Problem 24 - Elementary and Intermediate Algebra

Question

Simplify each complex fraction. See Examples 2 or \(4 .\) $$ \frac{\frac{2}{x}-\frac{1}{3}}{\frac{2}{3}+\frac{x}{5}} $$

Step-by-Step Solution

Answer

The simplified fraction is \( \frac{5(6-x)}{x(3x+10)} \).

Step 1: Identify the Complex Fraction

We have a complex fraction: \(\frac{\frac{2}{x}-\frac{1}{3}}{\frac{2}{3}+\frac{x}{5}} \).This fraction is complex because the numerator and denominator each contain fractions. The task is to simplify this into a single, non-complex fraction.

Step 2: Simplify the Numerator

The numerator is \( \frac{2}{x} - \frac{1}{3} \). To simplify that, find a common denominator:The least common denominator (LCD) of \(x\) and \(3\) is \(3x\). Rewriting each term:\[ \frac{2}{x} = \frac{2 \cdot 3}{3x} = \frac{6}{3x} \]\[ \frac{1}{3} = \frac{1 \cdot x}{3x} = \frac{x}{3x} \]Now, subtract:\[ \frac{6}{3x} - \frac{x}{3x} = \frac{6 - x}{3x} \].This gives the simplified numerator: \( \frac{6-x}{3x} \).

Step 3: Simplify the Denominator

The denominator is \( \frac{2}{3} + \frac{x}{5} \). Find a common denominator:The least common denominator (LCD) of \(3\) and \(5\) is \(15\). Rewrite each term:\[ \frac{2}{3} = \frac{2 \cdot 5}{15} = \frac{10}{15} \]\[ \frac{x}{5} = \frac{x \cdot 3}{15} = \frac{3x}{15} \]Add the terms:\[ \frac{10}{15} + \frac{3x}{15} = \frac{10 + 3x}{15} \].This gives the simplified denominator: \( \frac{10 + 3x}{15} \).

Step 4: Divide the Fractions

The entire fraction is: \[ \frac{\frac{6-x}{3x}}{\frac{10+3x}{15}} \].To divide by a fraction, multiply by its reciprocal:\[ \frac{6-x}{3x} \times \frac{15}{10+3x} \].Multiply across:\[ \frac{(6-x) \cdot 15}{3x \cdot (10+3x)} \].Simplify the expression by cancelling common factors if possible.

Step 5: Simplify the Expression

Distribute and simplify:Numerator:\[ (6 - x) \cdot 15 = 90 - 15x \].Denominator:\[ 3x \cdot (10 + 3x) = 30x + 9x^2 \].Thus, the fraction becomes:\[ \frac{90 - 15x}{9x^2 + 30x} \].Factor out the greatest common factor in the numerator and the denominator:The GCF of the numerator (\(90 - 15x\)) is 15, and the GCF of the denominator (\(9x^2 + 30x\)) is 3x.Factor these:\[ \frac{15(6-x)}{3x(3x + 10)} \].Simplify by cancelling common factors:\[ \frac{5(6-x)}{x(3x+10)} \].

Step 6: Conclusion: Final Result

The simplified form of the complex fraction \( \frac{\frac{2}{x}-\frac{1}{3}}{\frac{2}{3}+\frac{x}{5}} \) is \( \frac{5(6-x)}{x(3x+10)} \).

Key Concepts

Simplifying Fractions

Simplifying fractions is the process of rewriting a fraction in its simplest form. When dealing with complex fractions, which are fractions that have a fraction in the numerator, denominator, or both, simplifying can make them easier to understand and work with. For example, consider the fraction \( \frac{6-x}{3x} \). To ensure it's in its simplest form, we should look for any common factors that might be canceled out.

• In the given example, there are no common factors between the terms in the numerator \(6-x\) and the denominator \(3x\), so it is already simplified.

Simplifying fractions ensures that the result is more manageable and less prone to errors in further calculations.

Least Common Denominator

The least common denominator (LCD) is a key concept in fraction operations. It refers to the smallest number that each denominator can divide into without leaving a remainder. Finding the LCD is essential when adding, subtracting, or comparing fractions because it allows for the fractions to be expressed with a common denominator.In our original problem, we had to find the LCD for both the numerator and denominator of the complex fraction to simplify it:

• For the numerator \( \frac{2}{x} - \frac{1}{3} \), the LCD of \(x\) and \(3\) is \(3x\).
• For the denominator \( \frac{2}{3} + \frac{x}{5} \), the LCD of \(3\) and \(5\) is \(15\).

After rewriting each term with the LCD, the fraction expressions can be combined through addition or subtraction. This step is critical in simplifying complex fractions effectively.

Fraction Operations

Operations involving fractions include addition, subtraction, multiplication, and division. These operations are foundational to working with complex fractions.To divide a complex fraction, such as \( \frac{\frac{6-x}{3x}}{\frac{10+3x}{15}} \), you perform a critical operation: multiplying by the reciprocal. Instead of dividing, we flip the second fraction and change the operation to multiplication:

• Change \( \frac{10+3x}{15} \) to its reciprocal: \( \frac{15}{10+3x} \).
• Now multiply: \( \frac{6-x}{3x} \times \frac{15}{10+3x} \).

This transforms the operation into a simpler multiplication task. Then, each fraction's numerators and denominators are multiplied together, resulting in a single fraction that can be simplified further. Understanding these fundamental operations enables you to handle complex fractions correctly and efficiently.