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M10-35
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16 Sep 2014, 00:43
16 Sep 2014, 00:43
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In a football competition, the scoring system is: 3 points for a win, 1 point for a draw, and 0 points for a loss. With each team playing 20 matches in total, if a team has scored 9 points after 5 games, what is the least number of the remaining matches it must win to reach the 40-point mark by the end of the tournament?
A. 6
B. 7
C. 8
D. 9
E. 10
A. 6
B. 7
C. 8
D. 9
E. 10
Re M10-35
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16 Sep 2014, 00:43
16 Sep 2014, 00:43
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Expert Reply
Official Solution:
In a football competition, the scoring system is: 3 points for a win, 1 point for a draw, and 0 points for a loss. With each team playing 20 matches in total, if a team has scored 9 points after 5 games, what is the least number of the remaining matches it must win to reach the 40-point mark by the end of the tournament?
A. 6
B. 7
C. 8
D. 9
E. 10
To reach a total of 40 points, the team needs to gain an additional 31 points over the next 15 matches. To achieve this with the fewest wins, the team should aim for as many draws as possible, as a draw still provides points. Let's denote the number of draws as \(d\) and the number of victories as \(v\). Hence, we can establish the following system of equations:
\(d + v = 15\) (Total matches remaining)
\(d + 3v \ge 31\) (Points needed from these matches)
\(d + 3(15-d) \ge 31\)
\(d + 45-3d \ge 31\)
\(d \leq 7\)
Since \(d + v = 15\), then \(d \leq 7\) implies \(v \geq 8\). Thus, \(v\) must be at least 8. Therefore, the team needs a minimum of 8 victories (and 7 draws) to reach the 40-point mark by the end of the tournament.
Answer: C
In a football competition, the scoring system is: 3 points for a win, 1 point for a draw, and 0 points for a loss. With each team playing 20 matches in total, if a team has scored 9 points after 5 games, what is the least number of the remaining matches it must win to reach the 40-point mark by the end of the tournament?
A. 6
B. 7
C. 8
D. 9
E. 10
To reach a total of 40 points, the team needs to gain an additional 31 points over the next 15 matches. To achieve this with the fewest wins, the team should aim for as many draws as possible, as a draw still provides points. Let's denote the number of draws as \(d\) and the number of victories as \(v\). Hence, we can establish the following system of equations:
\(d + v = 15\) (Total matches remaining)
\(d + 3v \ge 31\) (Points needed from these matches)
\(d + 3(15-d) \ge 31\)
\(d + 45-3d \ge 31\)
\(d \leq 7\)
Since \(d + v = 15\), then \(d \leq 7\) implies \(v \geq 8\). Thus, \(v\) must be at least 8. Therefore, the team needs a minimum of 8 victories (and 7 draws) to reach the 40-point mark by the end of the tournament.
Answer: C
Re: M10-35
[#permalink]
19 Oct 2023, 11:33
19 Oct 2023, 11:33
Expert Reply
I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
