-20}$, we can move the negative sign out front to get $-\frac{12}{20}$, which then reduces to $-\frac{3}{5}$.