A total of four students belong to all three teams, the baseball, basketball and football team. These four students represent 10% of the baseball team, 25% of the basketball team and 8 1/3 % of the football team. There are 20 students who are just on two teams. What percent of all the different players on the three teams only play on one team?
Let's call the total number of players on the baseball team "B", the total number of players on the basketball team "K", and the total number of players on the football team "F". We know that:
- 4 students belong to all three teams
- These 4 students represent 10% of the baseball team, 25% of the basketball team, and 8 1/3 % of the football team
From this, we can set up three equations:
0.1B = 4
0.25K = 4
0.0833F = 4
Solving for B, K, and F, we get:
B = 40
K = 16
F = 48
Now we know the total number of players on each team. We also know that there are 20 students who are on two teams, but we don't know how many of these are on each combination of teams. Let's call the number of students on both the baseball and basketball teams "BB", the number of students on both the baseball and football teams "BF", and the number of students on both the basketball and football teams "KF". We know that:
- BB + BF + KF = 20
We don't have enough information to solve for BB, BF, and KF individually. However, we can use the principle of inclusion-exclusion to find the total number of students who play at least one of the three sports:
Total = B + K + F - (BB + BF + KF) + 4
Plugging in the values we found earlier, we get:
Total = 40 + 16 + 48 - (BB + BF + KF) + 4
Total = 108 - (BB + BF + KF)
So, we know that the total number of students who play at least one of the three sports is 108. We also know that the number of students who play at least two sports is 20. Therefore, the number of students who play only one sport is:
Total - number who play at least two sports = 108 - 20 = 88
So, out of the 100 students, 88 play only one sport, 4 play all three sports, and the remaining 8 play two sports each. We don't know how many play which combination of two sports, but we do know that:
BB + BF + KF = 8
Therefore, there are 8 students who play two sports, and we can't determine how many play each combination.
Let's assume that the number of players on the baseball team is B, the number of players on the basketball team is K, and the number of players on the football team is F.
From the given information, we can create a system of equations:
B + K + F - 4 = Total number of players (since the 4 students are counted in each team)
4 = 0.1B = 0.25K = 0.0833F (since the four students belong to all three teams and represent the given percentages of each team)
20 = Number of players on exactly two teams
To solve for B, K, and F, we can use the second equation and solve for one variable in terms of the others. Let's solve for B:
4 = 0.1B
B = 40
Similarly, we can solve for K and F:
4 = 0.25K
K = 16
4 = 0.0833F
F = 48
Now we know that there are 40 players on the baseball team, 16 players on the basketball team, and 48 players on the football team. We can use this information and the first equation to find the total number of players:
B + K + F - 4 = Total number of players
40 + 16 + 48 - 4 = 100
Therefore, there are 100 players in total
Now let's find the number of players who play on exactly one team. Let's start with the baseball team:
Number of players on the baseball team who play on exactly one team = 0.9B - 20 (since the 4 students who play on all three teams are counted)
Number of players on the baseball team who play on exactly one team = 0.9(40) - 20 = 16
Similarly, we can find the number of players on the basketball and football teams who play on exactly one team:
Number of players on the basketball team who play on exactly one team = 0.75K - 20 = 2
Number of players on the football team who play on exactly one team = 0.9167F - 20 = 28
Therefore, the total number of players who play on exactly one team is:
16 + 2 + 28 = 46
Finally, we can find the percentage of all the different players on the three teams who only play on one team:
Percentage of players who play on exactly one team = (Number of players who play on exactly one team / Total number of players) x 100
Percentage of players who play on exactly one team = (46 / 100) x 100 = 46%
Therefore,
46% of all the different players on the three teams only play on one team.
To find out the percentage of players who only play on one team, we need to calculate the total number of students on each team.
Let's assume:
- The total number of students on the baseball team is B.
- The total number of students on the basketball team is K.
- The total number of students on the football team is F.
According to the given information:
- 4 students belong to all three teams.
- These 4 students represent 10% of the baseball team, 25% of the basketball team, and 8 1/3 % of the football team.
From this, we can create three equations:
1. 0.10B = 4 (10% of the baseball team)
2. 0.25K = 4 (25% of the basketball team)
3. 0.0833F = 4 (8 1/3 % of the football team)
Let's solve these equations to find the total number of students on each team.
1. 0.10B = 4
Divide both sides by 0.10:
B = 4 / 0.10
B = 40 students on the baseball team
2. 0.25K = 4
Divide both sides by 0.25:
K = 4 / 0.25
K = 16 students on the basketball team
3. 0.0833F = 4
Divide both sides by 0.0833:
F = 4 / 0.0833
F ≈ 48 students on the football team
Now, let's find the total number of students who play on exactly two teams. The given information states that there are 20 students who are on exactly two teams.
To find the number of students who only play on one team:
- Subtract 4 from the total number of players on each team, as these 4 students belong to all three teams.
- Subtract 20 students who play on exactly two teams.
For the baseball team:
40 - 4 - 20 = 16 students who only play on this team.
For the basketball team:
16 - 4 - 20 = -8. Since there can't be a negative number of students, there is a mistake in the given information or calculation.
For the football team:
48 - 4 - 20 = 24 students who only play on this team.
Since we encountered an inconsistency with the basketball team, we cannot accurately determine the percentage of players who only play on one team. Please double-check the given information or calculations provided.