Directions : Each of the questions given below consists of a question followed by three statements. You have to study the question and the statements and decide which of the statements is/are necessary to answer the question. What is Ravi's present age ? I. The present age of Ravi is half of that of his father. II. After 5 years, the ratio of Ravi's age to that of his father's age will be 6 : 11. III. Ravi is 5 years younger than his brother.

### Concept & Logic This is a data sufficiency problem based on linear equations. To find a unique numerical value for an unknown variable, you need a system of independent equations equal to the number of unknown variables involved. ### Step-by-Step Solution Let Ravi's age be $R$, his father's age be $F$, and his brother's age be $B$. Let's translate the statements into equations: * **Statement I:** $R = \frac{1}{2}F \Rightarrow F = 2R$ * **Statement II:** $\frac{R + 5}{F + 5} = \frac{6}{11}$ * **Statement III:** $R = B - 5$ Now let's evaluate the combinations given in the options: * **Combining I and II:** We have two independent equations involving exactly two variables ($R$ and $F$). We can substitute $F = 2R$ into the second equation: $$ \frac{R + 5}{2R + 5} = \frac{6}{11} $$ Solving this gives $11(R + 5) = 6(2R + 5)$, which simplifies to $11R + 55 = 12R + 30 \Rightarrow R = 25$. Because we can find a unique age for Ravi, statements I and II together are sufficient. * **Combining II and III:** We have two equations but three variables ($R$, $F$, $B$). This cannot be solved for a unique value. * **Combining I and III:** We have two equations but three variables ($R$, $F$, $B$). This cannot be solved for a unique value. ### Exam Strategy & Shortcut **Variable Counting Method:** In data sufficiency, you rarely need to compute the final answer. Just count the variables and equations. Statements I and II deal exclusively with two entities (Ravi and his father) and provide two distinct mathematical relationships. Two variables plus two independent equations means it is solvable. Statement III introduces a third, unrelated entity (the brother) without providing a third equation to tie him back to the father, making it mathematically useless for finding Ravi's age without statement I. ### Common Pitfall Students often waste valuable exam time fully calculating the final age ($25$ years). Remember, the question only asks *which statements are necessary*, not what the actual answer is. Stop calculating as soon as you logically confirm that the system of equations is solvable! ### Final Answer **Therefore, the correct answer is I and II only.**