Simplify each complex fraction. See Examples 5 and 6. y r —— x r

Verified step by step guidance

1. Identify the complex fraction: \( \frac{\frac{y}{r}}{\frac{x}{r}} \).
2. Recognize that dividing by a fraction is equivalent to multiplying by its reciprocal.
3. Rewrite the complex fraction as a multiplication: \( \frac{y}{r} \times \frac{r}{x} \).
4. Cancel out the common factor \( r \) in the numerator and the denominator.
5. Simplify the expression to \( \frac{y}{x} \).

Key Concepts

Complex Fractions

A complex fraction is a fraction where the numerator, the denominator, or both contain fractions themselves. To simplify complex fractions, one typically finds a common denominator for the inner fractions and then simplifies the overall expression. This process often involves multiplying both the numerator and denominator by the least common denominator (LCD) to eliminate the inner fractions.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. This can be achieved by dividing both the numerator and denominator by their greatest common divisor (GCD). In the context of complex fractions, simplifying the inner fractions first is crucial before addressing the overall fraction.

Algebraic Manipulation

Algebraic manipulation refers to the techniques used to rearrange and simplify algebraic expressions. This includes operations such as factoring, distributing, and combining like terms. In the case of complex fractions, effective algebraic manipulation is essential for isolating terms and simplifying the expression, ensuring that the final result is clear and concise.