A9.30 PM
B10.45 PM
C11 PM
D11.48 PM
Answer:
D. 11.48 PM
Read Explanation:
Calculate Individual Rates:
Pipe A fills the tank in 10 hours. Its rate is 1/10 of the tank per hour.
Pipe B fills the tank in 15 hours. Its rate is 1/15 of the tank per hour.
Pipe C empties the tank in 12 hours. Its rate is -1/12 of the tank per hour (negative sign indicates emptying).
Determine Tank Capacity (using LCM):
Find the LCM of 10, 15, and 12.
LCM(10, 15, 12) = 60. Assume the tank capacity is 60 units.
In this case:
Rate of A = 60/10 = 6 units/hr
Rate of B = 60/15 = 4 units/hr
Rate of C = -60/12 = -5 units/hr
Calculate Work Done Before All Pipes Operate:
Pipe A starts at 1 PM.
Pipe B starts at 2 PM.
Pipe C starts at 3 PM.
Between 1 PM and 3 PM (2 hours), only pipe A is working.
Work done by A in 2 hours = 2 hrs * (1/10 tank/hr) = 2/10 = 1/5 tank.
Alternatively, using units: 2 hrs * 6 units/hr = 12 units. Remaining capacity = 60 - 12 = 48 units.
Calculate Combined Rate When All Pipes are Operational:
From 3 PM onwards, all three pipes A, B, and C are working.
Combined rate = Rate of A + Rate of B + Rate of C
Combined rate = (1/10) + (1/15) + (-1/12) tank/hr
To add these fractions, find a common denominator (LCM of 10, 15, 12 is 60):
Combined rate = (6/60) + (4/60) - (5/60) = (6 + 4 - 5) / 60 = 5/60 = 1/12 tank/hr.
Alternatively, using units: Combined rate = 6 + 4 - 5 = 5 units/hr.
Calculate Remaining Time to Fill the Tank:
At 3 PM, 1/5 of the tank is already filled (or 12 units are filled).
Remaining portion to be filled = 1 - (1/5) = 4/5 tank.
Alternatively, remaining units = 60 - 12 = 48 units.
Time required = Remaining Work / Combined Rate
Time = (4/5 tank) / (1/12 tank/hr) = (4/5) * 12 = 48/5 hours.
Alternatively, Time = 48 units / 5 units/hr = 9.6 hours.
Convert Time and Determine Final Filling Time:
9.6 hours needs to be converted into hours and minutes.
9.6 hours = 9 hours + 0.6 hours
0.6 hours * 60 minutes/hour = 36 minutes.
So, it takes 9 hours and 36 minutes from 3 PM to fill the rest of the tank.
Add this duration to the time when all pipes started working together (3 PM):
3:00 PM + 9 hours 36 minutes = 12:36 AM (next day).
Correction: Let's re-check the calculation of remaining work if we use the LCM method. At 3 PM, A has worked for 2 hours and B for 1 hour.
Work done by A from 1 PM to 3 PM: 2 hrs * 6 units/hr = 12 units.
Work done by B from 2 PM to 3 PM: 1 hr * 4 units/hr = 4 units.
Total filled by 3 PM = 12 + 4 = 16 units.
Remaining capacity = 60 - 16 = 44 units.
Time to fill remaining 44 units at a combined rate of 5 units/hr = 44 units / 5 units/hr = 8.8 hours.
Convert 8.8 hours: 8 hours + 0.8 hours * 60 minutes/hour = 8 hours and 48 minutes.
This time is calculated from 3 PM (when all pipes started working).
So, the tank will be filled at: 3:00 PM + 8 hours 48 minutes = 11:48 PM.