Question

A water tank can be filled by a tap in 6 hours while it can be emptied by another tap in 8 hours. If both the taps are opened simultaneously then after how much time will the tank get filled?A. 14 hoursB. 31 hoursC. 18 hoursD. 24 hours.

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Hint: In this particular problem find the effective rate of filling the water tank by using the formula (water filled in 1 hour – water emptied in 1 hour). And then find the time taken to fill 1 litre of water and after that multiply that with the amount of water to get the total time required to fill the water tank.

Let the capacity of the water tank be L litres.
So, as we know that when the only first tap is opened then the time taken to fill the tank is 6 hours.
So, water filled in 1 hour by the first tap will be $\dfrac{L}{6}$ litres/hour.
And when only the second tap is opened then it takes 8 hours to empty the water tank.
So, water emptied by the second tap in 1 hour will be $\dfrac{L}{8}$ litres/hour.
So, now if both the taps are open simultaneously then the amount of water filled in 1 hour will be equal to the amount of water filled by the first tap in one hour – the amount of water emptied by the second tap in one hour.
So, effective rate of filling the water tank will be $\dfrac{L}{6} - \dfrac{L}{8} = \dfrac{{4L - 3L}}{{24}} = \dfrac{L}{{24}}$ litres/hour.
Now if the amount of water that can be filled in one hour is $\dfrac{L}{{24}}$, then the time taken to fill one litre of water must be equal to $\dfrac{1}{{\dfrac{L}{{24}}}} = \dfrac{{24}}{L}$ hours.
So, now to completely fill the water tank we had to fill L litres of water in the tank as it is assumed above that the capacity of the tank is L litres.
So, if the 1 litre of water is filled in $\dfrac{{24}}{L}$ hours then L litres of water must be filled in $L \times \dfrac{{24}}{L} = 24$ hours.

So, it takes 24 hours to fill the water tank completely if both the taps are opened simultaneously.

Note: Whenever we face such types of problems then first, we have to assume that the capacity of the water tank is L litres and then we divide L by the time taken by each tap to fill or empty the tank to find the amount of water filled or emptied in one hour. After that we subtract the capacity of emptying from the capacity of filling the water tank. Then we can reverse that to get the time taken to fill one litre of water. Multiply that by L to get the time taken to completely fill the water tank. This will be the easiest and efficient way to find the solution of the problem.