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Four friends–Amelie, Benedikt, Chiara, and Dominic–played four rounds of a card game together. In each round, each player scored a positive integer number of points. The number of points each player scored in each round is recorded in the table, except for Chiara's score in Round 3, which the scorekeeper forgot to record and is represented by \(\mathrm{x}\). The winner of the game was the player whose total score (the sum of the scores for all 4 rounds) was greatest. The friends correctly remember that Chiara scored the greatest number of points among the 4 friends in exactly 2 of the 4 rounds, and she scored a greater number of points in Round 3 than she did in each of the other 3 rounds. The position of \(\mathrm{x}\) and \(\mathrm{x + 154}\) in the table sorts does not indicate anything about their values.
| Player | Round 1 | Round 2 | Round 3 | Round 4 | Total |
|---|---|---|---|---|---|
| Amelie | 23 | 46 | 25 | 38 | 132 |
| Benedikt | 47 | 52 | 61 | 65 | 225 |
| Chiara | 51 | 55 | x | 48 | x + 154 |
| Dominic | 21 | 53 | 55 | 41 | 170 |
For each of the following statements, select Consistent if the statement is consistent with the information provided. Otherwise, select Not consistent.
Chiara's Round 3 score was 43.
Chiara's Round 3 score was 58.
Chiara won the game.
Let's start by understanding what we're working with. This table shows game scores for multiple players across 4 rounds, with Chiara's Round 3 score missing.
Looking at the raw data, we can immediately identify several key insights:
| Player | Round 1 | Round 2 | Round 3 | Round 4 | Total |
| Chiara | 51 | 55 | x | 48 | \(154+\mathrm{x}\) |
| Benedikt | 49 | 50 | 61 | 65 | 225 |
| [other players with lower scores] |
Key Insights:
This last point is crucial - it immediately tells us that \(\mathrm{x} < 61\) (Benedikt's Round 3 score).
The question asks us to determine which statements are consistent with the information given. We know Chiara scored highest in exactly 2 rounds, and we've already identified those rounds (1 and 2).
Let's establish the boundaries for x (Chiara's Round 3 score):
Now let's examine each statement.
Statement 1 Translation:
Original: "Chiara's Round 3 score was 43."
What we're looking for:
In other words: Can Chiara's unknown score be 43?
Let's check if \(\mathrm{x} = 43\) works with what we know:
Therefore, a score of 43 is cannot be consistent with all the information we have.
Statement 1 is Not consistent
Statement 2 Translation:
Original: "Chiara's Round 3 score was 58."
What we're looking for:
In other words: Can Chiara's unknown score be 58?
Let's check if \(\mathrm{x} = 58\) works with what we know:
Therefore, a score of 58 is perfectly consistent with all the information we have.
Statement 2 is Consistent
Statement 3 Translation:
Original: "Chiara won the game."
What we're looking for:
In other words: Is it possible for Chiara to win with any valid value for x?
Chiara's total score would be \(154 + \mathrm{x}\).
Benedikt's total score is 225.
For Chiara to win: \(154 + \mathrm{x} > 225\)
Solving for x: \(\mathrm{x} > 71\)
But we already established that \(\mathrm{x} < 61\) (since Benedikt had the highest score in Round 3).
Since x cannot simultaneously be > 71 and < 61, there is no value of x that would allow Chiara to win the game.
Statement 3 is Not Consistent
Evaluating the three statements:
Remember: In table analysis questions, the fastest approach often comes from recognizing what the constraints immediately tell you about the data, not from performing calculations on every value in the table.
Chiara's Round 3 score was 43.
Chiara's Round 3 score was 58.
Chiara won the game.