# 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.

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).

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