Question

One pump can fill a water tank in \[40\] minutes and another pump takes \[30\] minutes. How long will it take to fill the water tank if both the pumps work together?

Answer 1

Hint: Given, one pump can fill a water tank in \[40\] minutes and another pump takes \[30\] minutes. We will assume the volume of the tank to be \[V\]. Then we will find the rate of filling of the first and second pump. Assume that the time taken by both the pumps to fill the same tank of volume \[V\] be \[t\]. We will find the volume filled by both pumps together in time \[t\] and will equate it to the volume \[V\], then we will simplify it to find the required result.

Complete step by step answer:
Given, one pump can fill a water tank in \[40\] minutes and another pump takes \[30\] minutes. Filling rate of a pump will be equal to the volume of the tank divided by time taken by the pump to fill it. So, we can write,
Filling rate of first pump \[ = \] \[\dfrac{V}{{40}}\]
Filling rate of second pump \[ = \] \[\dfrac{V}{{30}}\]

Now, let the time taken by the both the pump together to fill the same tank of volume \[V\] be \[t\]. Now, the volume filled by both pumps together in time \[t\] will be equal to the volume of the water tank. So, we get
\[ \Rightarrow \left( {\dfrac{V}{{40}} + \dfrac{V}{{30}}} \right)t = V\]
Taking \[V\] common from the LHS, we get
\[ \Rightarrow V\left( {\dfrac{1}{{40}} + \dfrac{1}{{30}}} \right)t = V\]

Cancelling the common term \[V\] from both the sides, we get
\[ \Rightarrow \left( {\dfrac{1}{{40}} + \dfrac{1}{{30}}} \right)t = 1\]
Taking LCM of LHS, we get
\[ \Rightarrow \left( {\dfrac{{3 + 4}}{{120}}} \right)t = 1\]
\[ \Rightarrow \left( {\dfrac{7}{{120}}} \right)t = 1\]
Multiplying both the sides by \[\dfrac{{120}}{7}\], we get
\[ \Rightarrow t = \dfrac{{120}}{7}\]
\[ \therefore t = 17.1428\]

Therefore, it takes \[17.1428\] minutes to fill the water tank if both the pumps work together.

Note: Here, the volume of the water tank will not be changing i.e., volume of the tank is constant. In this case we are not interested in the dimensions of the tank. The tank may be in any shape but here we only required volume terms to find the filling rate of the first and the second tank.