Introduction / Context:
Another ranking Data Sufficiency problem. We need Suman's top rank given up to three statements relating Suman and Deepak and the class size of 40.
Given Data / Assumptions:
- Total students = 40.
- Conversion: top_rank = total - bottom_rank + 1.
- “Below from the top” adds to the rank number; “above from the bottom” subtracts from the bottom rank number.
Concept / Approach:
Determine Deepak's absolute rank from one statement, then use the relative difference to get Suman's rank. Check which pairs of statements are sufficient.
Step-by-Step Solution:
From II: Deepak bottom rank = 23 → Deepak top rank = 40 - 23 + 1 = 18. Combine I + II: Suman is 3 below Deepak from top → Suman top rank = 18 + 3 = 21. (Sufficient.) Combine II + III: Suman is 3 above Deepak from bottom → Suman bottom rank = 23 - 3 = 20 → Suman top rank = 40 - 20 + 1 = 21. (Sufficient.) Combine I + III (without II): Let Deepak top rank be x ⇒ Suman top rank = x + 3. From bottom: Deepak bottom = 40 - x + 1; Suman bottom = 40 - (x + 3) + 1 = 38 - x. Statement III says Suman is 3 above Deepak from bottom ⇒ Suman bottom = Deepak bottom - 3. This simplifies to 38 - x = (40 - x + 1) - 3 ⇒ 38 - x = 38 - x, which is an identity. Hence I + III do not determine a unique rank.
Verification / Alternative check:
Both (I + II) and (II + III) compute Suman = 21 from the top consistently.
Why Other Options Are Wrong:
- Any two of the three / All three: Too strong; I + III alone are not enough.
- Only I and II or Only II and III: Each is sufficient, but the best characterization is “Only II and either I or III”.
Common Pitfalls:
Forgetting rank conversions; assuming I + III fix a value when they only impose a tautology without II's absolute reference.
Final Answer:
Only II and either I or III.
Discussion & Comments