Question

A pipe can fill a cistern in $ 9 $ hours. Due to a leak in its bottom, the cistern fills up in $ 10 $ hours. If the cistern is full, how much time will it be emptied by the leak?

Answer 1

Hint: A pipe that has a leak should take more time to complete filling the cistern, so when difference in the time taken by two scenarios are considered, the time taken by leaked pipe has to be subtracted from the time taken by the original pipe to fill the cistern. We make use of the unitary method to completely solve this question.

Complete step by step solution:
Let us see what is given to us:
The total time (in hours) with which the original pipe can completely fill the cistern is $ = 9 $ hours
The time taken by leaked pipe to completely fill the cistern is $ = 10 $ hours
Time taken to empty the completely filled cistern (by the leaked pipe) will be $ = ? $ hours
Now we proceed to solving the question using unitary method of solving;
Let us consider each part of the cistern to be the concerned unit.
We can rewrite the given statements as follows:
In $ 1 $ hour, the original pipe can fill this much part of cistern $ = \dfrac{1}{9} $
Similarly, in $ 1 $ hour, the leaked pipe can fill this much a part of cistern $ = \dfrac{1}{{10}} $
Then;
We know that to find the time taken to empty a part of the cistern by leaked pipe is the difference between the part filled by original pipe in an hour and the part filled by leaked pipe in an hour:
 $ \Rightarrow $ Leaked pipe can empty this much of part of the tank in $ 1 $ hour $ = \dfrac{1}{9} - \dfrac{1}{{10}} $
Simplifying by taking the LCM, we get:
 $ \Rightarrow $ Leaked pipe can empty this much of part of the tank in $ 1 $ hour $ = \dfrac{1}{{90}} $
 $ \Rightarrow $ Also the leaked pipe can empty this much of part of the tank in $ 90 $ hours $ = 1 $
This means that in $ 90 $ hours the complete cistern can be emptied, so:
$ \therefore $ The complete cistern will take $ 90 $ hours to get emptied by the leaked pipe.
So, the correct answer is “ $ 90 $ hours”.

Note: We use a unitary method here so there are a few things to be kept in mind while using this method. When an equation is given, rearrange the equation in such a way that we get the unit in the required terms. The left side contains the values provided in the question but the right hand will be the computed values