Problem 39 - Intermediate Algebra

Question

Simplify each complex fraction. $$ \frac{1+\frac{x}{y}}{1-\frac{x}{y}} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{y+x}{y-x} \).

Step 1: Identify the Complex Fraction

The given expression is \( \frac{1+\frac{x}{y}}{1-\frac{x}{y}} \). It is a complex fraction because both the numerator and denominator themselves contain fractions.

Step 2: Find the Least Common Denominator (LCD)

The fraction parts of the numerator and the denominator have \( y \) as their denominators. Thus, the Least Common Denominator (LCD) for these fractions is \( y \).

Step 3: Simplify the Numerator and Denominator Separately

Multiply both the numerator and denominator by the LCD \( y \) to eliminate the smaller fractions: - Numerator: \( y(1 + \frac{x}{y}) = y + x \)- Denominator: \( y(1 - \frac{x}{y}) = y - x \)

Step 4: Rewrite the Simplified Expression

After simplifying the numerator and the denominator, the expression becomes \( \frac{y+x}{y-x} \).

Key Concepts

Numerator and Denominator

In any fraction, the top part is known as the numerator, while the bottom part is the denominator. In a simple fraction like \( \frac{3}{4} \), 3 is the numerator and 4 is the denominator. They both play significant roles in how fractions behave. When we talk about complex fractions like \( \frac{1 + \frac{x}{y}}{1 - \frac{x}{y}} \), things get a little more intricate because both the numerator and denominator themselves contain fractions. Understanding Complex FractionsWhen dealing with complex fractions, it's crucial to identify the separate components. Here:

• The numerator is \( 1 + \frac{x}{y} \).
• The denominator is \( 1 - \frac{x}{y} \).

This means each part has its structure, consisting of whole numbers and a fraction. Tackling each separately helps in simplifying the complex fraction.

Least Common Denominator (LCD)

The Least Common Denominator (LCD) is vital when working with fractions that have different denominators. It simplifies adding, subtracting, or comparing fractions, allowing us to write them with the same denominator. In the exercise, both fractions in the complex numerator and denominator share the same denominator of \( y \).Importance of LCD

• Helps eliminate smaller fractions within a complex fraction.
• Makes it easier to simplify or manipulate fractions in calculations.

In our specific problem, the LCD for both the numerator and denominator is \( y \), which aids in clarifying and simplifying the expression.

Simplification Process

The simplification process in complex fractions involves making the fraction as straightforward as possible. In our exercise, once we identify the LCD, which is \( y \), we use it to simplify both the numerator and the denominator.Steps in the Simplification

• Multiply: Multiply both the numerator and the denominator by the LCD to eliminate the smaller fractions within them. For example, the numerator becomes \( y(1 + \frac{x}{y}) = y + x \).
• Simplify: Do the same for the denominator, resulting in \( y(1 - \frac{x}{y}) = y - x \).
• Reformulate: Rewrite the simplified expression as \( \frac{y+x}{y-x} \).

The goal is to make it simpler to work with by reducing it to a simpler expression, eliminating fractional components within the main fraction.