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Todd recently competed in a dance contest. The contest had exactly \(\mathrm{4}\) judges, each of whom gave Todd an integer rating from \(\mathrm{1}\) to \(\mathrm{10}\). Todd received a mean rating of \(\mathrm{7.5}\). Did any judge give Todd a rating of \(\mathrm{10}\)?
We need to determine if any judge gave Todd a rating of 10.
This is a yes/no question. We need sufficient information to answer either:
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.
Statement 1 tells us: One judge gave Todd a rating of 2
If one rating is 2, then the remaining 3 ratings must sum to 28 (since \(30 - 2 = 28\)).
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:
With one judge giving a 2, a valid rating set could be: {2, 9, 9, 10}
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.
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
Let's see if we can construct valid rating sets with and without a 10:
Scenario 1 - Without a 10:
Scenario 2 - With a 10:
Both scenarios satisfy all our constraints but give different answers to whether any judge gave a 10.
Statement 2 is NOT sufficient. We cannot determine whether any judge gave Todd a 10.
This eliminates choices B and D.
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."