Pipe X can fill a tank in 9 hours and Pipe Y can fill it in 21 hours. If they are opened on alternate hours and Pipe X is opened first, in how many hours shall the tank be full?

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Step 1: Calculate the rate of filling for each pipe per hour. - Pipe X fills the tank in 9 hours, so rate of X = 1/9 tank per hour. - Pipe Y fills the tank in 21 hours, so rate of Y = 1/21 tank per hour. Step 2: Since pipes are opened alternately starting with Pipe X, the filling pattern is: - Hour 1: Pipe X works, fills 1/9 of the tank. - Hour 2: Pipe Y works, fills 1/21 of the tank. Step 3: Calculate the amount filled in 2 hours (one full cycle): = (1/9) + (1/21) = (7/63) + (3/63) = 10/63 of the tank. Step 4: Determine how many such 2-hour cycles are needed to get close to full tank: Let n be the number of 2-hour cycles. Total filled after n cycles = (10/63) * n. Step 5: Find n such that total filled is just less than 1: (10/63) * n < 1 => n < 6.3. So after 6 cycles (12 hours), amount filled = (10/63)*6 = 60/63 = 20/21. Step 6: After 12 hours, the tank is 20/21 full. Next hour is Pipe X's turn (hour 13), which fills at 1/9 per hour. Remaining tank to fill = 1 - 20/21 = 1/21. Step 7: Time taken by Pipe X to fill remaining 1/21 tank: Time = (Remaining volume) / (Rate of X) = (1/21) / (1/9) = 9/21 = 3/7 hours. Step 8: Total time to fill the tank = 12 hours + 3/7 hours = 12 + 0.42857 = 12.42857 hours. Step 9: But the question states Pipe X starts first and pipes alternate every hour. So the filling hours are: - Hour 1: X - Hour 2: Y - Hour 3: X - Hour 4: Y - Hour 5: X - Hour 6: Y - Hour 7: X - Hour 8: Y - Hour 9: X - Hour 10: Y - Hour 11: X - Hour 12: Y - Hour 13: X (partial hour) Step 10: So total time = 12 + 3/7 = 10 3/7 hours. Correction: The above calculation shows 12 3/7 hours, but the question options suggest 10 3/7 is correct. Let's re-check the calculation. Recalculation: Amount filled in 2 hours = 10/63. Number of 2-hour cycles to reach near full: (10/63)*n < 1 => n < 6.3. After 5 cycles (10 hours), amount filled = (10/63)*5 = 50/63. Remaining = 1 - 50/63 = 13/63. Next hour (hour 11) is Pipe X's turn: Pipe X fills at 1/9 per hour. Time to fill remaining 13/63 tank = (13/63) / (1/9) = (13/63) * 9 = 13/7 = 1 6/7 hours. Total time = 10 + 1 6/7 = 11 6/7 hours, which is not an option. Try after 6 cycles (12 hours): Amount filled = 60/63 = 20/21. Remaining = 1/21. Next hour (hour 13) Pipe X works: Time = (1/21) / (1/9) = 9/21 = 3/7 hours. Total time = 12 + 3/7 = 12 3/7 hours. This matches option 2 (12 3/7), so option 2 is correct. Final Answer: 12 3/7 hours.

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