Question

A tap can fill a tank in $24$ hours and an outlet can empty the full tank in $30$ hours. In how many hours the empty tank will be filled, if both the tap and the outlet are opened simultaneously?

Answer 1

Hint: We will assume the volume of tank as $1$ unit. From the given data we can say that A tap takes $24$ hours of time to fill the tank of capacity $1$ unit. At the same time, we can also say that an outlet takes $30$ hours to empty the tank of capacity $1$ unit. From these two statements we will calculate how much part of the tank will fill when both the tap and outlet are opened for an Hour. From this value we can find the time required to fill the tank when both the tap and outlet are opened simultaneously.

Complete step by step answer:
Given that, A tap can fill a tank in $24$ hours. Assume the capacity of the tank as $1$ unit. Then the part of tank filled when the tap is opened in an hour is
${{f}_{1hr}}=\dfrac{1}{24}$
At the same time an outlet can empty the tank of capacity $1$ unit in $30$ hours. Then the part of tank empty when the outlet is opened for an hour is
${{e}_{1hr}}=\dfrac{1}{30}$
When the both tap and outlet are opened simultaneously for an hour then the water remaining in the tank is equal to the difference of tank filled in one hour when the tap is opened and tank empty in an hour when the outlet is opened. Mathematically
$\begin{align}
  & {{c}_{1hr}}={{f}_{1hr}}-{{e}_{1hr}} \\
 & =\dfrac{1}{24}-\dfrac{1}{30} \\
 & =\dfrac{30-24}{24\times 30} \\
 & =\dfrac{6}{24\times 30} \\
 & =\dfrac{1}{120}
\end{align}$
From the above equation we can say that when the tap and outlet are opened for an hour then the capacity of the tank remains $\dfrac{1}{120}$.
Let us assume that the tank of capacity $1$ unit will be filled in $x$ hours when the tap and outlet are opened simultaneously. Then
$\begin{align}
  & x\times {{c}_{1hr}}=1 \\
 & x\times \dfrac{1}{120}=1 \\
 & x=120
\end{align}$

Hence the tank will be filled in $120$ hours.

Note: While taking the difference of the capacity of tank filled when tap is opened to the capacity of the tank when outlet is opened please remember that we need to subtract the value of capacity of tank empty when the outlet is opened from the capacity of the tank when tap is opened because outlet removes the water from the tank so there is a decrease in the capacity of tank so we will subtract.