Lesson Objectives

- Demonstrate an understanding of how to find the LCD for a group of fractions
- Learn how to simplify a complex fraction by simplifying the numerator and denominator separately
- Learn how to simplify a complex fraction using the LCD method

## How to Simplify a Complex Fraction

A complex fraction is a fraction that contains at least one fraction in its numerator or denominator. As an example: $$\frac{\frac{1}{9}}{\frac{2}{7}}$$ How can we simplify a complex fraction? There are two methods we can use for this process. The first case is to simplify the numerator and denominator separately and then perform the main division. In our above example, 1/9 is the numerator of the complex fraction and 2/7 is our denominator. There isn't anything to simplify in either numerator or denominator, so we could really write this problem as: $$\frac{1}{9}\div \frac{2}{7}=\frac{1}{9}\cdot \frac{7}{2}=\frac{7}{18}$$ Let's take a look at a few examples and then we will learn a faster approach using the LCD of the fractions.

Example 1: Simplify each: $$\frac{\frac{4}{17}}{\frac{2}{51}}$$ There isn't anything to simplify in either numerator or denominator. Let's set up our division as: $$\frac{4}{17}\div \frac{2}{51}=\frac{4}{17}\cdot \frac{51}{2}=6$$ Example 2: Simplify each: $$\frac{\frac{1}{4}+\frac{2}{7}}{\frac{1}{6}+\frac{3}{7}}$$ We will simplify the numerator and denominator separately: $$\frac{1}{4}+ \frac{2}{7}=\frac{15}{28}$$ $$\frac{1}{6}+ \frac{3}{7}=\frac{25}{42}$$ Now our problem becomes: $$\frac{\frac{15}{28}}{\frac{25}{42}}$$ Let's set up our division as: $$\frac{15}{28}\div \frac{25}{42}=\frac{15}{28}\cdot \frac{42}{25}=\frac{9}{10}$$ Example 3: Simplify each: $$\frac{\frac{2}{9}-\frac{1}{12}}{\frac{2}{3}-\frac{7}{18}}$$ We will simplify the numerator and denominator separately: $$\frac{2}{9}- \frac{1}{12}=\frac{5}{36}$$ $$\frac{2}{3}- \frac{7}{18}=\frac{5}{18}$$ Now our problem becomes: $$\frac{\frac{5}{36}}{\frac{5}{18}}$$ Let's set up our division as: $$\frac{5}{36}\div \frac{5}{18}=\frac{5}{36}\cdot \frac{18}{5}=\frac{1}{2}$$

Example 4: Simplify each using the LCD method: $$\frac{\frac{12}{13}}{\frac{4}{39}}$$ First, we find the LCD of all fractions involved. We have 12/13 in our numerator and 4/39 in our denominator:

LCD » LCM(13, 39) = 39

Multiply the numerator and denominator of the complex fraction by the LCD: $$\frac{\frac{12}{13}\cdot \frac{39}{1}}{\frac{4}{39}\cdot \frac{39}{1}}$$ Simplify: $$\require{cancel}\frac{\frac{12}{\cancel{13}1}\cdot \frac{\cancel{39}3}{1}}{\frac{4}{\cancel{39}1}\cdot \frac{\cancel{39}1}{1}}$$ $$\frac{\cancel{12}3 \cdot 3}{\cancel{4}1}=9$$ Example 5 Simplify each using the LCD Method: $$\frac{\frac{3}{10}+ \frac{6}{15}}{\frac{19}{20}- \frac{2}{3}}$$ First, we find the LCD of all fractions involved. We have 3/10, 6/15, 19/20, and 2/3:

LCD » LCM(3, 10, 15, 20) = 60

Multiply the numerator and denominator of the complex fraction by the LCD: $$\frac{\left(\frac{3}{10}+ \frac{6}{15}\right) \cdot \frac{60}{1}}{\left(\frac{19}{20}- \frac{2}{3}\right) \cdot \frac{60}{1}}$$ We need to use the distributive property here. 60 will be distributed to each term or fraction in the numerator and the denominator: $$\frac{\frac{3}{10}\cdot \frac{60}{1}+ \frac{6}{15}\cdot \frac{60}{1}}{\frac{19}{20}\cdot \frac{60}{1}- \frac{2}{3}\cdot \frac{60}{1}}$$ Simplify: $$\frac{\frac{3}{\cancel{10}1}\cdot \frac{\cancel{60}6}{1}+ \frac{6}{\cancel{15}1}\cdot \frac{\cancel{60}4}{1}}{\frac{19}{\cancel{20}1}\cdot \frac{\cancel{60}3}{1}- \frac{2}{\cancel{3}1}\cdot \frac{\cancel{60}20}{1}}$$ $$\frac{18 + 24}{57 - 40}=\frac{42}{17}=2\frac{8}{17}$$

