I'll solve this manufacturing problem step-by-step using the Fractal Reasoning Matrix approach. ## 1. PROBLEM ARCHITECTURE Let me analyze the essential elements of this production problem: - Factory produces 100 widgets per hour - 5% of widgets are defective (meaning 95% are non-defective) - Need to determine how many hours to produce 940 non-defective widgets The key relationship here is that total production time depends on both the production rate and the defect rate. The constraint is that we need exactly 940 non-defective widgets. The main question to answer: How many total widgets must be produced to get 940 non-defective ones, and how long will that take? ## 2. COGNITIVE SEGMENTATION Breaking this down into components: 1. Calculate the effective production rate of non-defective widgets per hour 2. Determine the total number of widgets (both defective and non-defective) needed 3. Calculate the required production time in hours ## 3. SYSTEMATIC PROCESSING ### Component 1: Effective production rate of non-defective widgets - Total production rate = 100 widgets/hour - Defect rate = 5% - Non-defective rate = 95% = 0.95 - Effective production rate of non-defective widgets = 100 × 0.95 = 95 non-defective widgets/hour ### Component 2: Total widgets needed - Target number of non-defective widgets = 940 - If x = total widgets produced, then: - Number of non-defective widgets = 0.95x - Setting up the equation: 0.95x = 940 - Solving for x: x = 940 ÷ 0.95 = 989.47... widgets ### Component 3: Required production time - Total widgets needed = 989.47... widgets - Production rate = 100 widgets/hour - Time required = Total widgets ÷ Production rate - Time required = 989.47... ÷ 100 = 9.8947... hours ## 4. INTEGRATIVE UNIFICATION Combining these components: - The factory produces 95 non-defective widgets per hour - To get 940 non-defective widgets, we need to produce 989.47... total widgets - At a rate of 100 widgets per hour, this will take 9.8947... hours Since we can't produce a partial widget, we need to round up to ensure we get at least 940 non-defective widgets. This gives us 10 hours. ## 5. SOLUTION VALIDATION Let's verify this solution: - In 10 hours, the factory produces: 10 × 100 = 1,000 total widgets - With a 5% defect rate, the number of defective widgets is: 1,000 × 0.05 = 50 widgets - Therefore, the number of non-defective widgets is: 1,000 - 50 = 950 widgets - Since 950 > 940, our requirement of at least 940 non-defective widgets is satisfied Alternatively: - In 10 hours, the number of non-defective widgets = 10 × 95 = 950 widgets - This exceeds our requirement of 940 non-defective widgets In conclusion, it will take 10 hours to produce at least 940 non-defective widgets.