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 given question. Read both the statements and give answer. How old will $C$ be after 10 years ? I. Five years ago, the average age of $A$ and $B$ was 15 years. II. Average age of $A$, $B$ and $C$ today is 20 years.

### Concept & Strategy To find someone's age in the future ($C + 10$), we must first determine their present age ($C$). We can find a single individual's value from a group average if we know the total sum of the group and the sum of everyone else in that group. ### Step-by-Step Solution **Target:** Find the value of $(C + 10)$. **Evaluating Statement I:** * Five years ago, the average age of $A$ and $B$ was 15. * Sum of their ages five years ago = $15 \times 2 = 30$. * Present sum of their ages ($A + B$) = $30 + 5 + 5 = 40$. * We have no information about $C$. * Statement I alone is not sufficient. **Evaluating Statement II:** * The present average age of $A$, $B$, and $C$ is 20. * Present sum ($A + B + C$) = $20 \times 3 = 60$. * Without knowing the combined age of $A$ and $B$, we cannot isolate $C$. * Statement II alone is not sufficient. **Evaluating Statements I and II Together:** * From Statement II, $A + B + C = 60$. * From Statement I, $A + B = 40$. * Substituting the value of $(A + B)$: $$40 + C = 60$$ $$C = 20$$ * $C$'s age after 10 years will be $20 + 10 = 30$. We found a unique answer. * Both statements are required. ### Exam Strategy & Shortcut Map the variables. Statement II gives the total block $(A, B, C)$. Statement I gives the sub-block $(A, B)$ shifted in time. By age-shifting the $(A, B)$ block to the present, you can subtract it from the total block to isolate $C$. Since the logic perfectly aligns without any missing variables, you can confidently check "both" without calculating the final age of 30. ### Common Pitfall A frequent mistake is forgetting to add 5 years for *both* $A$ and $B$ when bringing their past sum to the present. If you only add 5 once, you get a present sum of 35 instead of 40, throwing off the calculation. Always age every individual in the group. ### Final Answer **Therefore, the correct answer is If the data in both Statements I and II together are necessary to answer the question.**