Directions : Each of the questions given below consists of a statement and / or a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statement(s) is / are sufficient to answer the question. Read both the statements and choose the appropriate option. How old is C now? I. Three years ago, the average of A and B was 18 years. II. With C joining them now, the average becomes 22 years.

### Concept & Logic To find the age of a specific person (C) joining a group, you need to find the difference between the sum of the ages of the new group (A + B + C) and the sum of the ages of the old group (A + B) at the same point in time (present). Average formulas are the key to finding these sums. ### Step-by-Step Solution * **Evaluating Statement I alone:** "Three years ago, the average of A and B was 18 years." Sum of A and B three years ago = $18 \times 2 = 36$ years. To find their present sum, we must add 3 years for A and 3 years for B. Present sum of A and B = $36 + 3 + 3 = 42$ years. This tells us nothing about C. *Statement I alone is NOT sufficient.* * **Evaluating Statement II alone:** "With C joining them now, the average becomes 22 years." This means the present average of A, B, and C is 22. Present sum of A, B, and C = $22 \times 3 = 66$ years. Without knowing the sum of A and B, we cannot isolate C. *Statement II alone is NOT sufficient.* * **Evaluating Statements I and II together:** From I, we know the present sum of A + B = 42. From II, we know the present sum of A + B + C = 66. By subtracting the first sum from the second, we can easily find C's present age: $$C = (A + B + C) - (A + B)$$ $$C = 66 - 42 = 24$$ years. Since combining both gives us the answer, both are necessary. ### Exam Strategy & Shortcut When dealing with averages of groups, focus entirely on the **Sum**. The moment you realize Statement I gives you the present sum of (A+B) and Statement II gives you the present sum of (A+B+C), you instantly know subtraction will yield C. You do not need to calculate $66 - 42 = 24$ to mark the correct Data Sufficiency option. ### Common Pitfall The most dangerous pitfall is bringing past sums to the present incorrectly. Students often find the sum 3 years ago ($18 \times 2 = 36$) and then just add 3 to get the present sum ($36 + 3 = 39$). You MUST add the time gap for *every* individual in the group ($36 + 3 \times 2 = 42$). ### Final Answer **Therefore, the correct answer is The data in both Statements I and II together are necessary to answer the question.**