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Two taps can fill a cistern in 2 hours and 51 hours respectively. A third tap can empty it in 51 hours. How long (in hours) will it take to fill the empty cistern, if all of them are opened together?
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Since the time taken by the second fill pipe and the emptying pipe are equal (51 hours), they effectively cancel each other out. Only the first pipe (2 hours) remains effective, filling the tank in 2 hours? Wait, let's re-calculate: 21тАЛ+511тАЛтИТ511тАЛ=21тАЛ. Thus, time = 2 hours.
Tap A fills in 2 hours, Tap B fills in 51 hours, and Tap C empties in 51 hours.
T=A1тАЛ+B1тАЛтИТC1тАЛ1тАЛ
Since the time taken by the second fill pipe and the emptying pipe are equal (51 hours), they effectively cancel each other out. Only the first pipe (2 hours) remains effective, filling the tank in 2 hours? Wait, let's re-calculate: 21тАЛ+511тАЛтИТ511тАЛ=21тАЛ. Thus, time = 2 hours.
Students often misinterpret the emptying tap or assume the filling time is the average; always focus on the work-rate sum: RatenetтАЛ=R1тАЛ+R2тАЛтИТR3тАЛ.
Define Rates
Calculate the work done per hour by each tap. Tap A rate = 1/2 per hour. Tap B rate = 1/51 per hour. Tap C rate = тИТ1/51 per hour (as it empties).
RateAтАЛ=21тАЛ,RateBтАЛ=511тАЛ,RateCтАЛ=тИТ511тАЛ
Calculate Net Rate
Sum the rates since all taps are open together: RatenetтАЛ=21тАЛ+511тАЛтИТ511тАЛ.
RatenetтАЛ=21тАЛ
Calculate Time
The time taken to fill the cistern is the reciprocal of the net rate.
Time=RatenetтАЛ1тАЛ=2┬аhours
D is correct because the net rate of the three taps together is 1/2 tank per hour, leading to a total time of 2 hours.
This is a fundamental concept in Time and Work problems; whenever an emptying pipe is involved, ensure you subtract its rate from the total capacity filling rate.
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