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Two pipes P and Q can fill a tank in 9 hours and 6 hours respectively, while a third pipe R can empty the tank in 12 hours. Initially, P and R are open for 6 hours. Then P is closed and Q is opened. After 6 more hours R is closed. The total time taken to fill the tank (in hours) is ____ A. 13.50 B. 14.00 C. 15.50 D. 14.50

GATE 2019 · General Aptitude · Work Time · medium

Answer: 14 hours, which matches option B.

  1. Phase 1: P and R open for 6 hours: Net rate = 1/9 - 1/12 = 4/36 - 3/36 = 1/36 per hour. Work in 6 hrs = 6 x (1/36) = 6/36 = 1/6.
  2. Phase 2: P closed, Q opened; Q and R open for 6 more hours: Net rate = 1/6 - 1/12 = 2/12 - 1/12 = 1/12 per hour. Work in 6 hrs = 6 x (1/12) = 6/12 = 1/2.
  3. Cumulative fill after 12 hours: W_total = 1/6 + 1/2 = 1/6 + 3/6 = 4/6 = 2/3. Remaining = 1 - 2/3 = 1/3.
  4. Phase 3: R closed, Q alone fills remaining 1/3: t_3 = (1/3) / (1/6) = (1/3) x 6 = 2 hours.
  5. Total time: T_total = 6 + 6 + 2 = 14 hours.