Problem 84 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

\( \frac{x+y}{x-y} \)

Step 1: Identify the complex fraction

We are given a complex fraction: \( \frac{\frac{1}{y}+\frac{1}{x}}{\frac{1}{y}-\frac{1}{x}} \). The fraction in the numerator is \( \frac{1}{y}+\frac{1}{x} \), and the fraction in the denominator is \( \frac{1}{y}-\frac{1}{x} \). These individual fractions need to be simplified.

Step 2: Obtain a common denominator for the numerator

To simplify the numerator \( \frac{1}{y} + \frac{1}{x} \), find a common denominator. The least common denominator of \( y \) and \( x \) is \( xy \). Rewriting, we have \( \frac{x}{xy} + \frac{y}{xy} = \frac{x+y}{xy} \).

Step 3: Obtain a common denominator for the denominator

Similarly, for the denominator \( \frac{1}{y} - \frac{1}{x} \), the least common denominator is \( xy \). Rewriting the expression gives \( \frac{x}{xy} - \frac{y}{xy} = \frac{x-y}{xy} \).

Step 4: Simplify the complex fraction

The complex fraction now becomes:\[\frac{\frac{x+y}{xy}}{\frac{x-y}{xy}}\]Since both the numerator and the denominator have the same denominator \( xy \), they can be canceled out, giving:\[\frac{x+y}{x-y}\].

Step 5: Final result

After canceling the common denominators, the simplified form of the given complex fraction is \( \frac{x+y}{x-y} \).

Key Concepts

Understanding Complex Fractions

Complex fractions are fractions where the numerator, denominator, or both contain fractions themselves. These can appear tricky at first glance but are manageable with the right approach. In our example, the complex fraction is \( \frac{\frac{1}{y}+\frac{1}{x}}{\frac{1}{y}-\frac{1}{x}} \). Here, the numerator and the denominator themselves contain fractions.
Dealing with complex fractions involves working through a series of steps to simplify them into something more understandable. This idea revolves around turning the complex into simpler forms using fraction operations and finding common denominators. Often, once simplified, you'll see a familiar or easier fraction form emerge.
Remember, solving complex fractions may initially appear daunting, but breaking them down into smaller, simpler steps makes them more straightforward.

Finding the Least Common Denominator

To simplify fractions within a complex fraction, the first thing we need is a common denominator. This is crucial for combining or separating fractions. The least common denominator (LCD) of fractions is the smallest number that both denominators multiply into without leaving a remainder.
In our example:

• The fractions are \( \frac{1}{y} \) and \( \frac{1}{x} \).
• The least common denominator is \( xy \), because it is the smallest expression that both \( y \) and \( x \) divide evenly into.

Once you find the LCD, rewrite each fraction so they share this same denominator. This step is crucial because it allows you to effectively perform operations such as addition or subtraction inside the complex fraction.

Simplification Process

Simplification involves reducing a complex fraction into its simplest form. This process makes the fraction easier to work with and understand. Once common denominators are established, the simplification process continues by performing arithmetic operations.
For the numerator \( \frac{1}{y} + \frac{1}{x} \), we

• Rewrite as \( \frac{x}{xy} + \frac{y}{xy} \).
• This combines to \( \frac{x + y}{xy} \).

For the denominator \( \frac{1}{y} - \frac{1}{x} \):

• Rewrite as \( \frac{x}{xy} - \frac{y}{xy} \).
• This simplifies to \( \frac{x-y}{xy} \).

The next step is to cancel out the common denominator \( xy \) from the overall fraction, simplifying the expression to \( \frac{x+y}{x-y} \). This is an example of how powerful simplification steps can vastly change the appearance of a mathematical expression.

Basic Fraction Operations

Mastering basic fraction operations is key to manipulating and simplifying not just complex fractions, but all fractions. Common operations include addition, subtraction, multiplication, and division. Each has unique requirements for combining fractions.

• **Addition/Subtraction**: Requires a common denominator.
• **Multiplication**: Simplify across numerators and denominators directly.
• **Division**: Involves multiplying by the reciprocal of the second fraction.

In our context, we performed addition and subtraction within the context of the complex fraction, all while maintaining a common denominator. It's also important to not overlook division at the final step of simplification, which in effect was simplified through canceling the common denominator. With practice, these operations become second nature, helping you tackle varied fraction problems with confidence.