\(6w = 40\), so \(w = \frac{20}{3}\).

["# Solving the Equation (6w = 40): Step-by-Step Guide", "Understanding how to solve a simple linear equation like (6w = 40) is a fundamental skill in algebra and can help build a solid foundation for more advanced mathematical concepts. In this article, we’ll break down the process of solving (6w = 40) step by step and explain why the solution is (w = \frac{20}{3}).", "## What Does the Equation (6w = 40) Mean?", "The equation (6w = 40) means that six times an unknown value (w) equals 40. To find (w), we need to isolate the variable—a process that involves performing inverse operations on both sides of the equation.", "## Step-by-Step Solution", "### Step 1: Divide Both Sides by 6\nTo solve for (w), divide every term in the equation by 6:", "[\n6w = 40\n]", "[\n\frac{6w}{6} = \frac{40}{6}\n]", "On the left side, the 6 cancels out:", "[\nw = \frac{40}{6}\n]", "### Step 2: Simplify the Fraction\nNow simplify (\frac{40}{6}) by reducing the fraction to its lowest terms.", "Find the greatest common divisor (GCD) of 40 and 6, which is 2:", "[\n\frac{40 \div 2}{6 \div 2} = \frac{20}{3}\n]", "So,", "[\nw = \frac{20}{3}\n]", "### Step 3: Convert to Decimal (Optional)\nFor practical applications, you may prefer a decimal form. Dividing 20 by 3 gives:", "[\nw \approx 6.\overline{6}\n]", "This repeating decimal highlights that (w) is an irrational fraction, often left as a simplified fraction in equations.", "## Why This Solution Matters", "Solving equations like (6w = 40) helps develop logical thinking and algebraic proficiency. The solution (w = \frac{20}{3}) is an exact value, which is preferred in mathematics over decimal approximations when possible. Using fractions ensures precision and clarity in representing quantities.", "## Real-World Application Example", "Imagine you’re dividing 40 units of material equally among 6 groups. The equation (6w = 40) models this scenario, where (w = \frac{20}{3}) represents the size of each group. Understanding this helps in budgeting, mixing recipes, project planning, and many other daily tasks.", "## Conclusion", "The equation (6w = 40) leads directly to the solution (w = \frac{20}{3})—a clear example of isolating a variable to reveal a precise value. Mastering such steps strengthens your mathematical toolkit and prepares you for more complex problems. Whether you're solving in class, work, or personal projects, recognizing how to solve linear equations is invaluable.", "---", "Keywords: solve (6w = 40), (w = \frac{20}{3}), linear equation, algebra, fractional solution, equation solving steps, how to isolate (w), solve equations with fractions, math practice for students."]