Problem 36 - Elementary and Intermediate Algebra

Question

Simplify each complex fraction. See Examples 3 or \(5 .\) $$ \frac{\frac{y}{x}+3 y}{y+\frac{2 y}{x}} $$

Step-by-Step Solution

Answer

The simplified form of the complex fraction is \( \frac{1 + 3x}{x + 2} \).

Step 1: Combine the Fractions in the Numerator

First, combine terms in the numerator \( \frac{y}{x} + 3y \). Notice that both terms have a common denominator of \( x \). Rewrite \( 3y \) as \( \frac{3yx}{x} \). Thus, the new form of the numerator is \( \frac{y + 3yx}{x} = \frac{y(1 + 3x)}{x} \).

Step 2: Combine the Fractions in the Denominator

Now, simplify the denominator \( y + \frac{2y}{x} \). Here, \( y \) can be rewritten as \( \frac{yx}{x} \). Thus, the expression becomes \( \frac{yx + 2y}{x} = \frac{y(x + 2)}{x} \).

Step 3: Rewrite the Complex Fraction

The original complex fraction can now be rewritten with the new numerator and denominator as follows: \[ \frac{\frac{y(1 + 3x)}{x}}{\frac{y(x + 2)}{x}} \].

Step 4: Simplify the Complex Fraction

To simplify \( \frac{\frac{y(1 + 3x)}{x}}{\frac{y(x + 2)}{x}} \), multiply by the reciprocal of the denominator: \[ \frac{y(1 + 3x)}{x} \cdot \frac{x}{y(x + 2)} \].

Step 5: Cancel Common Terms

Upon multiplication, the \( x \) terms cancel out, so do the \( y \) terms, leaving the simplified form \[ \frac{1 + 3x}{x + 2} \].

Key Concepts

Complex Fractions

Complex fractions are fractions where either the numerator, the denominator, or both, are themselves fractions. They can seem daunting at first, but with a bit of practice, they become much easier to handle. Imagine they are like two layers of fractions stacked on top of each other.

• Numerator might contain a fraction.
• Denominator might contain a fraction.
• Both can contain fractions.

To simplify a complex fraction, you want to get rid of the smaller fractions inside first. It's like organizing the chaos step-by-step. In this process, you'll often work to create a single fraction in the numerator and another single fraction in the denominator. Once simplified, you can treat the entire setup like a simple division problem - the fraction on top divided by the fraction on the bottom.

Fraction Simplification

Fraction simplification involves making a fraction as simple as possible. The goal is to have the smallest possible numbers, making the fraction easier to understand and work with. It's like tidying up your room so you can find things more easily.
When simplifying fractions:

• Identify common factors in the numerator and denominator.
• Rewrite terms so they share a common denominator, which allows for easy addition or subtraction.
• Cancel out any terms that appear in both the numerator and the denominator.

By simplifying fractions early on, you avoid dealing with cumbersome numbers later in the problem. It streamlines your calculations and saves you energy, just like organizing your workspace helps you focus better on the task at hand.In terms of our exercise, the simplification process allowed for easy cancellation of common terms, eventually breaking down the complex fraction into a simple expression \( \frac{1 + 3x}{x + 2} \).

Intermediate Algebra

Intermediate algebra is a key step in your math journey, bridging the basics you first learn to the more advanced concepts you'll tackle later. This level covers a range of topics including expressions, equations, functions, and their respective operations. It's like the transporter moving you from math foundations to complex problem-solving worlds.

• Understanding how to manipulate expressions.
• Learning to solve various types of equations, including quadratic and rational ones.
• Exploring functions and their properties.

In the context of the given exercise, intermediate algebra skills include correctly identifying and combining fractions, as well as multiplying them correctly. It tests your ability to identify patterns and simplify expressions using algebraic techniques, which is crucial for solving more intricate problems. This complexity combo pack prepares you for even higher mathematics by honing your ability to transform, simplify, and manipulate expressions effectively.