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Two taps can fill a cistern in 2 hours and 47 hours respectively. A third tap can empty it in 47 hours. How long (in hours) will it take to fill the empty cistern, if all of them are opened together?
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Notice Taps B and C have identical rates (47 h to fill vs 47 h to empty). They completely cancel each other out (+471тАЛтИТ471тАЛ=0). Thus, net rate equals Tap A alone, which takes 2 hours.
Tap A filling time = 2 hours, Tap B filling time = 47 hours, Tap C emptying time = 47 hours
TnetтАЛ1тАЛ=TAтАЛ1тАЛ+TBтАЛ1тАЛтИТTCтАЛ1тАЛ
Notice Taps B and C have identical rates (47 h to fill vs 47 h to empty). They completely cancel each other out (+471тАЛтИТ471тАЛ=0). Thus, net rate equals Tap A alone, which takes 2 hours.
Assuming the question asks to add all time values without accounting for the negative work of the outlet pipe C.
Define individual work rates
Let the capacity of the cistern be the LCM of 2 and 47, which is 94┬аunits. Calculate the efficiency/rate of each tap.
Capacity=LCM(2,47)=94┬аunits
Calculate rate per hour for each tap
Tap A (inlet) fills 294тАЛ=+47┬аunits/hr. Tap B (inlet) fills 4794тАЛ=+2┬аunits/hr. Tap C (outlet) empties 4794тАЛ=тИТ2┬аunits/hr.
RateAтАЛ=+47,RateBтАЛ=+2,RateCтАЛ=тИТ2
Calculate net rate when all taps are opened together
Combine the rates of filling and emptying taps: 47+2тИТ2=47┬аunits/hr. Notice that Tap B and Tap C cancel each other out completely.
Net┬аRate=47+2тИТ2=47┬аunits/hr
Calculate total time required
Divide the total capacity by the net filling rate to find total time.
Time=Net┬аRateTotal┬аCapacityтАЛ=4794тАЛ=2┬аhours
D is correct because the inlet rate of 2 units/hour from Tap B and outlet rate of 2 units/hour from Tap C completely cancel out, leaving the net filling time equal to Tap A's time of 2 hours.
When two components in a system perform identical work in opposite directions (e.g., downstream vs upstream components or filling vs emptying taps with identical time parameters), always cancel them out first to reduce calculation time.
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