Three Valves

Each valve A, B, and C, when open, releases water into a tank at its own constant rate. With all three valves open, the tank fills in 1 hour; with only valves A and C open, it takes 1.5 hours; and with only valves B and C open, it takes 2 hours. The number of hours required with only valves A and B open is:

  1. 1.1
  2. 1.15
  3. 1.2
  4. 1.25
  5. 1.75


Show/Hide Solution

Notation:  A  takes  a  hours to fill the tank alone, →   A fills T/a in 1 hour.
B  takes  b  hours to fill the tank alone, →   B fills T/b in 1 hour.
C  takes  c  hours to fill the tank alone, →   C fills T/c in 1 hour.
(T is the capacity of the tank)

Now, all 3 together fill the tank in one hour: T/a + T/b + T/c = T  (1)
A and C take 1½ hours, or 3/2 hours, so in one hour:  T/a + T/c = 2T/3 (2)
B and C take 2 hours, so in one hour: T/b + T/c = T/2 (3)

(1) – (2) gives us T/b = T/3, so b = 3 hours.
(3) gives us T/3 + T/c = T/2, so T/c = T/6 and c = 6 hours.
(2) gives us T/a + T/6 = 2T/3, so T/a = T/2 and a = 2 hours.

Interim check: (1) T/2 + T/3 + T/6 = T.

In one hour, A and B fill T/2 + T/3 = 5T/6, so they need 6/5 or 1.2 hours to fill the tank. The answer is (c).

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