Between X and Y, who is the better dancer, and which of the given statements is or are sufficient to answer this question?

Introduction / Context:
This is a data sufficiency question. You are not asked to actually decide who is the better dancer, but only whether the given statements provide enough information to answer that question logically and definitely.

Given Data / Assumptions:

• Main question: Between X and Y who is the better dancer?
• Statement 1: X had given more dance auditions.
• Statement 2: Y had given more stage performances.
• We must judge sufficiency, not make assumptions about talent.

Concept / Approach:
In data sufficiency, a statement is sufficient only if it allows you to answer the main question with certainty, without needing additional information or assumptions. More auditions or more performances may indicate higher activity but do not necessarily prove greater skill or quality.

Step-by-Step Solution:
Step 1: Consider statement 1 alone. X has given more auditions, but this does not guarantee that X is the better dancer. X may be less talented and therefore needs more attempts, or may simply be more active. Step 2: Consider statement 2 alone. Y has given more stage performances, but this also does not prove that Y is better. Y could have more opportunities, or the performances could be of a lower level. Step 3: Consider both statements together. Knowing that X has more auditions and Y has more performances still does not allow us to rank their dancing ability with certainty. We do not know judges' opinions, quality ratings, or any objective measure of skill. Step 4: Therefore neither statement alone, nor both together, is sufficient to decide who is the better dancer.

Verification / Alternative check:
Construct two scenarios. In one scenario, X is highly talented but shy and auditions rarely, while Y auditions and performs many times with average skill. In another scenario, Y is extremely talented and therefore performs a lot, while X keeps auditioning but does not get selected. Both scenarios fit the statements but lead to different answers about who is better, showing insufficiency of the data.

Why Other Options Are Wrong:
Option A assumes that having more auditions and performances together is enough, which is not logically justified. Options C and D each assume one of the statements is sufficient, which ignores many possibilities. Option E suggests that together the statements are enough, but even combined they do not give a clear criterion for dancing skill.

Common Pitfalls:
A common error is to equate quantity of activity with quality of performance. In data sufficiency, you must resist such assumptions and rely strictly on what the statements guarantee.

Final Answer:
The correct evaluation is that neither statement 1 nor statement 2 is sufficient to determine who is the better dancer.