Problem 52 - Algebra for College Students

Question

Simplify each complex fraction. $$ \frac{1+\frac{3}{x}}{1-\frac{6}{x}} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{x+3}{x-6} \).

Step 1: Identify the Complex Fraction

The given expression is \( \frac{1+\frac{3}{x}}{1-\frac{6}{x}} \). It is a complex fraction because it contains fractions in the numerator and the denominator. Our goal is to simplify it to a single fraction.

Step 2: Multiply by the Reciprocal

To simplify, multiply the numerator and the denominator by the least common denominator (LCD) of the smaller fractions, which is \( x \), to eliminate them:\[\frac{1+\frac{3}{x}}{1-\frac{6}{x}} \times \frac{x}{x} = \frac{x(1+\frac{3}{x})}{x(1-\frac{6}{x})}.\] This results in:\[\frac{x+3}{x-6}.\]

Step 3: Simplified Fraction Result

Now, the fraction \( \frac{x+3}{x-6} \) is the simplified form of the original complex fraction. Check for any further simplification, such as common factors, which in this case, there are none.

Key Concepts

Simplifying Fractions

Simplifying fractions is a crucial skill in mathematics, especially when dealing with complex expressions. A complex fraction, like \( \frac{1+\frac{3}{x}}{1-\frac{6}{x}} \), contains a fraction within its numerator, denominator, or both. Simplifying these fractions involves a few clear steps to transform them into a more manageable form. The main aim is to eliminate the smaller fractions within the complex fraction so it becomes a single, simpler fraction.

• **Identify Fractions:** Notice that both the numerator and the denominator have a fraction, \( \frac{3}{x} \) and \( \frac{6}{x} \), respectively.
• **Eliminate Fractions:** To simplify, aim to remove these smaller fractions by multiplying the entire complex fraction by a common term that cancels them out.

In this context, always look for opportunities to combine like terms or cancel factors, which could further simplify your fraction even after eliminating the smaller fractions.

Least Common Denominator

The least common denominator (LCD) is the smallest number that each of the denominators in your expression can divide into without leaving a remainder. In simplifying complex fractions, like \( \frac{1+\frac{3}{x}}{1-\frac{6}{x}} \), the LCD is essential because it allows us to remove the fractions within the complex fraction.

• **Identify the LCD:** Look at the denominators of the smaller fractions within the complex fraction. Here, \( x \) is the denominator for both \( \frac{3}{x} \) and \( \frac{6}{x} \), so \( x \) is the LCD.
• **Use the LCD:** Multiply both the numerator and the denominator of the complex fraction by this LCD. It helps transform the complex fraction into a single fraction by eliminating the smaller fractions.

By applying the LCD, you simplify the process of handling fractions within fractions, making it possible to express them in a single, straightforward form.

Reciprocal of a Fraction

The reciprocal of a fraction is simply the inverse of that fraction. If you have a fraction \( \frac{a}{b} \), its reciprocal would be \( \frac{b}{a} \). This concept is particularly useful when simplifying complex fractions, as it assists in canceling out unwanted terms.

• **Reciprocal Role in Simplification:** When you multiply a number by its reciprocal, the result is 1. Therefore, by multiplying by the reciprocal, you often simplify without changing the value of the expression.
• **Use in the Given Problem:** In the context of \( \frac{1+\frac{3}{x}}{1-\frac{6}{x}} \), multiplying the numerator and denominator by the reciprocal of their common terms (here it's part of multiplying by the LCD) helps to simplify the expression significantly.

Understanding reciprocals can make the process of simplifying complex fractions far smoother, as it provides a clear pathway to eliminate unwanted divisions within an expression.