# 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

 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).