Problem 32 - Elementary and Intermediate Algebra

Question

Simplify each complex fraction. See Examples 3 or \(5 .\) $$ \frac{\frac{1}{y}+3}{\frac{3}{y}-2} $$

Step-by-Step Solution

Answer

The simplified fraction is \( \frac{1 + 3y}{3 - 2y} \).

Step 1: Identify the Complex Fraction

You are given a complex fraction \( \frac{\frac{1}{y}+3}{\frac{3}{y}-2} \). A complex fraction has fractions in the numerator and/or the denominator.

Step 2: Find a Common Denominator for the Numerator

In the numerator, you have \( \frac{1}{y} + 3 \). Rewrite 3 as \( \frac{3y}{y} \) to have a common denominator: \( \frac{1}{y} + \frac{3y}{y} = \frac{1 + 3y}{y} \).

Step 3: Simplify the Denominator

In the denominator, you have \( \frac{3}{y} - 2 \). Rewrite 2 as \( \frac{2y}{y} \) so that the terms have a common denominator: \( \frac{3}{y} - \frac{2y}{y} = \frac{3 - 2y}{y} \).

Step 4: Rewrite the Complex Fraction

Substitute the simplified expressions back into the complex fraction: \( \frac{\frac{1 + 3y}{y}}{\frac{3 - 2y}{y}} \).

Step 5: Simplify by Cancelling the Common Denominator

The \( y \) in the numerator and the denominator cancels out: \( \frac{1 + 3y}{3 - 2y} \). The expression simplifies to this result.

Key Concepts

Simplifying Fractions

Learning how to simplify fractions is an essential skill in algebra. It helps you transform complex expressions into simpler forms. When simplifying fractions, the goal is to make the fraction as easy to understand as possible by eliminating any unnecessary parts. This often involves reducing the fraction to its simplest form.

• **Finding the Greatest Common Divisor (GCD)**: This is the largest number that can divide both the numerator and the denominator without leaving a remainder. Once you identify the GCD, you divide both the numerator and the denominator by this number to simplify the fraction.
• **Canceling Common Terms**: In algebraic expressions, you can also simplify by canceling out terms that appear in both the numerator and denominator.
• **Simplifying Complex Fractions**: As seen in the exercise, simplifying complex fractions involves using common denominators to combine terms and reduce the expression to its simplest form.

In the exercise at hand, we simplified complex fractions by canceling out the common denominator after finding equivalent expressions for both the numerator and the denominator. This resulted in a much cleaner and simplified fraction.

Common Denominator

A common denominator is crucial when you're dealing with addition or subtraction of fractions. It allows you to combine the fractions into a single, simpler fraction.
To find a common denominator for two or more fractions:

• List the denominators of the fractions you are working with.
• Determine the least common multiple (LCM) of those denominators. This becomes your common denominator.
• Rewrite each fraction with the common denominator by multiplying the numerator and denominator by the same factor to ensure the fraction retains its original value.

In the given problem, the terms in both the numerator and the denominator needed common denominators to be combined efficiently. By rewriting integers as fractions with the original denominators, simplifying by finding common denominators became straightforward, helping the complex fraction reduce neatly.

Algebraic Expressions

Algebraic expressions are a central part of algebra and involve a combination of numbers, variables, and arithmetic operators.
An algebraic expression can represent real-life situations and help solve problems. In simplifying complex fractions, you often deal with algebraic expressions.
To work with algebraic expressions effectively:

• **Identify Variables and Constants**: Be clear about which symbols represent variables and which represent fixed numbers.
• **Use Arithmetic Operations**: Combine like terms and use arithmetic operations to simplify expressions.
• **Apply Algebraic Properties**: Use properties such as distributive, associative, and commutative laws to simplify and rearrange terms for easier manipulation.

In the example exercise, you had to manage algebraic expressions while simplifying the complex fraction. The expression involved variables like \( y \), which required rewriting terms to ensure they shared a common denominator, aiding in simplifying the expression to its lowest form.