Lesson 3: Lowest Common Denominator

New Concept

To add or subtract algebraic fractions with different denominators, you first need the Lowest Common Denominator (LCD). This allows you to rewrite them as equivalent like fractions, making them easy to combine.

What’s next

You're now ready to see how it's done. Next, you'll tackle interactive examples on finding the LCD and adding and subtracting algebraic fractions.

Property

To build a fraction is to convert it to an equivalent form with a larger denominator. This is done by multiplying the numerator and denominator by the same (nonzero) quantity, called the building factor. This is an application of the fundamental principle of fractions:

Examples

• To build the fraction $\frac{3}{4}$ to have a denominator of 28, the building factor is 7. So, $\frac{3 \cdot 7}{4 \cdot 7} = \frac{21}{28}$.

Property

To Add or Subtract Algebraic Fractions:

1. Find the lowest common denominator (LCD) for the fractions.
2. Build each fraction to an equivalent one with the LCD as denominator.
3. Add or subtract the resulting like fractions: Add or subtract their numerators, and keep the same denominator.
4. Reduce the sum or difference if necessary.

Examples

• To compute $\frac{x}{3} + \frac{y}{5}$, the LCD is 15. We get $\frac{5x}{15} + \frac{3y}{15} = \frac{5x+3y}{15}$.

Property

The lowest common denominator (LCD) for two or more algebraic fractions is the simplest algebraic expression that is a multiple of each denominator. To find the LCD:

1. Factor each denominator completely.
2. For each factor, include the most copies of that factor that appears in any single denominator.
3. Multiply together the factors of the LCD.

Examples

• The LCD for $\frac{1}{6a^2b}$ and $\frac{5}{9ab^3}$ is $18a^2b^3$. We need factors of $2 \cdot 3^2 \cdot a^2 \cdot b^3$.