Abe, Beth, Carl and Duncan are four siblings, among which Abe and Carl

Detailed Solution

Step-I: Given Info

We are told about four siblings Abe, Beth, Carl and Duncan such that Abe and Carl are twins and Beth and Duncan are also twins. We are also given that the product of the present ages of the four siblings is 900. Further we are told that Beth is older than Abe and we are asked to find the age of Duncan

Step-II: Interpreting the Question Statement

Since Abe and Carl are twins, their ages would be same, let’s assume it to be \(x\). Similarly, since Beth and Duncan are twins, their ages would be same, let’s assume it to be \(y\).

We are told that Beth is older than Abe, i.e. \(y > x\) and the product of the ages of the siblings is 900, so we can write \(x^ 2 * y^2 = 900\).

We can observe here that 900 is written as product of two squares, since 900 can be prime factorized as \(900 = 2^2 * 3^2 * 5^2\), the possible set of values of (\(x, y\)) can be:

• (1, 30) or
• (2, 15) or
• (3, 10) or
• (5,6)

Let’s proceed to the solutions to see if we can get a unique value of \(x\) with this understanding.

Step-III: Statement I

Statement tells us that \(y\) \(–\) \(x\) is a prime number. Let’s evaluate our possible cases to see if we can find a unique value for \(x\):

• (1, 30) –>= 29-> Prime
• (2, 15) –>= 13 -> Prime
• (3, 10) – > = 7 -> Prime
• (5,6) – > = 1 -> Not Prime

We observe here that, there are three possible values for \(x\), hence statement-I is not sufficient to arrive at the answer.

Step-IV: Statement II

Statement-II tells us that had Carl been born four years earlier, the difference between Duncan’s age and Carl’s age would have been a prime number. Since Carl’s present age is \(x\), had he been born four years earlier, his present age would be (\(x +4\)).
The Statement tells us that \(y\) \(–\) \((x +4)\) is a prime number. Let’s evaluate our possible cases to see if we can find a unique value for \(x\).

• (1, 30) –>= 25-> Not Prime
• (2, 15) –>= 9 -> Not Prime
• (3, 10) – > = 3 -> Prime
• (5,6) – > = 3 -> Prime

We observe here that there are two possible values for \(x\), hence statement-II is not sufficient to arrive at the answer.

Step-V: Combining Statements I & II

Statement-I gives us the possible values of (\(x, y\)) as (1, 30), (2, 15) and (3, 10). Statement-II gives us the possible values of (\(x, y\)) as (3, 10) and (5,6).

Combining statement-I & II give us only possible option for values of (\(x, y\)) which is (3, 10).
Thus combination of St-I & II is sufficient to answer the question. Hence, the correct answer is Option C

Key Takeaways

1. Prime factorize a number to understand the ways in which a number can be represented

atom - you were right in describing why Statement-II alone is not sufficient but you did not consider the combinations of statement- I & II.

Regards
Harsh