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A cistern has two taps attached to it. Tap B can empty the cistern in 45 minutes. But Tap A can fill the cistern in just 30 minutes. Rohit started both taps unknowingly but realised his mistake after 30 minutes. He immediately closed Tap B. Now after this, in how much time will the cistern be filled?
15 min
20 min
30 min
45 min
20 min
Calculate LCM of 30 and 45 as 90 units. Tap A efficiency = +3, Tap B = -2. Net efficiency = +1. After 30 mins, 30 units are filled. B is closed, A fills remaining 60 units at rate 3, so 60/3 = 20 mins.
Tap A fills in 30 minutes, Tap B empties in 45 minutes. Both opened for 30 minutes, then B is closed.
Total┬аCapacity=LCM┬аof┬аtime┬аperiods, Work=Efficiency├ЧTime
Calculate LCM of 30 and 45 as 90 units. Tap A efficiency = +3, Tap B = -2. Net efficiency = +1. After 30 mins, 30 units are filled. B is closed, A fills remaining 60 units at rate 3, so 60/3 = 20 mins.
Many students forget that Tap B is a negative work (emptying) tap and add its capacity instead of subtracting it from Tap A.
Calculate efficiencies
Assuming the total capacity of the cistern is the LCM of 30 and 45, which is 90 units. The efficiency of Tap A is 90/30=3 units/min and Tap B is 90/45=2 units/min (emptying).
EfficiencyAтАЛ=3,EfficiencyBтАЛ=тИТ2
Determine net work in 30 minutes
With both taps open, the net filling rate is 3тИТ2=1 unit/min. After 30 minutes, the total quantity filled is 1├Ч30=30 units.
Work30minтАЛ=30┬аunits
Calculate remaining work
The remaining capacity to be filled is 90тИТ30=60 units. Now, Tap B is closed, so only Tap A is working.
Remaining=60┬аunits
Calculate time for final part
Tap A fills at a rate of 3 units/min. Time required to fill the remaining 60 units is 60/3=20 minutes.
Time=360тАЛ=20┬аmin
B is correct because the remaining capacity of 60 units is filled by Tap A at a rate of 3 units per minute, taking exactly 20 minutes.
This concept of 'Net Efficiency' is identical to solving speed-distance problems where one object moves against another; always subtract the opposing force or rate.
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