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. What was the ratio between the ages of $P$ and $R$ four years ago? I. The ratio between the present ages of $P$ and $Q$ is $3 : 4$. II. The ratio between the present ages of $Q$ and $R$ is $4 : 5$.

### Concept & Shifting Ratios Across Time While we can combine two ratios to find a compound ratio for the present time, subtracting a constant number of years from the numerator and denominator fundamentally changes the ratio. We need the exact ages to find a past or future ratio. ### Step-by-Step Solution * **Analyzing Statement I:** * Present ages: $P : Q = 3 : 4$. We know nothing about $R$. * Statement I is not sufficient. * **Analyzing Statement II:** * Present ages: $Q : R = 4 : 5$. We know nothing about $P$. * Statement II is not sufficient. * **Combining Both Statements:** * We have $P : Q = 3 : 4$ and $Q : R = 4 : 5$. * Since the common term $Q$ has the same ratio value (4) in both, we can combine them directly: $P : Q : R = 3 : 4 : 5$. * Let their present ages be $3x$, $4x$, and $5x$. * The question asks for the ratio of $P$ and $R$ four years ago. * Four years ago, $P$'s age was $3x - 4$ and $R$'s age was $5x - 4$. * The required ratio is $\frac{3x - 4}{5x - 4}$. * This ratio changes depending on the value of $x$. Without an actual age or a sum of ages to find $x$, we cannot determine the past ratio. ### Exam Strategy & Shortcut Whenever a question asks for a ratio involving ages in the past or future, and only provides present ratios (without any actual years or sums), the data will always be insufficient. ### Common Pitfall A common mistake is simply subtracting 4 from the ratio numbers directly (e.g., $3 - 4$ and $5 - 4$), which makes no mathematical sense. Ratios are multipliers, not absolute values. ### Final Answer Therefore, the correct answer is **if the data even in both Statements I and II together are not sufficient to answer the question**.