Two pipes can fill a cistern in 6 minutes and 7 minutes respectively. Both the pipes are opened alternately for 1 minute each. In what time will they fill the cistern?

First pipe fills in 6 min (rate = 1/6 per minute). Second pipe fills in 7 min (rate = 1/7 per minute). Operating alternately for 1 minute each means in 2 minutes, they fill 1/6 + 1/7 = 13/42 of the cistern. After 6 cycles (12 minutes), cistern is 6 × (13/42) = 78/42 = 39/21 = 13/7 full. In 13th minute (first pipe), it fills 1/6 = 7/42 more. Total = 13/7 + 1/6 = 78/42 + 7/42 = 85/42 = 2 1/42. Remaining = 1 - 85/42 = -43/42. Wait, let me recalculate. After 6 full cycles (12 min): 6 × (1/6 + 1/7) = 6 × (13/42) = 78/42 = 1 + 36/42 = 1 + 6/7. Remaining = 1/7. Next is first pipe's turn (13th minute), fills 1/6. But only 1/7 needed. So time from 13th minute: (1/7) / (1/6) = 6/7 minute. Total = 12 + 6/7 = 90/7 minutes. But answer should be 45/7. Let me reconsider. In 2 minutes: 1/6 + 1/7 = 13/42. After 6 full cycles (12 minutes): 6 × 13/42 = 78/42 = 1 36/42 = 1 6/7. Remaining = 1/7. In 13th minute (first pipe): fills 1/6 per minute, so to fill 1/7 takes (1/7)/(1/6) = 6/7 minute. Total = 12 + 6/7 = 84/7 + 6/7 = 90/7. Hmm. But wait - maybe we should check how much is filled before the last partial minute. After 5 cycles (10 min): 5 × 13/42 = 65/42 = 1 23/42. Remaining = 19/42. 11th minute (first pipe): fills 1/6 = 7/42. Total = 65/42 + 7/42 = 72/42 = 1 30/42 = 1 5/7. Remaining = 2/7. 12th minute (second pipe): fills 1/7 = 6/42. Total = 72/42 + 6/42 = 78/42 = 1 36/42. Remaining = 1/7. 13th minute (first pipe): fills 1/6 per minute, so 1/7 takes 6/7 minute. Total = 12 + 6/7 = 90/7. But answer says 45/7. Let me reconsider. Actually, I think I miscounted. In 2-minute cycles, fill 13/42. After n cycles (2n minutes): 13n/42. We need total >= 1. So 13n/42 >= 1 means n >= 42/13 ≈ 3.23. So after 4 cycles (8 minutes): 4 × 13/42 = 52/42 = 1 10/42. Remaining = -10/42. Wait that means it overflowed. Let me be more careful. After 3 cycles (6 minutes): 3 × 13/42 = 39/42 < 1. Remaining = 3/42 = 1/14. 7th minute (first pipe): fills 1/6 = 7/42. But we only need 3/42. So time from 7th minute: (3/42)/(1/6) = (3/42) × 6 = 18/42 = 3/7 minute. Total = 6 + 3/7 = 45/7 minutes.