Problem 14 - Elementary and Intermediate Algebra

Question

Simplify each complex fraction. See Example \(1 .\) $$ \frac{\frac{y}{x}}{\frac{x}{x y}} $$

Step-by-Step Solution

Answer

The simplified form of the complex fraction is \( \frac{y^2}{x} \).

Step 1: Understand the Problem

We have a complex fraction \( \frac{\frac{y}{x}}{\frac{x}{x y}} \). This means we need to simplify a fraction that is made up of other fractions. We need to simplify it to a single fraction.

Step 2: Rewrite the Complex Fraction

Change the division of fractions to multiplication of reciprocal. The complex fraction \( \frac{\frac{y}{x}}{\frac{x}{x y}} \) can be rewritten as \( \frac{y}{x} \times \frac{xy}{x} \).

Step 3: Multiply the Fractions

Multiply the fractions: \( \frac{y}{x} \times \frac{xy}{x} = \frac{y \cdot xy}{x \cdot x} \).

Step 4: Simplify the Multiplication

Calculate the multiplication: \( \frac{y \cdot xy}{x \cdot x} = \frac{y^2 \cdot x}{x^2} \).

Step 5: Simplify the Resulting Fraction

Cancel common terms from numerator and denominator. Here, \( x \) in the numerator and denominator cancels out, giving \( \frac{y^2}{x} \).

Key Concepts

Simplifying Fractions

Understanding how to simplify fractions is a crucial skill in algebra and arithmetic. A fraction consists of a numerator (the top part) and a denominator (the bottom part). Simplifying means to reduce the fraction to its simplest form. This involves making the numerator and the denominator as small as possible while still maintaining the same value. To simplify, follow these steps:

• Find the greatest common divisor (GCD) of both the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.
• Keep in mind that the fraction should still represent the same number, just in the simplest form.

Breaking down complex fractions involves additional steps as they are made up of fractions within fractions. The general rule is to change the division into multiplication by the reciprocal, thereby transforming them into simpler fractions that can readily be multiplied and reduced. This process requires careful attention to common factors to ensure completeness in simplification. Once identified, cancel these common terms to arrive at the optimal form. This approach not only clarifies the relationships between numerical components but also facilitates easier calculation using simpler forms.

Multiplication of Fractions

Knowing how to multiply fractions is vital, especially when dealing with complex fractions. It allows you to take the operation from a complex form to a manageable one. When multiplying fractions, the process is straightforward:

• Multiply the numerators of the fractions together to get the new numerator.
• Multiply the denominators of the fractions together to get the new denominator.

Unlike addition and subtraction, multiplication of fractions does not require a common denominator. This makes the multiplication process simpler and often quicker.Let's take a practical viewpoint from the exercise problem: you start with a complex fraction \[ \frac{\frac{y}{x}}{\frac{x}{xy}} \]which can be transformed into a multiplication \[ \frac{y}{x} \times \frac{xy}{x} \].Multiplying these, you focus on multiplying straight across to combine the fractions effectively. Once multiplication is completed, simplifying follows, which helps in reducing the product to the most concise possible form.

Algebraic Expressions

Algebraic expressions involve numbers, variables (letters representing numbers), and operations (such as addition or multiplication). Variables are the foundation of algebra, enabling the representation of generalized numbers. Understanding how these components work together is essential for simplifying expressions and solving equations.In the context of simplifying complex fractions, an expression can include variable terms as in the original problem \( \frac{y}{x} \) and \( \frac{x}{xy} \).These are expressions that contain relationships among variables and need to be managed using algebraic rules to simplify.To work with expressions, remember:

• Combine like terms when adding or subtracting expressions.
• Follow the order of operations (parentheses, exponents, multiplication and division, addition and subtraction).
• Factor expressions when possible to simplify or solve equations.

Simplifying these algebraic expressions in complex fractions typically involves identifying common factors and canceling them out, thereby providing a cleaner, more interpretable expression. This process not only makes future calculations easier but also reinforces the fundamental rules of algebra, making problem-solving intuitive and effective.