The new ratio is \( rac{3x}{2x + 10} = rac{3}{4} \).

["Solving the Equation: The New Ratio $ \dfrac{3x}{2x + 10} = \dfrac{3}{4} $", "Understanding and solving equations involving ratios is a fundamental skill in algebra, and the equation\n$$\n\dfrac{3x}{2x + 10} = \dfrac{3}{4}\n$$\nis a classic example that appears frequently in math coursework and standardized tests. In this article, we’ll walk through step-by-step solving techniques, explore practical applications, and highlight why mastering such problems enhances your mathematical fluency.", "---", "Step 1: Cross-Multiplication to Eliminate Fractions", "To simplify the equation, the most effective immediate step is cross-multiplication. Multiply both the numerator of the left side by 4 and the denominator of the right by (2x + 10):", "$$\n4 \cdot (3x) = 3 \cdot (2x + 10)\n$$", "This gives:", "$$\n12x = 3(2x + 10)\n$$", "---", "Step 2: Expand and Simplify", "Distribute the 3 on the right-hand side:", "$$\n12x = 6x + 30\n$$", "Now subtract (6x) from both sides to isolate terms involving (x):", "$$\n12x - 6x = 30\n$$\n$$\n6x = 30\n$$", "---", "Step 3: Solve for (x)", "Divide both sides by 6:", "$$\nx = \dfrac{30}{6} = 5\n$$", "---", "Step 4: Verification by Substitution", "Always verify your solution by substituting (x = 5) into the original ratio:", "Left side:\n$$\n\dfrac{3(5)}{2(5) + 10} = \dfrac{15}{10 + 10} = \dfrac{15}{20} = \dfrac{3}{4}\n$$\nRight side:\n$$\n\dfrac{3}{4}\n$$\nBoth sides match, confirming that (x = 5) is the correct solution.", "---", "Why This Ratio Equation Matters", "Equations of this form often arise in real-world contexts, such as:\n- Chemistry: balancing ratios in chemical reactions\n- Economics: analyzing cost-to-revenue relationships\n- Physics: comparing rates and proportions\n- Test prepping: standardized exams regularly test ratio and proportion skills", "Solving such problems improves logical reasoning and algebraic manipulation—essential competencies for STEM fields and beyond.", "---", "Summary of Key Steps:", "1. Cross-multiply: ( 3x \cdot 4 = 2x + 10 \cdot 3 )\n2. Simplify both sides to: ( 12x = 6x + 30 )\n3. Subtract (6x): (6x = 30)\n4. Solve: (x = 5)\n5. Verify by plugging back into the original equation", "---", "Practice Problems to Master the Concept", "To reinforce your understanding, try solving similar equations:\n- ( \dfrac{2x}{3x + 6} = \dfrac{2}{5} )\n- ( \dfrac{5x}{x - 4} = \dfrac{10}{3} )\n- ( \dfrac{x + 2}{2x - 1} = \dfrac{1}{3} )", "---", "Conclusion", "The equation ( \dfrac{3x}{2x + 10} = \dfrac{3}{4} ) is not just an abstract math problem— It’s a gateway to developing analytical thinking and problem-solving abilities. With precise algebraic steps and verification, you can confidently solve these ratios. Keep practicing, and watch your math skills grow!", "---", "Keywords: algebra, solving equations, ratio equation, ( \dfrac{3x}{2x+10} = \dfrac{3}{4} ), math tutorial, solving for (x), equation solving, middle school math, high school algebra.", "---", "Ready to tackle more equations? Explore advanced techniques and tips on mastering algebra ratios!"]