P and Q together can fill a cistern with water in 24 hours. If P alone can fill the cistern with water in 72 hours, then in how many hours will Q alone fill three-fourth of the same cistern with water?

The correct option is 27.

Let's break down the solution step-by-step to understand why 27 hours is the correct time for Q alone to fill three-fourth of the cistern.

Step 1: Determine the rate at which P and Q work together
Let the total capacity of the cistern be represented as 1 unit.
P and Q together can fill the cistern in 24 hours. Therefore, their combined rate of work (work done in 1 hour) is:
$$\text{Combined Rate (P + Q)}=\frac{1}{24}\text{of the cistern per hour}$$

Step 2: Determine the rate at which P works alone
P alone can fill the cistern in 72 hours. Therefore, P's individual rate of work is:
$$\text{Rate of P}=\frac{1}{72}\text{of the cistern per hour}$$

Step 3: Calculate the rate at which Q works alone
To find the rate of Q alone, we subtract the rate of P from the combined rate of P and Q:
$$\text{Rate of Q}=\text{Combined Rate (P + Q)}-\text{Rate of P}$$
Substituting the values we have:
$$\text{Rate of Q}=\frac{1}{24}-\frac{1}{72}$$
To subtract these fractions, we find a common denominator, which is 72:
$$\text{Rate of Q}=\frac{3-1}{72}=\frac{2}{72}=\frac{1}{36}$$
So, Q alone can fill the entire cistern in 36 hours.

Step 4: Calculate the time taken by Q to fill three-fourth of the cistern
We need to find the time Q takes to fill 3/4 of the cistern.
Using the formula:
$$\text{Time}=\frac{\text{Work to be done}}{\text{Rate of work}}$$
Substituting the work to be done (3/4 of the cistern) and Q's rate of work (1/36 of the cistern per hour):
$$\text{Time taken by Q}=\frac{\frac{3}{4}}{\frac{1}{36}}=\frac{3}{4}\times 36$$
Simplifying the expression:
$$\text{Time taken by Q}=3\times 9=27\text{hours}$$

Thus, Q alone will fill three-fourth of the same cistern in 27 hours.