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 statement(s) is/are necessary to answer the question. Is the average of the largest and the smallest of four given numbers greater than the average of the four numbers? I. The difference between the largest and the second largest numbers is less than the difference between the second largest and the second smallest numbers. II. The difference between the largest and the second largest numbers is greater than the difference between the second smallest and the smallest numbers. III. The difference between the largest and the second smallest numbers is greater than the difference between the second largest and the smallest numbers.

## Concept & Logic To evaluate inequalities involving averages, translate the word problem into a clean algebraic inequality using sorted variables. This allows you to directly compare the question's requirement against the statements. $$ \text{Let the sorted numbers be } a < b < c < d $$ ## Step-by-Step Solution * **Translating the Main Question:** Is the average of the largest ($d$) and smallest ($a$) greater than the average of all four? $$ \frac{a + d}{2} > \frac{a + b + c + d}{4} $$ Multiply both sides by $4$: $$ 2(a + d) > a + b + c + d $$ $$ 2a + 2d > a + b + c + d $$ Subtract $a$ and $d$ from both sides: $$ a + d > b + c $$ Rearrange to compare differences (as hinted by the statements): $$ d - c > b - a $$ The core question simplifies exactly to: **Is $(d - c) > (b - a)$?** * **Evaluate Statement I:** Difference between largest and second largest ($d - c$) is less than difference between second largest and second smallest ($c - b$). $d - c < c - b$. This does not relate to $(b - a)$ at all. Insufficient. * **Evaluate Statement II:** Difference between largest and second largest ($d - c$) is greater than difference between second smallest and smallest ($b - a$). $d - c > b - a$. This is an exact match for our simplified core question. It directly answers "Yes". Statement II alone is sufficient. * **Evaluate Statement III:** Difference between largest and second smallest ($d - b$) is greater than difference between second largest and smallest ($c - a$). $d - b > c - a$. Rearrange this inequality by adding $b$ and subtracting $c$ on both sides: $d - c > b - a$. This also mathematically simplifies to exactly match our core question. Statement III alone is sufficient. ## Exam Strategy & Shortcut Never try to plug in numbers for complex inequality data sufficiency questions; the permutations will eat up your time and cause errors. Always algebraically simplify the core question stem first. You will often find that the complex-sounding statements are just algebraic rearrangements of the simplified question stem. ## Common Pitfall Students often get lost in the heavy text ("second largest", "second smallest") and try to visualize a number line. This leads to confusion and logical gaps. Assigning variables ($a, b, c, d$) and translating the English directly into algebra is the only reliable method to avoid this trap. ## Final Answer **Therefore, the correct answer is Either II or III.**