Directions: Each of the questions below consists of a statement and/or a question and two statements labelled I and II given below it. You have to decide whether the data provided in the statements is/are sufficient to answer the question. Read both the statements and Give answer (a) if the data in Statement I alone are sufficient to answer the question, while the data in Statement II alone are not sufficient to answer the question; Give answer (b) if the data in Statement II alone are sufficient to answer the question, while the data in Statement I alone are not sufficient to answer the question; Give answer (c) if the data either in Statement I or in Statement II alone are sufficient to answer the question; Give answer (d) if the data even in both Statements I and II together are not sufficient to answer the question; and Give answer (e) if the data in both Statements I and II are necessary to answer the question. The ages of Vinay and Sameer are in the ratio of 7 : 6. What is the age of Sameer? I. The ages of Vinay and Ajay are in the ratio of 7 : 4. II. After 5 years the ratio of Vinay's and Sameer's ages will be 8 : 7. (Bank P.O., 2008)

### Concept & Data Sufficiency Logic This is a Data Sufficiency problem involving ratios and ages. The goal is to determine if we can uniquely find the unknown variable (Sameer's age) using the given statements, without necessarily solving the entire equation. ### Step-by-Step Solution - From the main question, let Vinay's present age be $7x$ and Sameer's present age be $6x$. - **Evaluating Statement I:** This introduces a new person (Ajay) and gives the ratio of Vinay to Ajay as $7:4$. This does not provide a way to find the actual value of $x$. Statement I alone is not sufficient. - **Evaluating Statement II:** It states that after 5 years, the ratio of Vinay's and Sameer's ages will be $8:7$. - We can set up the following equation: $$ \frac{7x + 5}{6x + 5} = \frac{8}{7} $$ - Cross-multiplying gives: $7(7x + 5) = 8(6x + 5) \Rightarrow 49x + 35 = 48x + 40 \Rightarrow x = 5$. - Since we have successfully found $x$, we can determine Sameer's age ($6 \times 5 = 30$). - Therefore, Statement II alone provides enough information to solve the problem. ### Exam Strategy & Shortcut In Data Sufficiency exams, do not waste time calculating the final answer. As soon as you establish a single solvable linear equation with one variable from Statement II, you can confidently mark it as sufficient and move on. ### Common Pitfall A frequent mistake is fully solving for the final age (30 years) and losing precious exam time. Remember, the question only asks *if* you can solve it, not *what* the answer is. ### Final Answer Therefore, the correct answer is **if the data in Statement II alone are sufficient to answer the question, while the data in Statement I alone are not sufficient to answer the question**.