Todd recently competed in a dance contest. The contest had exactly 4 judges, each of whom gave Todd an integer...

Understanding the Question

We need to determine if any judge gave Todd a rating of 10.

Given Information

• 4 judges total
• Each judge gives an integer rating from 1 to 10
• Todd's mean rating = 7.5
• Therefore, total sum of ratings = \(4 \times 7.5 = 30\)

What We Need to Determine

This is a yes/no question. We need sufficient information to answer either:

• YES: At least one judge gave a 10, or
• NO: No judge gave a 10

The key insight: With exactly 30 points distributed among 4 integer ratings (each between 1-10), we're looking for whether a 10 must appear, can appear, or cannot appear in that distribution.

Analyzing Statement 1

Statement 1 tells us: One judge gave Todd a rating of 2

What This Means

If one rating is 2, then the remaining 3 ratings must sum to 28 (since \(30 - 2 = 28\)).

The Key Constraint

Here's the critical insight: We need to distribute 28 points among 3 ratings, where each rating is between 1 and 10.

Let's check if it's possible without using a 10:

• Maximum possible without a 10: three 9s = 27 points
• But we need: 28 points
• Since \(27 < 28\), we cannot reach the required sum without at least one 10

Verification Example

With one judge giving a 2, a valid rating set could be: {2, 9, 9, 10}

• Check: \(2 + 9 + 9 + 10 = 30\) ✓

Conclusion

Statement 1 is sufficient. We can definitively answer YES - at least one judge must have given Todd a 10.

[STOP - Sufficient!]

This eliminates choices B, C, and E.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us: Each of 3 judges gave Todd ratings of 8 or higher

What This Means

• Three ratings are ≥ 8 each
• One rating could be anything from 1 to 10
• The three high ratings contribute at least 24 points (if all are 8s)

Testing Different Scenarios

Let's see if we can construct valid rating sets with and without a 10:

Scenario 1 - Without a 10:

• Three judges give: \(8 + 8 + 8 = 24\)
• Fourth judge must give: \(30 - 24 = 6\)
• Valid rating set: {8, 8, 8, 6} ✓
• Contains a 10? NO

Scenario 2 - With a 10:

• Three judges give: \(8 + 9 + 10 = 27\)
• Fourth judge must give: \(30 - 27 = 3\)
• Valid rating set: {8, 9, 10, 3} ✓
• Contains a 10? YES

Both scenarios satisfy all our constraints but give different answers to whether any judge gave a 10.

Conclusion

Statement 2 is NOT sufficient. We cannot determine whether any judge gave Todd a 10.

This eliminates choices B and D.

The Answer: A

Statement 1 alone forces at least one rating to be 10 (mathematical necessity), while Statement 2 allows scenarios both with and without a 10 (multiple possibilities).

Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."