Perimeter: \(2(w + 2w) = 180 \Rightarrow 6w = 180 \Rightarrow w = 30\).
["How to Solve Perimeter Problems: A Step-by-Step Guide Using the Equation (2(w + 2w) = 180)", "Understanding perimeter calculations is a fundamental skill in geometry, especially when solving real-world problems involving shapes like rectangles. Whether you're designing a garden, laying flooring, or creating a fencing plan, calculating perimeter efficiently helps you make accurate decisions. In this article, we walk through a classic perimeter problem using algebra to demonstrate exactly how to solve equations like (2(w + 2w) = 180) step by step.", "---", "### What Is Perimeter?", "The perimeter of a polygon is the total length of all its sides. For a rectangle, the perimeter (P) is given by the formula:", "[\nP = 2(\ ext{length} + \ ext{width})\n]", "In this example, the width is denoted by ( w ), and the length is ( 2w ), doubling the width hints at a specific ratio between length and width that helps simplify the calculation.", "---", "### The Given Equation", "We are given the perimeter equation:", "[\n2(w + 2w) = 180\n]", "This setup assumes a rectangle where the length is twice the width. Let’s simplify the expression inside the parentheses first.", "---", "### Step 1: Simplify Inside the Parentheses", "Combine like terms:", "[\nw + 2w = 3w\n]", "Now substitute back:", "[\n2(3w) = 180\n]", "---", "### Step 2: Multiply", "Simplify the multiplication:", "[\n6w = 180\n]", "Now, to find ( w ), divide both sides by 6.", "---", "### Step 3: Solve for ( w )", "[\nw = \frac{180}{6} = 30\n]", "So, the width ( w = 30 ) units. Since the length is ( 2w ), compute:", "[\n2w = 2 \ imes 30 = 60\n]", "---", "### Final Answer", "The perimeter confirms the values:", "[\nP = 2(30 + 60) = 2 \ imes 90 = 180\n]", "Which matches the given condition. Therefore, the width is ( \boxed{30} ) and the length is ( 60 ), with a total perimeter of 180 units.", "---", "### Why This Approach Matters", "- Clear logic from formula to solution: Breaking down the perimeter equation step-by-step makes solving streamlined and less error-prone.\n- Practical application: This algebraic method applies to any rectangular shape where dimensions follow a proportional relationship.\n- Foundation for more complex problems: Understanding this basics aids in solving composite shapes and real-world layout scenarios.", "---", "### Summary", "Using the equation (2(w + 2w) = 180), we simplified, solved, and confirmed that the width (w = 30), providing an efficient way to determine rectangular perimeters through algebra. Mastering this step-by-step method empowers students and professionals alike to tackle geometry with confidence.", "---", "Keywords: perimeter equation, geometry solution steps, algebra perimeter problem, rectangular perimeter, how to solve perimeter, critical thinking math, solving linear equations, geometry basics, solve for width, perimeter calculation.", "---", "If you found this helpful, share it with others learning algebra and geometry!"]
