2430.11 – 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
Solution

Notation:

T is the volume of the tank.

A takes aa hours to fill the tank alone ⇝\leadsto A fills T/aT/a in 1 hour.

B takes bb hours to fill the tank alone ⇝\leadsto A fills T/bT/b in 1 hour.

C takes cc hours to fill the tank alone ⇝\leadsto A fills T/cT/c in 1 hour.

Now, all 3 together fill the tank in one hour:

 (1) T/a +T/b +T/c =T \text{ (1) }T/a + T/b + T/c = T 

A and C take 1½ hours, or 3/2 hours, so in one hour: 

 (2) T/a +T/c =2T/3\text{ (2) }T/a + T/c = 2T/3

B and C take 2 hours, so in one hour:

 (3) T/b +T/c =T/2\text{ (3) } T/b + T/c = T/2

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

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

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