Simplify each complex fraction. [ 1/(y+3) - 1/y ] / (1/y)

Verified step by step guidance

1. Identify the complex fraction: \(\frac{\frac{1}{y+3} - \frac{1}{y}}{\frac{1}{y}}\).
2. Simplify the numerator by finding a common denominator for the two fractions inside it. The common denominator is \(y(y+3)\), so rewrite each fraction as \(\frac{y}{y(y+3)} - \frac{y+3}{y(y+3)}\).
3. Combine the fractions in the numerator: \(\frac{y - (y+3)}{y(y+3)}\).
4. Simplify the numerator inside the fraction: \(y - (y+3) = y - y - 3 = -3\), so the numerator becomes \(\frac{-3}{y(y+3)}\).
5. Now divide this result by the denominator \(\frac{1}{y}\), which is equivalent to multiplying by its reciprocal: \(\frac{-3}{y(y+3)} \times \frac{y}{1}\). 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 by the reciprocal.

Common Denominator

When subtracting or adding fractions, a common denominator is needed to combine them. This involves finding the least common denominator (LCD) so the fractions can be expressed with the same denominator, allowing straightforward addition or subtraction.

Multiplying by the Reciprocal

Dividing by a fraction is equivalent to multiplying by its reciprocal. To simplify a complex fraction, you multiply the numerator by the reciprocal of the denominator, which helps eliminate the complex fraction structure and simplifies the expression.