GATE CE 2019 Set 2 — Question 9
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Question
An oil tank can be filled by pipe X in 5 hours and pipe Y in 4 hours, each pump working on its own. When the oil tank is full and the drainage hole is open, the oil is drained in 20 hours. If initially the tank was empty and someone started the two pumps together but left the drainage hole open, how many hours will it take for the tank to be filled? (Assume that the rate of drainage is independent of the Head)
Correct answer
(C) 2.50
Solution
To find the total time required to fill the oil tank, we can determine the net rate of filling per hour when both inlet pipes (X and Y) are open and the drainage hole is also open.Let the total capacity of the tank be represented as (or one full tank).
$ \text{Rate of X} = \frac{1}{5} \text{ tank/hour}$
$ \text{Rate of Y} = \frac{1}{4} \text{ tank/hour}$
$ \text{Rate of Drainage} = -\frac{1}{20} \text{ tank/hour}$
$ \text{Net Rate} = \text{Rate of X} + \text{Rate of Y} - \text{Rate of Drainage}$ $ \text{Net Rate} = \frac{1}{5} + \frac{1}{4} - \frac{1}{20}$ To simplify, find a common denominator, which is :
$ \text{Net Rate} = \frac{4}{20} + \frac{5}{20} - \frac{1}{20} = \frac{4 + 5 - 1}{20} = \frac{8}{20} = \frac{2}{5} \text{ tank/hour}$
$ T = \frac{1}{\text{Net Rate}} = \frac{5}{2} = 2.50 \text{ hours}$Thus, it will take hours to fill the tank.Correct Option: C
1.Rate of Pipe X:
Pipe X can fill the tank in hours. $ \text{Rate of X} = \frac{1}{5} \text{ tank/hour}$
2.Rate of Pipe Y:
Pipe Y can fill the tank in hours.$ \text{Rate of Y} = \frac{1}{4} \text{ tank/hour}$
3.Rate of Drainage Hole:
The drainage hole can empty the full tank in hours. Since it removes oil, its rate is negative.$ \text{Rate of Drainage} = -\frac{1}{20} \text{ tank/hour}$
4.Net Rate of Filling:
When both pumps are working and the drainage hole is open, the net rate of filling per hour is:$ \text{Net Rate} = \text{Rate of X} + \text{Rate of Y} - \text{Rate of Drainage}$ $ \text{Net Rate} = \frac{1}{5} + \frac{1}{4} - \frac{1}{20}$ To simplify, find a common denominator, which is :
$ \text{Net Rate} = \frac{4}{20} + \frac{5}{20} - \frac{1}{20} = \frac{4 + 5 - 1}{20} = \frac{8}{20} = \frac{2}{5} \text{ tank/hour}$
5.Time Required to Fill the Tank:
The time taken to fill the empty tank completely is the reciprocal of the net rate:$ T = \frac{1}{\text{Net Rate}} = \frac{5}{2} = 2.50 \text{ hours}$Thus, it will take hours to fill the tank.Correct Option: C
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