1. Problem 1: If 5 machines take 10 hours to produce 100 widgets, how long will 10 machines take to produce the same number? Formula: Work = Rate × Time Since 5 machines take 10 hours, total work = 5 × 10 = 50 machine-hours for 100 widgets. Rate per machine = 100 widgets / 50 machine-hours = 2 widgets per machine-hour. For 10 machines, time = Total work / Number of machines = 50 machine-hours / 10 = 5 hours. 2. Problem 2: A tap fills a tank in 6 hours, a second tap in 3 hours. How long together? Formula: Combined rate = Rate1 + Rate2 Rate1 = 1 tank / 6 hours = \frac{1}{6} Rate2 = 1 tank / 3 hours = \frac{1}{3} Combined rate = \frac{1}{6} + \frac{1}{3} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2} Time to fill together = \frac{1}{Combined rate} = 2 hours. 3. Problem 3: A job takes 8 hours for 4 people. After 2 hours, 2 more join. Total time? Total work = 4 people × 8 hours = 32 person-hours. Work done in first 2 hours = 4 × 2 = 8 person-hours. Remaining work = 32 - 8 = 24 person-hours. Now 6 people work, time = \frac{24}{6} = 4 hours. Total time = 2 + 4 = 6 hours. 4. Problem 4: Train travels 120 km in 2 hours. Distance in 5 hours? Speed = \frac{Distance}{Time} = \frac{120}{2} = 60 km/h. Distance in 5 hours = Speed × Time = 60 × 5 = 300 km. 5. Problem 5: 12 workers build 3 houses in 6 days. How many houses can 8 workers build in 12 days? Work rate = \frac{3 houses}{12 workers × 6 days} = \frac{3}{72} = \frac{1}{24} houses per worker-day. Total worker-days for 8 workers in 12 days = 8 × 12 = 96. Houses built = Rate × Worker-days = \frac{1}{24} × 96 = 4 houses. Final answers: 1. 5 hours 2. 2 hours 3. 6 hours 4. 300 km 5. 4 houses