Simplify each complex fraction. (2 - 2/y) / (2 + 2/y)

Verified step by step guidance

1. Identify the complex fraction: \( \frac{2 - \frac{2}{y}}{2 + \frac{2}{y}} \).
2. Find a common denominator for the terms in the numerator and the denominator. Here, the common denominator is \( y \).
3. Rewrite the numerator: \( 2 - \frac{2}{y} = \frac{2y}{y} - \frac{2}{y} = \frac{2y - 2}{y} \).
4. Rewrite the denominator: \( 2 + \frac{2}{y} = \frac{2y}{y} + \frac{2}{y} = \frac{2y + 2}{y} \).
5. Simplify the complex fraction by dividing the rewritten numerator by the rewritten denominator: \( \frac{\frac{2y - 2}{y}}{\frac{2y + 2}{y}} = \frac{2y - 2}{2y + 2} \).

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 eliminates the inner fractions by finding a common denominator or multiplying through by the least common multiple (LCM) of the denominators involved.

Finding a Common Denominator

Finding a common denominator is essential when adding, subtracting, or simplifying fractions. In the context of complex fractions, it involves determining a common base for the denominators of the inner fractions, allowing for easier manipulation and simplification of the overall expression.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing fractions to their simplest form by canceling common factors in the numerator and denominator. This process is crucial in complex fractions, as it helps to clarify the expression and make calculations more manageable.