To solve this problem, we need to use the concept of combined work rates. Each pipe has a certain rate at which it drains the pool, and when they work together, their combined rate will allow us to find how long it takes to drain the pool.

Step-by-step solution:

1. Pipe 1's rate:

   • Pipe 1 can drain the pool in 5 days, so its rate is $\frac{1}{5}$ of the pool per day.

2. Pipe 2's rate:

   • Pipe 2 can drain the pool in 7 days, so its rate is $\frac{1}{7}$ of the pool per day.

3. Combined rate:

   • When both pipes work together, their combined rate is the sum of their individual rates: $\text{Combined rate} = \frac{1}{5} + \frac{1}{7}$ To add these fractions, we find a common denominator: $\frac{1}{5} + \frac{1}{7} = \frac{7}{35} + \frac{5}{35} = \frac{12}{35}$ So, the combined rate of both pipes is $\frac{12}{35}$ of the pool per day.

4. Time to drain the pool together:

   • The time to drain the pool when both pipes are working together is the reciprocal of their combined rate: $\text{Time} = \frac{35}{12} \text{ days}$ Simplifying this fraction gives: $\frac{35}{12} \approx 2.92 \text{ days}$

Thus, it will take approximately 2.92 days for both pipes to drain the pool together.

Would you like more details or have any questions?

Here are five related questions you could explore:

1. If Pipe 1 takes 4 days and Pipe 2 takes 6 days, how long will they take together?
2. How much of the pool will Pipe 1 drain in 1 day if Pipe 2 is not working?
3. If both pipes drain 50% of the pool together, how long would that take?
4. What happens if Pipe 2 is slower, taking 10 days to drain the pool?
5. How would the calculation change if there were three pipes involved?

Tip: When adding work rates, always express them as fractions and find a common denominator for easier calculations.