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Magazine Chapter 7: Problem 24
Simplify each complex fraction. See Examples 2 or \(4 .\) $$ \frac{\frac{2}{x}-\frac{1}{3}}{\frac{2}{3}+\frac{x}{5}} $$
Short Answer
Expert verified
The simplified fraction is \( \frac{5(6-x)}{x(3x+10)} \).
Step by step solution
01
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.
02
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: \( \frac{6-x}{3x} \).
03
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: \( \frac{10 + 3x}{15} \).
04
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.
05
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:\[ \frac{5(6-x)}{x(3x+10)} \].
06
Conclusion: Final Result
The simplified form of the complex fraction \( \frac{\frac{2}{x}-\frac{1}{3}}{\frac{2}{3}+\frac{x}{5}} \) is \( \frac{5(6-x)}{x(3x+10)} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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 \(15\).
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: \( \frac{6-x}{3x} \times \frac{15}{10+3x} \).
