One pipe can fill the tank in 9 minutes while another pipe can empty completely filled tank in 90 minutes. If both the pipes are operated together on empty tank, how long (in minutes) will it take to fill half of the tank?

Correct option is A

Given:
One pipe can fill the tank in 9 minutes.
Another pipe can empty a completely filled tank in 90 minutes.
Both the pipes are operated together on an empty tank.
Solution:
Let the capacity of the tank be 1 unit.
The rate of the first pipe (filling) = $$\frac 19$$​(since it fills the tank in 9 minutes).
The rate of the second pipe (emptying) = $$\frac1{90}$$​(since it empties the tank in 90 minutes).
Efficiency of both pipe in 1 min = $$\frac{1}{9} - \frac{1}{90} = \frac{10}{90} - \frac{1}{90} = \frac{9}{90} = \frac{1}{10}$$​​
To fill half of the tank (i.e., 0.5 units), the time taken = $$\frac{0.5}{\frac{1}{10}} = 0.5 \times 10 = 5 \, \text{minutes}$$ 
Alternate Method:
Total Capacity of Tank = LCM of (10 and 90) = 90 unit
Efficiency of one pipe to fill tank per minute = $$\frac {90}{9}$$ = 10 unit per min.
​Efficiency of another pipe to empty tank per minute = $$\frac {90}{90}$$​ = -1 unit per min. 
Required Time to fill tank = $$\frac{\frac 12 \times 90} {10-1} = \frac{45}{9} = 5 min$$​​