Section 9.4: Complex Fractions
A complex fraction has a fraction in its numerator, denominator, or both. To simplify, multiply numerator and denominator by the least common denominator (LCD) of all small fractions.
Example 1
Simplify: \( \dfrac{\tfrac{1}{x}}{\tfrac{2}{x}} \)
\( \dfrac{1/x}{2/x} = \dfrac{1}{x} \times \dfrac{x}{2} = \dfrac{1}{2} \), restriction: \( x \neq 0 \).
Example 2
Simplify: \( \dfrac{\tfrac{3}{y} + \tfrac{2}{y+1}}{\tfrac{1}{y(y+1)}} \)
Numerator: \( \dfrac{3(y+1)+2y}{y(y+1)} = \dfrac{5y+3}{y(y+1)} \).
Denominator: \( \dfrac{1}{y(y+1)} \).
Divide: \( \dfrac{\tfrac{5y+3}{y(y+1)}}{\tfrac{1}{y(y+1)}} = (5y+3) \).
Restrictions: \( y \neq 0, -1 \).
Example 3
Simplify: \( \dfrac{\tfrac{x}{x+2}}{\tfrac{3}{x-2}} \)
\( \dfrac{x}{x+2} \times \dfrac{x-2}{3} = \dfrac{x(x-2)}{3(x+2)} \).
Restrictions: \( x \neq -2, 2 \).
Practice Problems
- \( \dfrac{\tfrac{4}{a}}{\tfrac{2}{a}} \)
- \( \dfrac{\tfrac{5}{x+1}}{\tfrac{1}{x}} \)
- \( \dfrac{\tfrac{2}{m} + \tfrac{3}{m+1}}{\tfrac{1}{m(m+1)}} \)
- \( \dfrac{\tfrac{y}{y-1}}{\tfrac{2}{y+1}} \)
- \( \dfrac{\tfrac{x+3}{x}}{\tfrac{4}{x+2}} \)