Lesson 3.8
Complex Fractions
A complex fraction is a fraction that has fractions in its numerator, denominator, or both — a fraction within a fraction. There are two clean methods to simplify them.
Introduction
Complex fractions look intimidating, but they're just a division problem in disguise. The "big fraction bar" means divide the top by the bottom. We have two strategies to clean them up.
Past Knowledge
All rational expression operations: adding (3.6-3.7), multiplying (3.3), dividing (3.4), and LCD (3.5).
Today's Goal
Simplify complex fractions using the LCD method or the division method.
Future Success
Complex fractions appear in calculus (difference quotients) and in many real-world rate problems.
Key Concepts
Method 1: LCD Multiply
Multiply every term in the numerator and denominator by the LCD of all the "little" fractions.
Find the LCD of ALL mini-fractions
Multiply every term by the LCD
The mini-fractions disappear!
Simplify what remains
Method 2: Rewrite as Division
Simplify the numerator into one fraction, simplify the denominator into one fraction, then divide.
Combine the numerator into a single fraction
Combine the denominator into a single fraction
Divide: Keep-Change-Flip
Cancel and simplify
Worked Examples
Example 1: LCD Method
BasicSimplify .
LCD of all mini-fractions =
Multiply every term by
Example 2: Division Method
IntermediateSimplify .
Rewrite as division
Keep-Change-Flip
Cancel and one
Example 3: Mixed Terms
AdvancedSimplify .
LCD of all mini-fractions =
Multiply every term by
Common Pitfalls
Multiplying Only Part of the Fraction
When using the LCD method, you must multiply every single term in both the big numerator and big denominator by the LCD.
Missing Non-Fraction Terms
In , the "1" is also a term! It gets multiplied by the LCD too: .
Real-Life Applications
In calculus, the difference quotient often produces complex fractions when is a rational function. Mastering simplification of complex fractions is essential preparation for derivatives.
Practice Quiz
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