Let's solve this problem step by step.

Step 1: Total work to be done

• 600 men can complete the work in 500 days.
• Total work required = $600 \times 500 = 300,000 \text{ man-days}$.

Step 2: Work completed in the first 100 days

• In the first 100 days, there are 600 men working.
• Work completed by 600 men in 100 days = $600 \times 100 = 60,000 \text{ man-days}$.

Step 3: Work completed in the next 100 days

• After the first 100 days, 50 men leave, so now 550 men remain.
• Work completed by 550 men in the next 100 days = $550 \times 100 = 55,000 \text{ man-days}$.

Step 4: Work completed in the next 100 days

• After the second 100 days, another 50 men leave, leaving 500 men.
• Work completed by 500 men in the next 100 days = $500 \times 100 = 50,000 \text{ man-days}$.

Step 5: Work completed in the next 100 days

• After the third 100 days, another 50 men leave, leaving 450 men.
• Work completed by 450 men in the next 100 days = $450 \times 100 = 45,000 \text{ man-days}$.

Step 6: Work completed in the next 100 days

• After the fourth 100 days, another 50 men leave, leaving 400 men.
• Work completed by 400 men in the next 100 days = $400 \times 100 = 40,000 \text{ man-days}$.

Step 7: Total work completed so far

Summing up the work completed in each phase: $60,000 + 55,000 + 50,000 + 45,000 + 40,000 = 250,000 \text{ man-days}.$ The total work remaining is: $300,000 - 250,000 = 50,000 \text{ man-days}.$

Step 8: Final phase of work

• Now, 350 men are left (since after the fifth 100 days, another 50 men leave).
• The remaining work to be done is 50,000 man-days.
• The number of days required to complete the remaining work by 350 men is: $\frac{50,000}{350} \approx 142.86 \text{ days}.$

Step 9: Total time taken

The total time taken to complete the work is: $100 + 100 + 100 + 100 + 100 + 142.86 = 642.86 \text{ days}.$

Thus, the total time taken to complete the road is approximately 643 days.

Would you like further details or clarification on any step?

Here are some related questions to deepen your understanding:

1. How would the total time change if 75 men left every 100 days?
2. What if, instead of leaving, additional workers joined every 100 days?
3. How much work (in man-days) remains after 400 days?
4. What is the minimum number of men required to complete the remaining work in 100 days?
5. How would the problem change if the work rate per man varied over time?

Tip: In problems involving man-days, always calculate the total work in terms of man-days first before considering the varying workforce.