Question

A pipe can fill a tank in 6 hours, while a leak not at the bottom of the tank can empty upto it in 3 hours. If both are operated simultaneously and initially the tank is full, when the tank will be full upto the height of the leak?
Statement 1: The leak is developed at a one-third height of the tank
Statement 2: The leak takes 3 hours to empty the tank upto its height

• A. Statement (1) alone is sufficient to answer the question
• B. Statement (2) alone is sufficient to answer the question
• C. Both the statements together are needed to answer the question
• D. Either statement (1) alone or statement (2) alone is sufficient to answer the question
• E. Neither statement (1) nor statement (2) suffices to answer the question

Answer 1

The Correct Option is A

To solve the problem, let's analyze the given statements and the information:

1. The pipe can fill the entire tank in 6 hours, which means the rate of filling is \frac{1}{6} of the tank per hour.
2. The leak can empty the full tank in 3 hours. This leak is positioned at one-third of the tank's height, based on Statement 1. Thus, it affects only one-third of the tank's volume.

Let's examine the sufficiency of each statement:

• Statement 1: The leak is located at one-third of the tank height, which indicates it can affect one-third of the tank's volume. Since the leak itself can empty the whole tank volume in 3 hours, it will take 1 hour to empty one-third of the tank. Given this information, we determine that the rate for the leak is \frac{1}{3} \text{ of the tank in 1 hour} .
• Thus, the effective rate at which the tank gets filled when both the pipe and leak are operational is \frac{1}{6} - \frac{1}{3} = -\frac{1}{6} . This means the tank will continue to lose water, implying that the situation given doesn't actually allow the tank to fill up to the level of the leak, given that it's already full initially. So, this fact alone gives us a complete picture.

Statement 2 states the leak takes 3 hours to empty the tank up to its height. Without knowing the height where the leak is placed, this statement alone does not allow us to determine when the tank will be full up to the height of the leak.

Finally, analyzing the sufficiency of each statement, we conclude:

• Statement 1 alone is sufficient to answer the question, as it gives the position (height) of the leak.
• Statement 2 alone does not provide enough information to determine or calculate when the tank will be full to the leak's height, as the leak's height isn't explicitly defined by volume ratio or height specifics.

Hence, the correct option is: Statement (1) alone is sufficient to answer the question.