A school sports team contains 68 student.33 do field events,40do track events,23 do swimming,14 do both field and track events,8 do both swimming and field events. If 15 students do field events only,10 do both swimming and track events, how many students do
1) swimming only
2) track events only
3) all 3 events
What is 33 to
To find the number of students who do swimming only, track events only, and all three events, we can use the principle of inclusion-exclusion.
First, we add up the number of students who participate in each event.
Number of students in field events = 33 + 14 + 8 + 10 + 15 = 80
Number of students in track events = 40 + 14 + 8 + 10 = 72
Number of students in swimming = 23 + 8 + 10 = 41
Now, we can calculate the number of students who do each event exclusively.
1) Swimming only = Number of students in swimming - (Number of students in swimming and field events) - (Number of students in swimming and track events) + (Number of students in all three events)
Swimming only = 41 - 8 - 10 + 0 = 23
2) Track events only = Number of students in track events - (Number of students in swimming and track events) - (Number of students in track events and field events) + (Number of students in all three events)
Track events only = 72 - 10 - 14 + 0 = 48
3) All three events = Number of students in all three events
All three events = 0 + 0 + 0 + 0 = 0
Therefore, the answers are:
1) Swimming only = 23 students
2) Track events only = 48 students
3) All three events = 0 students
If x do all three, then assuming all 68 students participate,
33+40+23 - (14+8+10) + x = 68
x = 4
Now you can completely fill in your Venn diagram and easily answer the questions.
1) Swimming only: To find the number of students who do swimming only, we need to subtract the students who do both swimming and field events, as well as the students who do all three events from the total number of students who do swimming. So, the number of students who do swimming only would be 23 - 8 - 10 = 5.
2) Track events only: Similar to the previous question, we need to subtract the students who do both track and swimming events, as well as the students who do all three events from the total number of students who do track events. Therefore, the number of students who do track events only would be 40 - 10 - 14 = 16.
3) All three events: To find the number of students who do all three events, we can subtract the number of students who do only one or two events from the total number of students. Therefore, the number of students who do all three events would be 68 - 5 - 16 - 15 - 14 - 8 - 10 = 0.
Looks like we have a situation where nobody is a superstar multi-sport athlete!