There are two inlets and one outlet to a cistern. One of the inlets takes 3 hours to fill up the cistern and the other inlet takes twice as much time to fill up the same cistern. Both of the inlets are turned on at 9:00 AM with the cistern completely empty, and at 10:30AM, the outlet is turned on and it takes 1 more hour to fill the cistern completely. How much time does the outlet working alone takes to empty the cistern when the cistern is full?
(A) 2 hours
(B) 2.5 hours
(C) 3 hours
(D) 3.5 hours
(E) 4 hours
Let the cistern = 12 gallons.
Since the faster inlet takes 3 hours to fill the cistern, the rate for the faster inlet = w/t = 12/3 = 4 gallons per hour.
Since the slower inlet takes twice as long -- 6 hours -- to fill the cistern, the rate for the slower inlet = w/t = 12/6 = 2 gallons per hour.
Combined rate for the two inlets = 4+2 = 6 gallons per hour.
In the 1.5 hours from 9am to 10:30am, the amount of fluid pumped IN by the two cisterns = r*t = (6)(3/2) = 9 gallons.
Remaining fluid to be pumped in = 12-9 = 3 gallons.
Since the cistern is filled after 1 more hour, the rate for the final hour = 3 gallons per hour.
The oultet decreases the hourly rate from 6 gallons per hour to 3 gallons per hour.
Implication:
The outlet's rate = 3 gallons per hour.
Thus:
Time for the outlet to empty the cistern = w/r = 12/3 = 4 hours.
The correct answer is
E.
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