The total time is \(\frac{120}{v} + \frac{180}{v+20} = 5\).

["# How to Solve the Equation:\n(\frac{120}{v} + \frac{180}{v+20} = 5)\nA Complete Guide with Step-by-Step Explanation and Real-World Application", "---", "If you’re tackling the equation\n[\n\frac{120}{v} + \frac{180}{v+20} = 5,\n]\nyou’re working with a rational equation — an equation involving fractions with variables in the denominator. Solving this properly is essential for math students, engineers, and anyone involved in problems requiring proportional reasoning. In this article, we’ll walk you through solving the equation step by step, explain its real-world meaning, and highlight key techniques for tackling similar problems.", "---", "## What This Equation Represents", "The equation\n[\n\frac{120}{v} + \frac{180}{v+20} = 5\n]\ncommonly appears in optimization and flow problems — for example, in physics when modeling combined rates, or in finance when analyzing time-dependent returns. Conceptually, it models the idea that two processes with different rates (denominators (v) and (v+20)) together take a total of 5 units to complete a task.", "Key Parameters:\n- (v): unknown time or rate of the first process\n- (v+20): a related rate, 20 units greater than the first\n- (5): total time or condition for completion", "---", "## Step-by-Step Solution", "### Step 1: Eliminate the denominators\nTo eliminate the fractions, find a common denominator. The least common denominator (LCD) of (v) and (v+20) is (v(v+20)). Multiply every term in the equation by this LCD:", "[\nv(v+20) \cdot \left( \frac{120}{v} + \frac{180}{v+20} \right) = 5 \cdot v(v+20)\n]", "Now distribute:", "[\n120(v+20) + 180v = 5v(v+20)\n]", "---", "### Step 2: Expand and simplify", "Expand both sides:", "Left:\n[\n120v + 2400 + 180v = 300v + 2400\n]", "Right:\n[\n5v^2 + 100v\n]", "Now write the full equation:", "[\n300v + 2400 = 5v^2 + 100v\n]", "Bring all terms to one side to form a quadratic:", "[\n0 = 5v^2 + 100v - 300v - 2400\n]", "Simplify:", "[\n5v^2 - 200v - 2400 = 0\n]", "Divide entire equation by 5 to simplify:", "[\nv^2 - 40v - 480 = 0\n]", "---", "### Step 3: Solve the quadratic equation", "Use the quadratic formula:\n[\nv = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Here, (a = 1), (b = -40), (c = -480):", "[\nv = \frac{40 \pm \sqrt{(-40)^2 - 4(1)(-480)}}{2}\n]\n[\nv = \frac{40 \pm \sqrt{1600 + 1920}}{2}\n]\n[\nv = \frac{40 \pm \sqrt{3520}}{2}\n]", "Simplify (\sqrt{3520}):\n(3520 = 16 \ imes 220 = 16 \ imes 4 \ imes 55 = 64 \ imes 55), so\n[\n\sqrt{3520} = \sqrt{64 \ imes 55} = 8\sqrt{55}\n]", "Thus,", "[\nv = \frac{40 \pm 8\sqrt{55}}{2} = 20 \pm 4\sqrt{55}\n]", "---", "### Step 4: Analyze the solution", "Since (v) represents time, it must be positive. Calculate approximate values:", "[\n\sqrt{55} \approx 7.416 \Rightarrow 4\sqrt{55} \approx 29.664\n]", "So,", "[\nv = 20 + 29.664 = 49.664 \quad \ ext{(valid)}\n]\n[\nv = 20 - 29.664 = -9.664 \quad \ ext{(invalid, discard)}\n]", "Therefore, the valid solution is:", "[\nv \approx 49.66 \quad \ ext{(exact: } 20 + 4\sqrt{55}\ ext{)}\n]", "---", "## Why This Matters — Real-World Application", "Solving such equations helps determine:", "- The optimal time to run two overlapping systems\n- Breakeven points in parallel processes\n- Flow rates in engineering designs (e.g., pipes, electrical circuits)\n- When multiple decay or growth rates combine to meet a target", "---", "## Key Takeaways & Formula", "- Always eliminate denominators by multiplying by the LCD\n- Simplify to a quadratic equation and apply the quadratic formula\n- Check for physical feasibility (positive, realistic values)\n- Simplify radicals when possible", "Final exact solution:\n[\nv = 20 + 4\sqrt{55}\n]", "For faster computation:\n[\nv \approx 49.66 \ ext{ (units of time)}\n]", "---", "## Final Thoughts", "Understanding how to solve rational equations like (\frac{120}{v} + \frac{180}{v+20} = 5) opens doors to modeling complex real-world systems. Mastering this technique strengthens algebraic skills and deepens analytical thinking.", "If you’re working on similar equations, remember:\n1. Identify common denominators\n2. Clear fractions\n3. Simplify into a solvable polynomial\n4. Solve and validate solutions", "Happy problem-solving!", "---", "# SEO Keywords\n- Solve rational equation\n- (\frac{120}{v} + \frac{180}{v+20} = 5<br/>\n- Step-by-step rational equation solution\n- Combining rational expressions\n- Algebraic problem-solving\n- Real-world applications of equations\n- Quadratic equation from rational equation\n- Time-related equation $$\frac{120}{v} + \frac{180}{v+20} = 5$$", "---", "References:\n- Khan Academy – Rational Equations\n- Paul’s Online Math Notes – Solving Rational Equations\n- Wolfram MathWorld – Quadratic Equations", "---", "Keywords Optimized for search engines positioning this article as a definitive guide for students, teachers, and professionals facing equations involving variable rates and combined time conditions."]