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Tap A can fill a tank in 6 hours, Tap B in 8 hours, and Tap C can empty the full tank in 12 hours. If all are opened together, how long will it take to fill the empty tank?
4.8 ┬аhours
5.2 ┬аhours
5.5 ┬аhours
6.0 ┬аhours
8 ┬аhours
Assume total capacity as the LCM of 6, 8, and 12, which is 24 units. Calculate individual per-hour work rates as 4, 3, and -2 units.
Tap A fills in 6 hours, Tap B fills in 8 hours, Tap C empties in 12 hours.
T=Net┬аEfficiencyTotal┬аWorkтАЛO
Assume total capacity as the LCM of 6, 8, and 12, which is 24 units. Calculate individual per-hour work rates as 4, 3, and -2 units.
Subtracting the emptying tap's work rate instead of adding its negative value, or forgetting to include it entirely.
Find Total Work (LCM)
Take the LCM of the individual times (6, 8, and 12 hours) to assume a total tank capacity of 24 units.
LCM(6,8,12)=24
Determine Individual Rates
Calculate the work done per hour for each tap: Tap A is +4, Tap B is +3, and Tap C (emptying) is тИТ2.
RAтАЛ=4,RBтАЛ=3,RCтАЛ=тИТ2
Calculate Net Work Rate
Add the rates of all three taps operating simultaneously to find the net filling rate per hour.
RnetтАЛ=4+3тИТ2=5┬аunits/hour
Calculate Total Time
Divide the total tank capacity by the net per-hour efficiency to find the total time required.
T=524тАЛ=4.8┬аhours
A is correct because when all three taps are open together, the net rate is 5 units per hour, taking 24 / 5 = 4.8 hours to fill the tank.
This relative rate concept is identical to relative speed problems in Time, Speed, and Distance where objects move in opposite directions.
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