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. The sum of the ages of P, Q and R is 96 years. What is the age of Q? I. P is 6 years older than R. II. The total of the ages of Q and R is 56 years.

### Concept & Logic To find the specific value of one variable in a three-variable system, you typically need to isolate that variable using the given conditions. In Data Sufficiency, we don't always have to find the exact answer, but we must prove whether finding the exact answer is possible using the provided data. ### Step-by-Step Solution * **Given:** The sum of the ages of P, Q, and R is 96. Let their ages be $P$, $Q$, and $R$. $$P + Q + R = 96$$ * **Evaluating Statement I alone:** $P = R + 6$ Substituting this into the main equation gives: $$(R + 6) + Q + R = 96$$ $$2R + Q = 90$$ We have two variables ($Q$ and $R$) and only one equation. We cannot find the exact value of $Q$. *Statement I alone is NOT sufficient.* * **Evaluating Statement II alone:** $$Q + R = 56$$ Substituting this into the main equation gives: $$P + (56) = 96 \Rightarrow P = 40$$ While we found $P$, we still only know that $Q + R = 56$. We cannot determine $Q$ uniquely. *Statement II alone is NOT sufficient.* * **Evaluating Statements I and II together:** From Statement II, we know $Q + R = 56$ and $P = 40$. From Statement I, we know $P = R + 6$. Substituting $P = 40$ into Statement I's equation: $$40 = R + 6 \Rightarrow R = 34$$ Now, substitute $R = 34$ into Statement II's equation: $$Q + 34 = 56 \Rightarrow Q = 22$$ We can find the exact age of $Q$. ### Exam Strategy & Shortcut In Data Sufficiency, once you see that combining both statements yields a solvable system of equations (e.g., finding $P$ from II, using $P$ to find $R$ from I, and then finding $Q$), you do **not** need to calculate the final values ($R=34, Q=22$). As soon as you realize the path to the solution is complete and unique, immediately mark "Both are necessary" to save precious seconds. ### Common Pitfall A frequent mistake is stopping after Statement II reveals the value of $P$. Students sometimes assume that finding *any* variable means the data is sufficient, forgetting the question specifically asks for $Q$. Always keep the final target variable in mind. ### Final Answer **Therefore, the correct answer is The data in both Statements I and II together are necessary to answer the question.**