A rectangular tank has two inlet pipes (A and B) and one outlet pipe C. Pipe A fills the tank twice as fast as pipe B, and pipe B fills the tank three times as fast as pipe C empties it. Anuj opens pipes A and B in the 1st hour, pipes B and C in the 2nd hour, and pipes A and C in the 3rd hour. He repeats this 3-hour cycle for 9 hours, after which he opens only the two inlet pipes (A and B), and the remaining part of the tank fills in 5 hours. What fraction of the tank's total capacity is filled by pipe A alone?
Answer and solution
Correct answer: B
Solution
Work rates A:B:C = 6:3:1 units/hour. 3-hour cycle net fill:
Hour 1 (A+B): $6+3=9$, Hour 2 (B+C): $3-1=2$, Hour 3 (A+C): $6-1=5$. Cycle total = 16 units.
After 3 cycles (9 hours): $3 \times 16 = 48$ units.
Then A+B only at rate 9 for 5 hours: $9 \times 5 = 45$ units.
Total capacity $= 48 + 45 = 93$ units.
A operated in: hours 1,3 of each cycle (6+6=12 per cycle × 3 = 36) + 5 extra hours (30). A total = $36 + 30 = 66$ units.
Fraction by A $= \dfrac{66}{93} = \dfrac{22}{31}$.
Hour 1 (A+B): $6+3=9$, Hour 2 (B+C): $3-1=2$, Hour 3 (A+C): $6-1=5$. Cycle total = 16 units.
After 3 cycles (9 hours): $3 \times 16 = 48$ units.
Then A+B only at rate 9 for 5 hours: $9 \times 5 = 45$ units.
Total capacity $= 48 + 45 = 93$ units.
A operated in: hours 1,3 of each cycle (6+6=12 per cycle × 3 = 36) + 5 extra hours (30). A total = $36 + 30 = 66$ units.
Fraction by A $= \dfrac{66}{93} = \dfrac{22}{31}$.