Directions: Each of the questions 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 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 together are necessary to answer the question. What is the usual speed of the train? I. The speed of the train is increased by 25 km/hr to reach the destination 150 km away in time. II. The train is late by 30 minutes. (M.B.A., 2002)

### Concept & Logic When a train adjusts its speed to compensate for a delay to arrive on time, we can model the situation with a time-difference equation connecting usual speed ($v$), increased speed, distance ($d$), and the delay time ($\Delta t$). $$ \frac{d}{v} - \frac{d}{v + \text{increase}} = \Delta t $$ ### Step-by-Step Solution Let the usual speed of the train be $v$ km/hr and the usual time taken be $t$ hours. The distance is given as 150 km. From Statement I: * The speed is increased by 25 km/hr. New speed = $(v + 25)$ km/hr. * It reached "in time", meaning it overcame some unstated delay. * Because the delay amount is not specified, we cannot formulate a complete equation. Statement I is not sufficient. From Statement II: * The train is late by 30 minutes (0.5 hours). * This statement alone gives no information regarding the train's speed or the distance it travels. Statement II is not sufficient. Combining Statements I and II: * The train started 30 minutes late, but reached its destination on time because it increased its speed by 25 km/hr. * We can set up the equation: $\frac{150}{v} - \frac{150}{v + 25} = 0.5$ * This resolves to a quadratic equation with a single variable $v$. Solving it will yield a unique positive value for the usual speed. Thus, both statements together are required. ### Exam Strategy & Shortcut Identify the standard framework: a time-speed-distance equation requires distance, speed change, and time change. Statement I supplies distance and speed change. Statement II supplies time change. Therefore, both are inherently required to solve the framework. ### Common Pitfall Trying to solve the quadratic equation to find the exact speed. It is time-consuming and unnecessary for determining data sufficiency. Knowing that the equation is solvable for a unique positive variable is enough. ### Final Answer Therefore, the correct answer is **if the data in both Statements I and II together are necessary to answer the question**.