A cistern has a leak which empties it in 8 hrs. A tap is turned on

Question

A cistern has a leak which empties it in 8 hrs. A tap is turned on which fills water into the cistern at 6 liters a minute, and it is now emptied in 12 hours. How many liters of water does the cistern hold?

A. 8640
B. 8740
C. 8840
D. 8540
E. 8940

A. 8640
B. 8740
C. 8840
D. 8540
E. 8940

This is a  PS Butler Question

Check the links to other Butler Projects:

Data Sufficiency Butler  
  Critical Reasoning Butler

MarkGMATMentor · Answer

This is an interesting variant of a work / rate problem where the second "worker" does negative work.

We'll use the formula Work = Rate * Time (W = RT). Here, the "work" consists of emptying the cistern. We want to know how many liters the cistern holds, which is the same as the amount of work done (the number of liters emptied). There are two scenarios:

Leak Only: 
The leak empties the cistern in 8 hours. If we set the work (the liters to be emptied) as w and the leak rate as r, you have:
w = r * 8

Leak and Tap: 
The leak still operates, but now a tap fills the cistern at 6 liters per minute, which impedes the work. Let's multiply that by 60 to get 360 liters per hour (keeping the units consistent). Together the tap and the leak take 12 hours to empty the cistern.

Remember, in work / rate problems, when multiple workers work together, you add their rates to find the combined rate. It's exactly the same here, except that the tap is actually  reversing  the work that the leak does. So, to find the combined rate of the leak and tap together, you need to  subtract the tap's rate from the leak's rate.

So, the combined rate is (r - 360) and the work / rate equation is:
w = (r - 360) * 12

Since the work (w) is the same in both scenarios, set the equations equal to each other and solve:
8r = 12(r - 360)
2r = 3(r - 360)
2r = 3r - 1080
r = 1080

Now plug into the first equation and solve for w:
w = 1080 * 8
w = 8640

The answer is A.

On work / rate problems, remember that you can add rates (don't add the times!). And when something is actually creating more work rather than doing work,  subtract  its rate instead.

Bunuel · Answer

Video Explanation:

https://www.youtube.com/watch?v=m_T_v5wqOS8

FerociousStorm · Answer

cistern capacity - 100x 
Leak capacity - 12.5x per hour 
Filling capacity 360 per hour. 
 
Leak and filling works simultaneously for 12 hours to empty cistern when it is fully filled . So 
 
360*12 - 12.5x *12 + 100x = 0 
 
x= 4320/50 
Cistern cap is 100x so - 4320/50 * 100= 8640

LibrarianTurtle · Answer

Lets consider capacity of cistern as N
If a leak empties it in 8 hours then its rate becomes=  N/8 . (N being total water capacity and it emptying it in 8 hours)
Rate of tap which fills the cistern=  \frac{6Liters}{1Minute}  (Liters per minute). Our units has to be compatible so convert it to hour (60 Minutes).  6*60= \frac{360Liters}{1Hour}. 
With the tap added the leak empties the cistern in 12 hours, meaning that leak has greater rate than the tap. 
So we subtract the rate of the tap from rate of the leak.  \frac{N}{8} - \frac{360}{1}=- \frac{(N-2880)}{8} 
So our rate for the leak equals  \frac{(N-2880)}{8} 
Ultimately total capacity divided by rate of leak =  \frac{N }{((N-2880)/8)} = 12  
 N =8640 
IMO

A cistern has a leak which empties it in 8 hrs. A tap is turned on

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

A cistern has a leak which empties it in 8 hrs. A tap is turned on

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards