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 the present age of Tanya ? I. The ratio between the present ages of Tanya and her brother Rahul is 3 : 4 respectively. II. After 5 years the ratio between the ages of Tanya and Rahul will be 4 : 5. III. Rahul is 5 years older than Tanya.

### Concept & Logic This is a two-variable problem involving Tanya ($T$) and Rahul ($R$). To find $T$, we need two independent conditions. ### Step-by-Step Solution Let's evaluate the statements by pairing them: * **Statement I:** $T : R = 3 : 4$ * **Statement II:** $(T + 5) : (R + 5) = 4 : 5$ * **Statement III:** $R - T = 5$ * **Using I and II:** Let present ages be $3x$ and $4x$. After 5 years: $\frac{3x + 5}{4x + 5} = \frac{4}{5}$. $15x + 25 = 16x + 20 \Rightarrow x = 5$. Tanya's age = $3 \times 5 = 15$. (Sufficient) * **Using II and III:** Let ages after 5 years be $4y$ and $5y$. The difference in ages is always constant. From III, difference is 5. So, $5y - 4y = 5 \Rightarrow y = 5$. Tanya's age in 5 years = $4 \times 5 = 20$. Present age = $20 - 5 = 15$. (Sufficient) * **Using I and III:** Let present ages be $3z$ and $4z$. From III, difference is 5. So, $4z - 3z = 5 \Rightarrow z = 5$. Tanya's age = $3 \times 5 = 15$. (Sufficient) ### Exam Strategy & Shortcut **Cross-Verification Shortcut:** Notice that Statement III ($R - T = 5$) is essentially the derived difference from the ratios in I ($3:4 \rightarrow$ difference of 1 unit) and II ($4:5 \rightarrow$ difference of 1 unit). Because all three statements are mathematically consistent and independent in pairs, any two will trigger the solution. No need to calculate all three pairs in the exam! ### Common Pitfall A common mistake is assuming that because Statement III looks simpler, it must be paired with a ratio, leading students to choose "I and III only" or "II and III only" while ignoring that the two ratios (I and II) combined also perfectly solve the problem. ### Final Answer **Therefore, the correct answer is Any two of the three.**