Rates Word Problem : Pipes cisterns

This aptitude practice question is a Pipes Cisterns problem solving question. Concept tested: How to frame equations for word problem in pipes and cistern accounting for some pipes filling and others emptying a tank.

Question

Pipe A usually fills a tank in 2 hours. On account of a leak at the bottom of the tank, it takes pipe A 30 more minutes to fill the tank. How long will the leak take to empty a full tank if pipe A is shut?

1. 2 hours 30 minutes
2. 5 hours
3. 4 hours
4. 10 hours

  Correct Answer      Choice (4). The leak at the bottom of the tank will empty the tank in 10 hours.

Explanatory Answer

Pipe A fills the tank normally in 2 hours.
Therefore, it will fill \\frac{1}{2}) of the tank in an hour.
Let the leak take x hours to empty a full tank when pipe A is shut.
Therefore, the leak will empty \\frac{1}{x}) of the tank in an hour.
The net amount of water that gets filled in the tank in an hour when pipe A is open and when there is a leak =
\\frac{1}{2}) - \\frac{1}{x}) of the tank. ----- (1)

When there is a leak, the problem states that it takes two and a half hours to fill the tank. i.e. \\frac{5}{2})hours.
Therefore, in an hour, \\frac{2} {5}^{th}) of the tank gets filled. ----- (2)
Equating (1) and (2), we get \\frac{1}{2}) - \\frac{1}{x}) = \\frac{2}{5})
=> \\frac{1}{x}) = \\frac{1}{2}) - \\frac{2}{5}) = \\frac{1}{10})
=> x = 10 hours.

The problem can also be solved without putting pen to paper as follows.
Pipe A takes 2 hours to fill the tank. Therefore, it fills half the tank in an hour or 50% of the tank in an hour.
When there is a leak it takes 2 hours 30 minutes for the tank to fill. i.e \\frac{5}{2})hours to fill the tank or \\frac {2}{5}^{th}) or 40% of the tank gets filled.
On account of the leak, (50 - 40)% = 10% of the water gets wasted every hour.
Therefore, the leak will take 10 hours to drain a full tank.

  The correct answer is Choice (4).