2. In a class of 48 students, 24 of them do Arts, 22 do Chemistry and 20 do Biology. All the students do at least one of the three subjects. 3 do all three subjects while 4 do Arts and Biology only, 3 do Arts and Chemistry only and 5 do Chemistry and Biology only.
a. Find the number of numbers of students that do
i. two subjects only
ii. exactly one subject
iii. at least two of the subjects
b. Represent the information on a complete Venn diagram. lease with full solution
To find the number of students that do:
a. i. two subjects only:
We need to find the students who do exactly two subjects. To calculate this, we subtract the students who do all three subjects and the students who do exactly one subject from the total number of students.
Number of students doing exactly two subjects = Total number of students - Number of students doing all three subjects - Number of students doing exactly one subject
Total number of students = 48
Number of students doing all three subjects = 3
To find the number of students doing exactly one subject, we need to subtract the number of students doing two subjects only from the number of students doing each subject individually.
Number of students doing Arts only = Number of students doing Arts and Chemistry only + Number of students doing Arts and Biology only + Number of students doing all three subjects
= 3 + 4 + 3 = 10
Number of students doing Chemistry only = Number of students doing Arts and Chemistry only + Number of students doing Chemistry and Biology only + Number of students doing all three subjects
= 3 + 5 + 3 = 11
Number of students doing Biology only = Number of students doing Arts and Biology only + Number of students doing Chemistry and Biology only + Number of students doing all three subjects
= 4 + 5 + 3 = 12
Number of students doing exactly one subject = Number of students doing Arts only + Number of students doing Chemistry only + Number of students doing Biology only
= 10 + 11 + 12 = 33
Number of students doing two subjects only = Total number of students - Number of students doing all three subjects - Number of students doing exactly one subject
= 48 - 3 - 33 = 12
Therefore, there are 12 students who do two subjects only.
a. ii. exactly one subject:
We have already calculated the number of students doing exactly one subject, and it is 33.
Therefore, there are 33 students who do exactly one subject.
a. iii. at least two of the subjects:
We need to find the students who do two subjects only and the students who do all three subjects.
Number of students doing at least two subjects = Number of students doing two subjects only + Number of students doing all three subjects
= 12 + 3 = 15
Therefore, there are 15 students who do at least two subjects.
b. To represent the information on a complete Venn diagram:
A Venn diagram is a graphical representation of sets and the relationships between them. In this case, we have three sets: Arts, Chemistry, and Biology.
To create a complete Venn diagram, we start by drawing three overlapping circles to represent each subject. Inside the circles, we fill in the number of students doing each subject:
In the Arts circle, we write 24.
In the Chemistry circle, we write 22.
In the Biology circle, we write 20.
Next, we fill in the overlapping regions:
In the region where Arts and Chemistry overlap, we write the number of students doing Arts and Chemistry only (3).
In the region where Arts and Biology overlap, we write the number of students doing Arts and Biology only (4).
In the region where Chemistry and Biology overlap, we write the number of students doing Chemistry and Biology only (5).
Finally, in the region where all three subjects overlap, we write the number of students doing all three subjects (3).
The remaining regions outside the overlapping circles represent the number of students doing exactly one subject:
In the region outside all three circles, we write the number of students doing none of the subjects (0).
In the regions outside the overlapping circles but inside the individual subject circles, we write the number of students doing exactly one subject: for Arts (10), Chemistry (11), and Biology (12).
This completes the Venn diagram representation of the given information.