For five children with ages a < b < e ; any two successive ages differ by 2 years.

Correct Option (2)

The problem defines five children with ages a < b < c < d < e, where any two successive ages differ by 2 years. This establishes the following relationships for their ages:

• b = a + 2
• c = a + 4
• d = a + 6
• e = a + 8

The objective is to determine the age of the youngest child, denoted by a.

Evaluation of Statement I:

Statement I provides the condition: "The age of the eldest is 3 times the youngest."

• This can be expressed as e = 3a.
• Substituting the expression for e from the initial setup (e = a + 8): a + 8 = 3a.
• Solving this linear equation for a: 2a = 8, which yields a = 4 years.

Therefore, Statement I alone is sufficient to uniquely determine the age of the youngest child.

Evaluation of Statement II:

Statement II provides the condition: "The average age of the children is 8 years."

• The average age is calculated as (a + b + c + d + e) / 5 = 8.
• Substituting the expressions for b, c, d, e in terms of a: (a + (a + 2) + (a + 4) + (a + 6) + (a + 8)) / 5 = 8.
• Simplifying the sum in the numerator: (5a + 20) / 5 = 8.
• Multiplying both sides by 5: 5a + 20 = 40.
• Solving for a: 5a = 20, which yields a = 4 years.

Therefore, Statement II alone is also sufficient to uniquely determine the age of the youngest child.

Since the question can be answered using either Statement I alone or Statement II alone, Option 2 is the correct choice.

Incorrect Options:

Options 1, 3, and 4 are incorrect because the analysis demonstrates that the age of the youngest child (a) can be precisely determined using Statement I independently, and also using Statement II independently. Consequently, it is not required to use both statements together, nor is either statement alone insufficient, nor is the question unanswerable.