To solve this problem, we need to determine the combined rate of both pumps when they are used simultaneously.

Step 1: Calculate the rate for each pump

• The small pump can drain the pool in 77 hours, so its rate is: $\text{Small pump rate} = \frac{1 \text{ pool}}{77 \text{ hours}} = \frac{1}{77} \text{ pools per hour}$
• The large pump can drain the pool in 44 hours, so its rate is: $\text{Large pump rate} = \frac{1 \text{ pool}}{44 \text{ hours}} = \frac{1}{44} \text{ pools per hour}$

Step 2: Add the rates

When both pumps are working together, their combined rate is the sum of their individual rates: $\text{Combined rate} = \frac{1}{77} + \frac{1}{44}$

To simplify this, find a common denominator. The least common denominator of 77 and 44 is 154. So:

$\frac{1}{77} = \frac{2}{154}, \quad \frac{1}{44} = \frac{3}{154}$ Thus, $\text{Combined rate} = \frac{2}{154} + \frac{3}{154} = \frac{5}{154} \text{ pools per hour}$

Step 3: Calculate the time to drain the pool

The time to drain the pool is the reciprocal of the combined rate: $\text{Time} = \frac{1}{\text{Combined rate}} = \frac{154}{5} = 30.8 \text{ hours}$

Step 4: Convert the decimal to minutes

To convert 0.8 hours into minutes: $0.8 \times 60 = 48 \text{ minutes}$

So, the time it will take to drain the pool with both pumps working together is approximately 30 hours and 48 minutes.

Would you like further clarification or details on any part of the solution? Here are a few related questions you might find interesting:

1. How would the time change if the rates of both pumps were doubled?
2. What would the time be if the small pump drained the pool in 100 hours and the large pump in 60 hours?
3. How do the rates of two pumps working together compare to when they're working separately?
4. How can this type of problem be applied to other real-life scenarios?
5. What would happen if both pumps had a malfunction and could only function at half of their normal rates?

Tip: Always remember that combining rates involves adding them together, and to calculate time, you can take the reciprocal of the combined rate!