The correct answer is option 3, which means either I or II is sufficient to find out the difference in their ages.

Let`s analyze each option to understand why.

Option 1: Only I - This means that knowing that after five years, the ratio of their ages would be 9:5 is sufficient. This information alone allows us to determine the difference in their ages.

Option 2: Only II - This means that knowing that before ten years, the ratio of their ages was 3:1 is sufficient. This information alone allows us to determine the difference in their ages.

Option 3: Either I or II - This means that either knowing the information in option I or option II is sufficient to find out the difference in their ages. Thus, if we know either that after five years, the ratio of their ages would be 9:5, or that before ten years, the ratio of their ages was 3:1, we can determine the difference in their ages.

Option 4: Both I and II - This means that we need to know both pieces of information (option I and II) to find out the difference in their ages. However, the question states that either option I or II is sufficient, making option 4 incorrect