P and Q together can fill a cistern with water in 24 hours. If P alone can fill the cistern with water in 96 hours, then in how many hours will Q alone fill three-fourth of the same cistern with water?

The correct option is 24.

Let's solve the problem step-by-step by determining the work rates of P and Q.

First, we write down the rate at which P and Q fill the cistern together.
Since P and Q together can fill the cistern in 24 hours, their combined rate is:

$\frac{1}{24}$

part of the cistern per hour.

Next, we write down the rate of P alone.
P alone can fill the cistern in 96 hours, so P's rate is:

$\frac{1}{96}$

part of the cistern per hour.

To find the rate of Q alone, we subtract P's rate from their combined rate:

$\text{Rate of Q}=\frac{1}{24}-\frac{1}{96}$

We find a common denominator, which is 96, to subtract the fractions:

$\text{Rate of Q}=\frac{4-1}{96}$

$\text{Rate of Q}=\frac{3}{96}=\frac{1}{32}$

This means that Q alone can fill the entire cistern in 32 hours.

Now, we need to find the time taken by Q to fill three-fourth of the cistern.
Since it takes 32 hours to fill the complete cistern, the time taken to fill three-fourth of it is:

$\text{Time}=\frac{3}{4}\times 32\text{hours}$

$\text{Time}=3\times 8=24\text{hours}$

Thus, Q alone will fill three-fourth of the cistern in 24 hours.