2. In a class of 48 students, 24 of them do Art, 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 Art 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
If you do your Venn diagram, it should be easy to see that you have
Art only: 24-(3+3+4) = 14
Biology only: 20-(4+3+5) = 8
Chem only: 22-(3+3+5) = 11
Now you can answer the questions.
To solve part a of the question, let's break it down step by step:
i. To find the number of students that do two subjects only, we need to subtract the number of students who do all three subjects from the total number of students who do at least one of the three subjects. In this case, we know that 3 students do all three subjects. So, the number of students that do two subjects only would be:
Total number of students who do at least one of the three subjects - Number of students who do all three subjects = 48 - 3 = 45
Therefore, there are 45 students who do two subjects only.
ii. To find the number of students that do exactly one subject, we need to subtract the number of students who do two subjects or all three subjects from the total number of students. In this case, we already know that there are 45 students who do two subjects only, so we just need to subtract the number of students who do all three subjects:
Total number of students - Number of students who do all three subjects = 48 - 3 = 45
Therefore, there are 45 students who do exactly one subject.
iii. To find the number of students that do at least two of the subjects, we need to add the number of students who do two subjects only and the number of students who do all three subjects:
Number of students who do two subjects only + Number of students who do all three subjects = 45 + 3 = 48
Therefore, there are 48 students who do at least two of the subjects.
Now, let's move on to part b of the question and represent the information on a complete Venn diagram:
Start by drawing a rectangle to represent the total number of students. Divide the rectangle into three overlapping circles, each representing one subject: Art, Chemistry, and Biology. Label the circles accordingly.
Next, fill in the given information:
- 20 students do Biology. Place the number 20 in the Biology circle.
- 22 students do Chemistry. Place the number 22 in the Chemistry circle.
- 24 students do Art. Place the number 24 in the Art circle.
- 3 students do all three subjects. Place the number 3 in the overlapping region of all three circles.
- 4 students do Art and Biology only. Place the number 4 in the overlapping region of Art and Biology.
- 3 students do Art and Chemistry only. Place the number 3 in the overlapping region of Art and Chemistry.
- 5 students do Chemistry and Biology only. Place the number 5 in the overlapping region of Chemistry and Biology.
Now, we can calculate the remaining numbers:
- To find the number of students who do exactly one subject:
- Subtract the numbers in the overlapping regions from the respective subjects: Art - Art and Biology - Art and Chemistry = 24 - 4 - 3 = 17
- Chemistry - Art and Chemistry - Chemistry and Biology = 22 - 3 - 5 = 14
- Biology - Art and Biology - Chemistry and Biology = 20 - 4 - 5 = 11
- Place the numbers 17, 14, and 11 in the non-overlapping regions of Art, Chemistry, and Biology, respectively.
- To find the number of students who do two subjects only:
- Add the numbers in the overlapping regions: Art and Biology + Art and Chemistry + Chemistry and Biology = 4 + 3 + 5 = 12
- Place the number 12 in the regions where two circles overlap, but not all three.
With this information, the Venn diagram is complete, and you can visualize the distribution of students studying different subjects.
a.
i. To find the number of students that do two subjects only, we need to subtract the number of students that do all three subjects from the total number of students that do two subjects.
Number of students doing two subjects only = Total number of students doing two subjects - Number of students doing all three subjects
Total number of students doing two subjects = Number of students doing Art and Biology only + Number of students doing Arts and Chemistry only + Number of students doing Chemistry and Biology only
Total number of students doing two subjects = 4 + 3 + 5 = 12
Number of students doing two subjects only = 12 - 3 (number of students doing all three subjects) = 9
Therefore, the number of students that do two subjects only is 9.
ii. To find the number of students that do exactly one subject, we need to subtract the number of students that do two subjects and the number of students that do all three subjects from the total number of students.
Number of students doing exactly one subject = Total number of students - Number of students doing two subjects - Number of students doing all three subjects
Total number of students = 48
Number of students doing exactly one subject = 48 - 9 (number of students doing two subjects only) - 3 (number of students doing all three subjects) = 36
Therefore, the number of students that do exactly one subject is 36.
iii. To find the number of students that do at least two subjects, we need to add the number of students doing two subjects only and the number of students doing 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
Number of students doing at least two subjects = 9 + 3 = 12
Therefore, the number of students that do at least two subjects is 12.
b.
Here is a representation of the information on a complete Venn diagram:
```
Art
/ \
/ \
/ \
3 / 1 \ 4
/ \
/--------\--------\
/ \ \
Art and Art Only Art and Biology Only
Bio Chem
\ /
\ /
\ /
\ /
\/
Chem
\/
Bio
```