Cisterns

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Cisterns

by RiyaR » Tue Oct 28, 2014 12:34 am
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?

2 hours
2.5 hours
3 hours
3.5 hours
4 hours

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by GMATGuruNY » Tue Oct 28, 2014 1:31 am
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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by [email protected] » Tue Oct 28, 2014 11:43 am
Hi RiyaR

Here, we're told that one inlet takes 3 hours to fill a cistern while another takes twice as long (meaning it takes 6 hours to fill a cistern).

Using the Work Formula, we can figure out how long it takes the two inlets, working together, to fill the cistern:

Work = AB/(A+B) = (3)(6)/(3+6) = 18/9 = 2 hours

This means that the two inlets will completely fill 1/2 the cistern per hour.

Starting at 9am, the cistern would be filled at 11am. However, the outlet removes water at such a rate that at 10:30am, it takes a full hour to fill the cistern (as opposed to the 1/2 hour that it would take if there was no outlet at all).

So, at 10:30am, the cistern is 3/4 full, but then the outlet is turned on and it starts draining water....an hour later, the cistern is full. Since the cistern will be 1/2 full after an hour, the outlet must be removing 1/4 of the tank during that time (3/4 + 1/2 - 1/4 = 1 = full).

Thus, the outlet removes 1/4 of the water in 1 hour.

The question asks how long it will take the outlet to empty the cistern when it's full. Since the outlet removes 1/4 of the water in 1 hour, it will remove the entire volume of water in 4 hours.

Final Answer: E

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