\[ \frac{1}{20} = \frac{1}{v} - \frac{1}{-30} \]

["Solving the Equation:\n[ \frac{1}{20} = \frac{1}{v} - \frac{1}{-30} ]", "---", "Understanding how to solve rational equations is essential in algebra, and this equation offers a great example of manipulating fractions to isolate a variable. In this guide, we’ll walk through solving\n[ \frac{1}{20} = \frac{1}{v} - \frac{1}{-30} ]\nstep-by-step, clarify its meaning, and explore its applications.", "---", "### Step 1: Simplify the Equation", "Start with the original equation:\n[ \frac{1}{20} = \frac{1}{v} - \frac{1}{-30} ]", "Note that subtracting a negative is the same as adding a positive, so rewrite (\frac{1}{-30}) as (-\frac{1}{30}):\n[ \frac{1}{20} = \frac{1}{v} + \frac{1}{30} ]", "---", "### Step 2: Isolate the Variable Term", "To isolate (\frac{1}{v}), subtract (\frac{1}{30}) from both sides:\n[ \frac{1}{20} - \frac{1}{30} = \frac{1}{v} ]", "Now simplify the left-hand side. Find a common denominator—12 is the least common multiple of 20 and 30.", "Convert:\n[\n\frac{1}{20} = \frac{3}{60},\quad \frac{1}{30} = \frac{2}{60}\n]\nSo:\n[\n\frac{3}{60} - \frac{2}{60} = \frac{1}{60}\n]", "Thus:\n[ \frac{1}{v} = \frac{1}{60} ]", "---", "### Step 3: Solve for (v)", "Take the reciprocal of both sides to solve for (v):\n[ v = 60 ]", "---", "### Step 4: Verify the Solution", "Plug (v = 60) back into the original equation:\n[\n\frac{1}{20} = \frac{1}{60} - \left(\frac{1}{-30}\right) = \frac{1}{60} + \frac{1}{30}\n]", "Compute:\n[\n\frac{1}{60} + \frac{2}{60} = \frac{3}{60} = \frac{1}{20}\n]", "The left side equals the right side, confirming the solution is correct.", "---", "### Practical Applications and Interpretation", "This equation models situations where rates or reciprocals are involved—such as work problems, inverse proportionality, or flow rates. For example, if (\frac{1}{20}) represents a combined rate, and one component is offset by (-\frac{1}{-30}) (i.e., a positive contribution), finding (v = 60) reveals the unknown rate or time.", "---", "### Summary", "Solving\n[ \frac{1}{20} = \frac{1}{v} - \frac{1}{-30} ]\nleads to the clear solution:\n[\n\boxed{v = 60}\n]\nThrough simplification, common denominators, and basic algebra, we isolate the variable and validate it, showcasing foundational algebra skills critical for more advanced math.", "---", "### Key Concepts Recap:", "- Reciprocal and negative fraction rules\n- Finding least common denominators\n- Isolation of variables in rational equations\n- Verification by substitution", "---", "### Related Reading", "- How to Solve Linear Equations with Rational Expressions\n- Common Reciprocal Misconceptions in Algebra\n- Applications of Inverse Proportions in Real Life", "---", "Mastering equations like this strengthens your mathematical fluency and prepares you for topics in physics, engineering, and economics where fractional reasoning applies.", "---\nKeywords: solve (\frac{1}{20} = \frac{1}{v} - \frac{1}{-30}), rational equations, algebraic solutions, step-by-step, (v = 60)"]