Problem 90 - Intermediate Algebra

Question

Simplify. \(\frac{\frac{1}{y}+\frac{1}{x}}{\frac{1}{y}-\frac{1}{x}}\)

Step-by-Step Solution

Answer

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

Step 1: Identify the expression components

The given expression is a fraction with a numerator and a denominator, both of which themselves are fractions. The numerator is \( \frac{1}{y} + \frac{1}{x} \) and the denominator is \( \frac{1}{y} - \frac{1}{x} \). Our task is to simplify the overall expression.

Step 2: Find a common denominator for numerator terms

For the numerator \( \frac{1}{y} + \frac{1}{x} \), find the least common denominator (LCD). The LCD of \( y \) and \( x \) is \( xy \). Thus, \( \frac{1}{y} + \frac{1}{x} \) becomes \( \frac{x}{xy} + \frac{y}{xy} = \frac{x + y}{xy} \).

Step 3: Find a common denominator for denominator terms

Similarly for the denominator \( \frac{1}{y} - \frac{1}{x} \), use the LCD \( xy \). Rewrite it as \( \frac{x}{xy} - \frac{y}{xy} = \frac{x - y}{xy} \).

Step 4: Simplify the complex fraction

Now substitute back into the original expression: \[ \frac{\frac{x + y}{xy}}{\frac{x - y}{xy}} \]. This complex fraction simplifies by multiplying the numerator \( \frac{x + y}{xy} \) by the reciprocal of the denominator \( \frac{xy}{x - y} \). This gives \[ \left(\frac{x + y}{xy}\right) \times \left(\frac{xy}{x - y}\right) = \frac{x + y}{x - y} \].

Step 5: Confirm simplification and final result

The simplification has reduced the original complex fraction to \( \frac{x + y}{x - y} \). There are no common factors between the numerator and the denominator; thus, \( \frac{x + y}{x - y} \) is the simplest form of the given expression.

Key Concepts

Least Common Denominator (LCD)

When we are simplifying expressions that involve fractions, finding the least common denominator (LCD) is a crucial step. The LCD is the smallest number or expression that can be exactly divided by each of the denominators involved in the given fractions. To find this, we often look at the denominators of the fractions we are dealing with and determine what they can collectively be expanded into.

Here's how it works:

• Identify each denominator in the expression. In this case, the denominators were \( y \) and \( x \).
• Determine the smallest expression that each original denominator can divide without leaving a remainder. Here, both \( y \) and \( x \) can divide \( xy \) completely, making \( xy \) the least common denominator.

Once you have the LCD, you can rewrite each fraction over this common denominator, which drastically simplifies the process of addition or subtraction of fractions within rational expressions.

Numerators and Denominators

Understanding numerators and denominators is fundamental to handling fractions effectively. In any fraction, the top part is called the numerator, and the bottom part is called the denominator. These components have specific roles:

• **Numerator:** Indicates how many parts of the whole we are considering.
• **Denominator:** Tells us into how many parts the whole is divided.

In expressions like our example, both the numerator and the denominator are themselves fractions: \( \frac{1}{y} + \frac{1}{x} \) as the numerator and \( \frac{1}{y} - \frac{1}{x} \) as the denominator. Managing these components requires manipulating both the numerators and denominators individually, then combining them in the final expression.

By discovering a common denominator for both the smaller fractions in the numerator and the denominator, we simplify these individual parts before addressing the fraction as a whole. This careful distinction and handling make the differences between parts clear, enabling straightforward calculation.

Rational Expressions

Rational expressions share similarities with numerical fractions but are made up of algebraic expressions. Simplifying rational expressions can involve several algebraic techniques, including finding the least common denominator, as we did here.

Rational expressions often appear as complex fractions, meaning there are fractions within fractions. The goal is to simplify these fractions to make them manageable. As demonstrated,

• The expression \( \frac{\frac{x + y}{xy}}{\frac{x - y}{xy}} \) was simplified by multiplying the numerator by the reciprocal of the denominator.
• This ensured that complicated divisions were reduced to simpler expressions: \( \frac{x + y}{x - y} \).

When dealing with these expressions, it's essential to keep track of both the operations between fractions and the simplification techniques that tidy up the expressions. Recognizing rational expressions and knowing how to manipulate them efficiently will make solving similar fraction-related problems much easier.