A set (X) of 20 pipes can fill 70% of a tank in 14 minutes. Another set (Y) of 10 pipes fills 3/8th of the tank in 6 minutes. A third set (Z) of 16 pipes can empty half of the tank in 20 minutes. If half of the pipes of set X are closed and only half of the pipes of set Y are open, and all pipes of the set (Z) are open, then how long will it take to fill 50% of the tank?

The correct option is 16 minutes.

Let us break down the problem step-by-step to calculate the work rate of each pipe set and then determine the total time required under the given conditions.

Step 1: Calculate the filling rate of 1 pipe of Set X
20 pipes of set X fill 70% (or 0.70) of the tank in 14 minutes.
Work done by 20 pipes of X in 1 minute = $\frac{0.70}{14}=0.05$ of the tank.
Work done by 1 pipe of X in 1 minute = $\frac{0.05}{20}=\frac{1}{400}$ of the tank.

Step 2: Calculate the filling rate of 1 pipe of Set Y
10 pipes of set Y fill $\frac{3}{8}$ (or 0.375) of the tank in 6 minutes.
Work done by 10 pipes of Y in 1 minute = $\frac{3}{8\times 6}=\frac{1}{16}$ of the tank.
Work done by 1 pipe of Y in 1 minute = $\frac{1}{16\times 10}=\frac{1}{160}$ of the tank.

Step 3: Calculate the emptying rate of 1 pipe of Set Z
16 pipes of set Z empty half of the tank (0.5) in 20 minutes.
Work done by 16 pipes of Z in 1 minute = $\frac{0.5}{20}=\frac{1}{40}$ of the tank (emptying).
Work done by 1 pipe of Z in 1 minute = $\frac{1}{40\times 16}=\frac{1}{640}$ of the tank (emptying).

Step 4: Determine the net rate under the new conditions
- Half of the pipes of Set X are open = $\frac{20}{2}=10$ pipes of X.
- Half of the pipes of Set Y are open = $\frac{10}{2}=5$ pipes of Y.
- All pipes of Set Z are open = 16 pipes of Z.

Now, compute the net fraction of the tank filled in 1 minute:
Net rate = (Rate of 10 pipes of X) + (Rate of 5 pipes of Y) - (Rate of 16 pipes of Z)

$\text{Net rate}=10\times \frac{1}{400}+5\times \frac{1}{160}-16\times \frac{1}{640}$

Simplifying each term:
$\text{Net rate}=\frac{1}{40}+\frac{1}{32}-\frac{1}{40}=\frac{1}{32}$

So, the open pipes fill $\frac{1}{32}$ of the tank per minute.

Step 5: Calculate the time needed to fill 50% of the tank
50% of the tank corresponds to $12$ of the tank capacity.

$\text{Time required}=\frac{1/2}{1/32}=\frac{32}{2}=16\text{minutes}$

Thus, it will take 16 minutes to fill 50% of the tank.