\[ \frac{60 + x}{200 + x} = 0.5 \]

["Solving the Equation: (\frac{60 + x}{200 + x} = 0.5)", "Finding the value of (x) in the equation (\frac{60 + x}{200 + x} = 0.5) is a common mathematical challenge that appears across algebra courses, standardized tests, and real-world applications. Whether you’re a student, educator, or professional, understanding how to solve this equation not only strengthens your algebraic skills but also provides insight into practical problem-solving.", "### Understanding the Equation", "The equation (\frac{60 + x}{200 + x} = 0.5) represents a proportion where a linear numerator is divided by a linear denominator, equaling 0.5 — a key input value in many proportional relationships.", "### Step-by-Step Solution", "To solve for (x), follow these algebraic steps:", "1. Start with the equation:", "[\n \frac{60 + x}{200 + x} = 0.5\n ]", "2. Multiply both sides by ((200 + x)) to eliminate the denominator:", "[\n 60 + x = 0.5(200 + x)\n ]", "3. Expand the right-hand side:", "[\n 60 + x = 100 + 0.5x\n ]", "4. Subtract (0.5x) from both sides:", "[\n 60 + x - 0.5x = 100\n ]", "[\n 60 + 0.5x = 100\n ]", "5. Subtract 60 from both sides:", "[\n 0.5x = 40\n ]", "6. Divide by 0.5:", "[\n x = \frac{40}{0.5} = 80\n ]", "### Verification", "Plugging (x = 80) back into the original equation confirms the solution:", "[\n\frac{60 + 80}{200 + 80} = \frac{140}{280} = 0.5\n]", "### Real-World Applications", "Equations like this arise in finance (e.g., determining break-even points), physics (ratio of distances or proportions), and data analysis (scaling factors). Solving for (x) helps model these scenarios mathematically.", "### Alternative Methods", "While algebraic manipulation is direct and reliable, you can also solve by cross-multiplication:", "[\n0.5(200 + x) = 60 + x \quad \Rightarrow \quad 100 + 0.5x = 60 + x\n]", "Then proceed identically to reach (x = 80).", "### Conclusion", "The equation (\frac{60 + x}{200 + x} = 0.5) demonstrates how proportion problems can be systematically solved using basic algebra. Mastering these steps builds a strong foundation for tackling more complex equations and real-life quantitative challenges. If you're interested in similar problems or need further practice, continue exploring ratio equations and proportional reasoning — skills invaluable across STEM fields.", "---", "Keywords: solve (\frac{60 + x}{200 + x} = 0.5), algebra equation solution, equation solving, proportion equation, step-by-step math tutorial, real-world math applications, linear equations."]