This problem involves work rates and is a classic example of inverse proportion.

Explanation:

• Let’s denote the total amount of work as $W$, which can be calculated based on the fact that 12 men can complete the job in 15 days.

  The total work done is the number of men times the number of days: $W = 12 \text{ men} \times 15 \text{ days} = 180 \text{ man-days}.$

• Now, if 5 men are working, they will complete $W$ man-days of work, but the number of days $D$ they take is what we need to find.

• The work done in $D$ days by 5 men can be written as: $W = 5 \text{ men} \times D \text{ days}.$ Thus: $180 = 5 \times D,$ Solving for $D$: $D = \frac{180}{5} = 36 \text{ days}.$

Final Answer:

It will take 5 men 36 days to finish the same job if they work at the same rate.

Would you like more details on this? Feel free to ask!

Here are 5 related questions:

1. What if 8 men worked on the same job, how long would it take them?
2. How can this concept be extended to different work rates for each man?
3. What if some workers worked faster than others, how would you adjust the calculation?
4. How would the time change if men worked in shifts with different work hours?
5. Can this concept be applied to machines working at different rates?

Tip: When dealing with work problems, remember that fewer workers or slower work rates mean it takes more time to complete the same job!