50 students showed which of the following sports Day live football volleyball and hoagie 24 live football 21 volleyball and 18 hawken I like football or three like volleyball and 205 like all the three spots but six like football and volleyball only
To solve this problem, we can use a Venn diagram. Let's start by drawing three overlapping circles representing football, volleyball, and hoagie.
[insert Venn diagram here]
We know that 50 students in total were surveyed. Let's fill in the information given:
- 24 students live football, so we put 24 in the football circle
- 21 students like volleyball, so we put 21 in the volleyball circle
- 18 students like hoagie, so we put 18 in the hoagie circle
- 3 students like both football and volleyball, so we put 3 in the overlapping region of the football and volleyball circles
- 6 students like both football and hoagie, but not volleyball. We put 6 in the overlapping region of the football and hoagie circles (outside the volleyball circle).
- We also know that 205 students like all three sports. We put 205 in the small overlapping region where all three circles intersect.
[insert updated Venn diagram here]
Now we need to figure out how many students only like each sport. We can start with football:
- To find the number of students who only like football, we need to subtract the students who like both football and another sport, as well as the students who like all three sports, from the total number who like football:
```
Total football fans = 24
Football fans who also like volleyball = 3
Football fans who also like hoagie = 6
Football fans who like all three = 205
Total football fans who only like football = 24 - 3 - 6 - 205 = -190
```
Wait a minute, we got a negative number! This means there must be an error in our diagram or calculations. Let's check:
- We see that the number of students who like all three sports is larger than the total number of students (since 205 is larger than 50). This means there must be some overlap between the groups we haven't accounted for yet. Specifically, the 205 students in the small overlapping region of the Venn diagram also belong to the larger circles for football, volleyball, and hoagie. We need to subtract them again from the counts for each sport:
```
Total football fans = 24
Football fans who also like volleyball = 3
Football fans who also like hoagie = 6
Football fans who like all three = 205 - 24 = 181
Total football fans who only like football = 24 - 3 - 6 - 181 = -166
```
Still negative! Let's try the same process for volleyball:
- To find the number of students who only like volleyball, we need to subtract the students who like both volleyball and another sport, as well as the students who like all three sports, from the total number who like volleyball:
```
Total volleyball fans = 21
Volleyball fans who also like football = 3
Volleyball fans who like all three = 205 - 21 = 184
Total volleyball fans who only like volleyball = 21 - 3 - 184 = -166
```
Hmmm, we're getting the same negative number again. This must mean that we made a mistake earlier. Let's go back and double-check the counts for each circle:
- For football: 24 - 3 - 6 - 205 = -190
- For volleyball: 21 - 3 - 184 = -166
- For hoagie: 18 - 6 - 205 = -193
We see that the count for hoagie is also negative, which means there must be an error in our Venn diagram. After some careful inspection, we realize that we made a mistake by not including the 6 students who like both football and hoagie in the overlapping region of the football and hoagie circles (inside the volleyball circle) as well. When we fix that error and redo the calculations, we get:
[insert corrected Venn diagram here]
- For football: 24 - 3 - 6 - 205 - 6 = -192
- For volleyball: 21 - 3 - 184 = -166
- For hoagie: 18 - 6 - 205 - 6 = -199
Now we have negative numbers for all three sports, which means something is definitely wrong. After checking our calculations and diagram once more, we realize that the problem may be unsolvable. This is because there are only 50 students in total, but the counts for each sport and combination of sports add up to more than 50. Specifically, the count for all three sports is 205, which is larger than the total number of students. This means that the data provided is inconsistent or incomplete, and we cannot determine the exact number of students who only like each sport.