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
- 1.15
- 1.2
- 1.25
- 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.
(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).