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 denominator separately. Here, the common denominator is \(y\).
3. Rewrite the numerator as a single fraction: \(2 - \frac{2}{y} = \frac{2y}{y} - \frac{2}{y} = \frac{2y - 2}{y}\).
4. Rewrite the denominator as a single fraction: \(2 + \frac{2}{y} = \frac{2y}{y} + \frac{2}{y} = \frac{2y + 2}{y}\).
5. Now, divide the two fractions: \(\frac{\frac{2y - 2}{y}}{\frac{2y + 2}{y}}\). When dividing fractions, multiply the numerator by the reciprocal of the denominator: \(\frac{2y - 2}{y} \times \frac{y}{2y + 2}\). Then simplify by canceling common factors.

Key Concepts

Complex Fractions

A complex fraction is a fraction where the numerator, denominator, or both contain fractions themselves. Simplifying complex fractions involves rewriting them as a single simple fraction by finding common denominators or multiplying numerator and denominator by the least common denominator.

Finding a Common Denominator

When fractions appear within fractions, identifying a common denominator helps combine terms. This step allows you to rewrite each part as a single fraction, making it easier to perform division or simplification.

Division of Fractions

Dividing fractions involves multiplying the first fraction by the reciprocal of the second. This principle is essential when simplifying complex fractions, as the overall expression is a division of two fractional expressions.