Drain Pipe A can empty a completely filled water reservoir in 50 hours. Inlet Pipe B and Pipe A operating together can fill the empty reservoir in (m + 15) hours. The efficiency of Pipe B is 50% greater than that of Inlet Pipe C. Pipes B and C operating together can fill the empty reservoir in 20 hours. What is the value of 'm'?

The correct answer is 85.

Step 1: Determine the rate of Drain Pipe A

Drain Pipe A can empty the reservoir in 50 hours. Since it empties the reservoir, its rate is negative:

${\mathrm{Rate}}_A=-\frac{1}{50}$

reservoir per hour.

Step 2: Determine the rates of Inlet Pipe B and Inlet Pipe C

The efficiency of Pipe B is 50% greater than that of Pipe C. Therefore:

${\mathrm{Rate}}_B=1.5\times {\mathrm{Rate}}_C=\frac{3}{2}{\mathrm{Rate}}_C$

Pipes B and C operating together can fill the empty reservoir in 20 hours:

${\mathrm{Rate}}_B+{\mathrm{Rate}}_C=\frac{1}{20}$

Substituting ${\mathrm{Rate}}_B=\frac{3}{2}{\mathrm{Rate}}_C$ into the combined equation:

$\frac{3}{2}{\mathrm{Rate}}_C+{\mathrm{Rate}}_C=\frac{1}{20}$

$\frac{5}{2}{\mathrm{Rate}}_C=\frac{1}{20}$

${\mathrm{Rate}}_C=\frac{1}{20}\times \frac{2}{5}=\frac{1}{50}$

Now, calculate the rate of Pipe B:

${\mathrm{Rate}}_B=\frac{3}{2}\times \frac{1}{50}=\frac{3}{100}$

reservoir per hour.

Step 3: Calculate the combined rate of Pipe B and Pipe A

When Inlet Pipe B and Drain Pipe A operate together, their combined net rate is:

${\mathrm{Rate}}_{B+A}={\mathrm{Rate}}_B+{\mathrm{Rate}}_A$

${\mathrm{Rate}}_{B+A}=\frac{3}{100}-\frac{1}{50}$

${\mathrm{Rate}}_{B+A}=\frac{3}{100}-\frac{2}{100}=\frac{1}{100}$

reservoir per hour.

Step 4: Find total time and solve for m

The time taken by Pipes B and A operating together to fill the reservoir is:

$\mathrm{Time}=\frac{1}{1/100}=100$

hours.

We are given that this time is equal to $(m+15)$ hours:

$m+15=100$

$m=100-15$

$m=85$

Therefore, the value of m is 85.