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 per hour: $A : B : C = 6 : 3 : 1$ (C empties).
Hour 1: $6 + 3 = 9$; hour 2: $3 - 1 = 2$; hour 3: $6 - 1 = 5$. One cycle $= 16$.
9 hours (3 cycles) $= 48$; next 5 hours with A + B $= 9 \times 5 = 45$
Capacity $= 48 + 45 = 93$
A runs 6 hours in the cycles $+ 5$ hours after: $6 \times 6 + 5 \times 6 = 66$
Fraction by A $= \dfrac{66}{93} = \dfrac{22}{31}$
Hour 1: $6 + 3 = 9$; hour 2: $3 - 1 = 2$; hour 3: $6 - 1 = 5$. One cycle $= 16$.
9 hours (3 cycles) $= 48$; next 5 hours with A + B $= 9 \times 5 = 45$
Capacity $= 48 + 45 = 93$
A runs 6 hours in the cycles $+ 5$ hours after: $6 \times 6 + 5 \times 6 = 66$
Fraction by A $= \dfrac{66}{93} = \dfrac{22}{31}$