Explanation:

- Initially, 250 soldiers have food for 600 days.

- After 60 days, food has been consumed for these 60 days, leaving food for 540 more days for 250 soldiers.

- Calculate the total amount of food in soldier-days: $250\times 540=135,000$ soldier-days.

- 50 soldiers leave, leaving 200 soldiers.

- Divide the remaining food (135,000 soldier-days) by the 200 soldiers to find the number of days it will last: $\frac{135,000}{200}=675$ days.

- The correct option is Option 3: 675 days.