Two pumps, X and Y, can independently fill a water reservoir in 40 minutes and 60 minutes, respectively. A drainage outlet, Z, evacuates water at a rate that is 25% more efficient than pump Y's filling rate. How long will it take to fill the reservoir if pumps X and Y and drainage outlet Z are all opened simultaneously?

The correct option is 48 min.

Step-by-Step Explanation:

Step 1: Determine the individual rates of pumps X and Y

Let the total capacity of the reservoir be 1 unit.

Filling rate of pump X = $\frac{1}{40}$ of the reservoir per minute.

Filling rate of pump Y = $\frac{1}{60}$ of the reservoir per minute.

Step 2: Calculate the evacuation rate of drainage outlet Z

Drainage outlet Z evacuates water at a rate that is 25% more efficient than pump Y's filling rate.

Evacuation rate of outlet Z = $\frac{1}{60}\times (1+0.25)=\frac{1}{60}\times \frac{5}{4}=\frac{5}{240}=\frac{1}{48}$ of the reservoir per minute.

Step 3: Calculate the net rate when all three are opened simultaneously

Since pumps X and Y fill the reservoir while outlet Z drains water, the net rate of filling per minute is:

$\text{Net Rate}=\text{Rate of X}+\text{Rate of Y}-\text{Rate of Z}$

$\text{Net Rate}=\frac{1}{40}+\frac{1}{60}-\frac{1}{48}$

Taking the Least Common Multiple (LCM) of 40, 60, and 48, which is 240:

$\text{Net Rate}=\frac{6+4-5}{240}=\frac{5}{240}=\frac{1}{48}$ of the reservoir per minute.

Step 4: Calculate the total time required

Time required to fill the reservoir = $\frac{1}{\text{Net Rate}}=\frac{1}{1/48}=48$ minutes.

Therefore, it will take 48 minutes to fill the reservoir completely when pumps X and Y and drainage outlet Z are all open.