Set up the equation: \( \frac{30 + x}{50 + x} = 0.8 \).

["# How to Set Up and Solve the Equation: ( \frac{30 + x}{50 + x} = 0.8 )", "When working with proportions or rates in algebra, one common task is setting up and solving equations involving fractions. In this article, we’ll focus on how to properly set up the equation ( \frac{30 + x}{50 + x} = 0.8 ), step-by-step, and explain how to solve it accurately. This type of equation often appears in math, economics, and real-world modeling scenarios.", "---", "## Step 1: Identify the Structure of the Equation", "The equation ( \frac{30 + x}{50 + x} = 0.8 ) is a proportional equation where:", "- The numerator ( 30 + x ) represents a linear expression growing with ( x ),\n- The denominator ( 50 + x ) also grows linearly,\n- The right-hand side ( 0.8 ) is a constant ratio between 0 and 1.", "This equation means that the ratio of (30 + x) to (50 + x) equals 0.8 — a common format in rate comparisons.", "---", "## Step 2: Write the Full Setup Equation", "To properly set up the equation, clearly express the ratio as a fraction set equal to the decimal:", "[\n\frac{30 + x}{50 + x} = 0.8\n]", "This setup reflects the problem accurately and prepares you for algebraic manipulation.", "---", "## Step 3: Eliminate the Fraction", "To simplify, multiply both sides by ( 50 + x ), which clears the denominator (assuming ( 50 + x <br/>\ne 0 ), a valid restriction):", "[\n30 + x = 0.8(50 + x)\n]", "This step transforms the equation from a rational expression into a linear equation — easier to solve.", "---", "## Step 4: Expand the Right Side", "Use distributive property on the right:", "[\n30 + x = 0.8 \ imes 50 + 0.8x = 40 + 0.8x\n]", "Now the equation is:", "[\n30 + x = 40 + 0.8x\n]", "---", "## Step 5: Collect Like Terms", "Move all terms involving ( x ) to one side and constants to the other:", "[\nx - 0.8x = 40 - 30\n]", "[\n0.2x = 10\n]", "---", "## Step 6: Solve for ( x )", "Divide both sides by 0.2:", "[\nx = \frac{10}{0.2} = 50\n]", "---", "## Step 7: Verify the Solution", "Substitute ( x = 50 ) back into the original equation:", "Left side:\n[\n\frac{30 + 50}{50 + 50} = \frac{80}{100} = 0.8\n]", "Right side is 0.8 — correct.", "This confirms the solution is accurate.", "---", "## Conclusion", "Setting up the equation ( \frac{30 + x}{50 + x} = 0.8 ) involves clearly expressing the ratio, eliminating the fraction, and solving a simple linear equation. With careful steps—identifying terms, multiplying through, isolating variables—you can solve proportional equations confidently. This structure applies broadly to ratios, mixture problems, and percent-based modeling.", "---", "### Key Takeaways:", "- Always express ratios as fractions.\n- Multiply both sides by the denominator to eliminate fractions.\n- Combine like terms systematically.\n- Verify your solution by plugging back into the original equation.", "Mastering this method helps build strong algebraic skills essential for math, science, and real-life problem solving."]