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29 Jul 2024, 08:23
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The French team and the German team are playing a series of games against each other, where a win results in +3 points, a draw results in 0 points, and a loss results in -2 points. If each team played fewer than 10 games in the series and each team experienced all three outcomes, how many games did each team play?
(1) The French team scored 14 points in the series.
(2) The German team scored -6 points in the series.
(1) The French team scored 14 points in the series.
(2) The German team scored -6 points in the series.
29 Jul 2024, 08:23
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Official Solution:
The French team and the German team are playing a series of games against each other, where a win results in +3 points, a draw results in 0 points, and a loss results in -2 points. If each team played fewer than 10 games in the series and each team experienced all three outcomes, how many games did each team play?
Assuming the French team won \(x\) games, lost \(y\) games, and drew \(z\) games, the question asks for the value of \(x + y + z\), given that \(x + y + z\) is less than 10. Note that the French team winning \(x\) games and losing \(y\) games implies that the German team won \(y\) games and lost \(x\) games. Also, since each team experienced all three outcomes, \(x\), \(y\), and \(z\) must each be greater than 0.
(1) The French team scored 14 points in the series.
This implies that \(3x - 2y = 14\). By trial and error, we can find that only one set of \((x, y)\) such that \(x + y < 10\) satisfies this equation: (6, 2). For (6, 2), \(x + y = 8\), and since \(z > 0\), \(z\) must be 1 for \(x + y + z\) to be less than 10. Therefore, \(x = 6\), \(y = 2\), and \(z = 1\). Total games played by each team \(= 6 + 2 + 1 = 9\). Sufficient.
(2) The German team scored -6 points in the series.
This implies that \(3y - 2x = -6\). By trial and error, we can find that only one set of \((x, y)\) such that \(x + y < 10\) satisfies this equation: (6, 2). As with statement (1), for (6, 2), we can conclude that \(z\) must be 1. Therefore, \(x = 6\), \(y = 2\), and \(z = 1\). Total games played by each team \(= 6 + 2 + 1 = 9\). Sufficient.
Answer: D
The French team and the German team are playing a series of games against each other, where a win results in +3 points, a draw results in 0 points, and a loss results in -2 points. If each team played fewer than 10 games in the series and each team experienced all three outcomes, how many games did each team play?
Assuming the French team won \(x\) games, lost \(y\) games, and drew \(z\) games, the question asks for the value of \(x + y + z\), given that \(x + y + z\) is less than 10. Note that the French team winning \(x\) games and losing \(y\) games implies that the German team won \(y\) games and lost \(x\) games. Also, since each team experienced all three outcomes, \(x\), \(y\), and \(z\) must each be greater than 0.
(1) The French team scored 14 points in the series.
This implies that \(3x - 2y = 14\). By trial and error, we can find that only one set of \((x, y)\) such that \(x + y < 10\) satisfies this equation: (6, 2). For (6, 2), \(x + y = 8\), and since \(z > 0\), \(z\) must be 1 for \(x + y + z\) to be less than 10. Therefore, \(x = 6\), \(y = 2\), and \(z = 1\). Total games played by each team \(= 6 + 2 + 1 = 9\). Sufficient.
(2) The German team scored -6 points in the series.
This implies that \(3y - 2x = -6\). By trial and error, we can find that only one set of \((x, y)\) such that \(x + y < 10\) satisfies this equation: (6, 2). As with statement (1), for (6, 2), we can conclude that \(z\) must be 1. Therefore, \(x = 6\), \(y = 2\), and \(z = 1\). Total games played by each team \(= 6 + 2 + 1 = 9\). Sufficient.
Answer: D
25 Aug 2025, 04:41
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Just a rant: every time I solve such DS questions, a little part inside me dies. I missed the inference of "and each team experienced all three outcomes" and got E, taking 5 mins to confidently arrive at the wrong answer. Chat, I am cooked.
