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 given 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; Give answer (e) if the data in both Statements I and II together are necessary to answer the question. A train crosses a signal post in $x$ seconds. What is the length of the train? (NABARD, 2002) I. The train crosses a platform of 100 metres in $y$ seconds. II. The train is running at the speed of 80 km/hr.

### Concept & Data Sufficiency Logic To find a specific numerical value for the length of a train, all variables in the final equation must be known numerical values. A variable cannot be solved if the given parameters themselves are unknown variables. $$Distance = Speed \times Time$$ ### Step-by-Step Solution * **Question Stem:** A train crosses a signal post in $x$ seconds. Let the length of the train be $L$ and its speed be $S$. Thus, $L = S \times x \implies S = \frac{L}{x}$. * **Analyze Statement I:** The train crosses a 100m platform in $y$ seconds. * Equation: $Speed = \frac{L + 100}{y}$. * Equating speeds: $\frac{L}{x} = \frac{L + 100}{y} \implies L \cdot y = L \cdot x + 100x \implies L(y - x) = 100x \implies L = \frac{100x}{y - x}$. * Since $x$ and $y$ are unknown variables (not numerical values), we cannot find a numerical value for $L$. Statement I alone is insufficient. * **Analyze Statement II:** The train's speed is 80 km/hr (which is $80 \times \frac{5}{18}$ m/s). * Equation: $L = Speed \times x = (80 \times \frac{5}{18}) \times x$. * Again, because $x$ is an unknown variable, we cannot find a numerical value for $L$. Statement II alone is insufficient. * **Combine Statements I & II:** * We have $S = 80$ km/hr. We can find a relation between $x$ and $y$, but because we still don't have the actual numerical value of the time $x$, we can never calculate the definitive length $L$. * Therefore, even combining both statements, we cannot find the exact length of the train. ### Exam Strategy & Shortcut In Data Sufficiency, if the question asks "What is the length?" it expects a unique numerical number. Since the question stem and Statement I provide times as unknown variables ($x$ and $y$), and neither statement provides numerical values for them, you can immediately deduce that a numerical length is impossible to calculate. ### Common Pitfall A common mistake is solving the equations algebraically to get $L = \frac{100x}{y - x}$ and assuming the data is sufficient because a formula was derived. Data sufficiency generally requires a definitive numerical answer unless specified otherwise. ### 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;**.