a motor pump can fill a tank in 8 minutes and there is a tap that can empty it in 7 minutes. if the tank already have one-fourth filled and both the pump and tap is opened together, when will the tank emptied?

a motor pump can fill a tank in 8 minutes and there is a tap that can empty it in 7 minutes. if the tank already have one-fourth filled and both the pump and tap is opened together, when will the tank emptied?

12 hours ago 3
Nature

The pump fills the tank in 8 minutes, so its filling rate is 18\frac{1}{8}81​ of the tank per minute. The tap empties the tank in 7 minutes, so its emptying rate is 17\frac{1}{7}71​ of the tank per minute. When both the pump and the tap are opened together, the net rate of change in the tank's water level is the filling rate minus the emptying rate:

Net rate=18−17=7−856=−156 of the tank per minute\text{Net rate}=\frac{1}{8}-\frac{1}{7}=\frac{7-8}{56}=-\frac{1}{56}\text{ of the tank per minute}Net rate=81​−71​=567−8​=−561​ of the tank per minute

The negative sign means the tank is being emptied overall at a rate of 156\frac{1}{56}561​ of the tank per minute. Since the tank is initially one- fourth full (14\frac{1}{4}41​), the amount of water to be emptied before the tank is empty is:

14 of the tank\frac{1}{4}\text{ of the tank}41​ of the tank

The time ttt it will take to empty the tank from one-fourth full at the net emptying rate is:

t=14156=14×56=14 minutest=\frac{\frac{1}{4}}{\frac{1}{56}}=\frac{1}{4}\times 56=14\text{ minutes}t=561​41​​=41​×56=14 minutes

So, the tank will be emptied in 14 minutes when both the pump and the tap are opened together starting from one-fourth full. This matches the calculated logic and confirms the answer.

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