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. How long did the secretary's speech last? I. He spoke at an average of 50 words per minute. II. He would have spoken for 10 minutes extra, had his speech rate been 4 words less per minute.

## Concept & Logic This problem operates on the core concept of Work and Time, specifically adapted for a speech rate. The total number of words spoken is constant in both scenarios. $$ \text{Total Words} = \text{Speech Rate} \times \text{Time Duration} $$ ## Step-by-Step Solution * **Evaluating Statement I:** * We are given the rate: $50$ words per minute. * We do not know the total word count or the time duration. * Statement I alone is **not sufficient**. * **Evaluating Statement II:** * Let the original rate be $r$ and original time be $t$. * The new rate is $(r - 4)$ and the new time is $(t + 10)$. * We know $r \times t = (r - 4)(t + 10)$, but this is one equation with two variables ($r$ and $t$). * Statement II alone is **not sufficient**. * **Evaluating Both Statements Together:** * From Statement I, we know original $r = 50$. * Substitute $r = 50$ into the equation from Statement II: * $$ 50 \times t = (50 - 4) \times (t + 10) $$ * $$ 50t = 46(t + 10) $$ * $$ 50t = 46t + 460 $$ * $$ 4t = 460 \implies t = 115 \text{ minutes} $$ * By combining both statements, we can find the exact duration. ## Exam Strategy & Shortcut For Data Sufficiency problems involving dual scenarios (like a change in speed altering the time), recognize that setting up an equivalence equation ($\text{Words}_1 = \text{Words}_2$) usually requires both initial speed and initial time to be solvable. Once you see that Statement I gives the missing variable (speed) to solve Statement II's relative equation, you immediately know both are required. ## Common Pitfall A frequent mistake is assuming Statement II implies the normal speech rate is already known, leading to mistakenly selecting option (b). Always evaluate each statement strictly in isolation first. ## Final Answer **Therefore, the correct answer is If the data in both Statements I and II together are necessary to answer the question.**