Example 1: Simplify each: $$\frac{\frac{4}{17}}{\frac{2}{51}}$$ There isn't anything to simplify in either numerator or denominator. Let's set up our division as: $$\frac{4}{17}\div \frac{2}{51}=\frac{4}{17}\cdot \frac{51}{2}=6$$ Example 2: Simplify each: $$\frac{\frac{1}{4}+\frac{2}{7}}{\frac{1}{6}+\frac{3}{7}}$$ We will simplify the numerator and denominator separately: $$\frac{1}{4}+ \frac{2}{7}=\frac{15}{28}$$ $$\frac{1}{6}+ \frac{3}{7}=\frac{25}{42}$$ Now our problem becomes: $$\frac{\frac{15}{28}}{\frac{25}{42}}$$ Let's set up our division as: $$\frac{15}{28}\div \frac{25}{42}=\frac{15}{28}\cdot \frac{42}{25}=\frac{9}{10}$$ Example 3: Simplify each: $$\frac{\frac{2}{9}-\frac{1}{12}}{\frac{2}{3}-\frac{7}{18}}$$ We will simplify the numerator and denominator separately: $$\frac{2}{9}- \frac{1}{12}=\frac{5}{36}$$ $$\frac{2}{3}- \frac{7}{18}=\frac{5}{18}$$ Now our problem becomes: $$\frac{\frac{5}{36}}{\frac{5}{18}}$$ Let's set up our division as: $$\frac{5}{36}\div \frac{5}{18}=\frac{5}{36}\cdot \frac{18}{5}=\frac{1}{2}$$

### How to Simplify a Complex Fraction using the LCD Method

When working with complex fractions, we can use a faster approach:- Find the LCD of all fractions involved
- Multiply the numerator and denominator of the complex fraction by the LCD
- Simplify

Example 4: Simplify each using the LCD method: $$\frac{\frac{12}{13}}{\frac{4}{39}}$$ First, we find the LCD of all fractions involved. We have 12/13 in our numerator and 4/39 in our denominator:

LCD » LCM(13, 39) = 39

Multiply the numerator and denominator of the complex fraction by the LCD: $$\frac{\frac{12}{13}\cdot \frac{39}{1}}{\frac{4}{39}\cdot \frac{39}{1}}$$ Simplify: $$\require{cancel}\frac{\frac{12}{\cancel{13}1}\cdot \frac{\cancel{39}3}{1}}{\frac{4}{\cancel{39}1}\cdot \frac{\cancel{39}1}{1}}$$ $$\frac{\cancel{12}3 \cdot 3}{\cancel{4}1}=9$$ Example 5 Simplify each using the LCD Method: $$\frac{\frac{3}{10}+ \frac{6}{15}}{\frac{19}{20}- \frac{2}{3}}$$ First, we find the LCD of all fractions involved. We have 3/10, 6/15, 19/20, and 2/3:

LCD » LCM(3, 10, 15, 20) = 60

Multiply the numerator and denominator of the complex fraction by the LCD: $$\frac{\left(\frac{3}{10}+ \frac{6}{15}\right) \cdot \frac{60}{1}}{\left(\frac{19}{20}- \frac{2}{3}\right) \cdot \frac{60}{1}}$$ We need to use the distributive property here. 60 will be distributed to each term or fraction in the numerator and the denominator: $$\frac{\frac{3}{10}\cdot \frac{60}{1}+ \frac{6}{15}\cdot \frac{60}{1}}{\frac{19}{20}\cdot \frac{60}{1}- \frac{2}{3}\cdot \frac{60}{1}}$$ Simplify: $$\frac{\frac{3}{\cancel{10}1}\cdot \frac{\cancel{60}6}{1}+ \frac{6}{\cancel{15}1}\cdot \frac{\cancel{60}4}{1}}{\frac{19}{\cancel{20}1}\cdot \frac{\cancel{60}3}{1}- \frac{2}{\cancel{3}1}\cdot \frac{\cancel{60}20}{1}}$$ $$\frac{18 + 24}{57 - 40}=\frac{42}{17}=2\frac{8}{17}$$

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