Given:

Pipe A alone can fill the tank in 6 hours
Pipe B alone can empty the tank in 8 hours
Pipe C alone can fill the tank in 12 hours

Next crucial step, is to check whether the Total Work is defined with a specific Quantity. If not then to establish the given relations in the questions we can assume the work to be a VERY EASY TO WORK NUMBER.

EASY to Work number in most cases, for such question types, is usually the LCM of given rates.

In this case the LCM of rates (6,8,12) is 24. However, personally i would not use 24 as the Total capacity of the tank, i would rather choose 120(which is a multiple of 24) for ease of calculations.

Multiples of 10 are easier to work & they put me at ease. Imagine trying to solve this question under exam pressure, won't you welcome any possible comfort in the calculations? I definitely would.

Back to the question,

Assuming Total capacity of the Tank as 120 litres.

Pipe A alone can fill the Tank in 6 hours, hence rate of filling of A = 120/6 = 20 litres/hour

Pipe B alone can empty the Tank in 8 hours, hence rate of emptying of B = 120/8 = 15 litres/hour

Pipe C alone can fill the Tank in 12 hours, hence of rate of filling of C = 120/12 = 10 litres/hour

Now,

Activity 1
Pipe A is opened at 9 am, for one hour, till 10 am.

Pipe A alone for one hour will fill the Tank with 20 litres

Hence the Tank still has 120-20 = 100 litres left to be filled.

Activity 2
Pipe B is opened at 10 am, for one hour, till 11 am.

Pipe A filling the Tank & Pipe B Emptying the Tank for one hour.

Hence, 20 litres filled in & 15 litres emptied out in one hour, has a resultant effect of (20 - 15) = 5 litres filled in.

That means the Tank still has 100 - 5 = 95 litres to be filled in

Activity 3
Pipe C is opened at 11 am, now all pipes are open till the Tank is full. The three pipes together have to give a net effect of filling 95 litres.

Pipe A, B & C working together for one hour will have a resultant effect of (20-15+10) = 15 litres filled in one hour.

Hence in 6 hours, together they will fill 90 litres.

The remaining 5 litres will take (5*60)/15 = 20 mins, since all pipes together fill 15 litres per 60 mins

Therefore with all the pipes opened, the tank will fill the remaining 95 litres in 6 hours & 20 mins.

Now Total Time Taken to fill the Tank = 1 Hour (Pipe A) + 1 Hour (Pipe A+Pipe B) + 6 Hours 20 mins (Pipe A + Pipe B + Pipe C) = 8 hours 20 mins

Therefore the Time at which the Tank will be full is 9 am + 8 hours 20 mins = 5:20 pm.

Answer E. $$$$ $$$$