In a class of 100 students, 40 offer Biology and 10 offer Chemistry and Biology .10 offer Physics and Chemistry , and 30 offer Physic and Biology . Each student offer at least one of the three subjects. The number of students who offer Biology is equal to those who offer Physics only and the sum of the number of students who offer Biology and those who offer Chemistry is equal to the number of students who offer Physics.
(a) illustrate the information on a Venn diagram
(b)find the number of students who offer
(I) Chemistry only
(II)only two subjects
(III)only one subject
(IV)physics
(V)Chemistry
(VI)Chemistry and Physics only
Answer
i.20 ii.70 iii.20 iv.70
I. 20
II. 20
III. 70
IV. 70
V.30
Please I need a full answer to this question .
(a) Here is an illustration of the information on a Venn diagram:
```
_________
/ \
/ \
| Biology |
\___________/
|
________|_________________________
| | | |
| | | |
| Physics| Chemistry |
| | | |
|________|____________|_________|
|
| |
| |
Chemistry Physics
```
(b) Now, let's find the number of students who offer:
(I) Chemistry only:
Since the number of students who offer Chemistry and Biology is equal to the number of students who offer Physics, and 10 students offer Chemistry and Biology, we can subtract that from the total number of students who offer Chemistry to find the number of students who offer Chemistry only:
Number of students who offer Chemistry only = 10
(II) Only two subjects:
To find the number of students who offer only two subjects, we can add the number of students who offer Physics and Chemistry (10) and the number of students who offer Chemistry and Biology (10):
Number of students who offer only two subjects = 10 + 10 = 20
(III) Only one subject:
To find the number of students who offer only one subject, we need to subtract the number of students who offer Biology and Chemistry (10) and the number of students who offer Physics and Chemistry (10) from the total number of students (100):
Number of students who offer only one subject = 100 - 10 - 10 = 80
(IV) Physics:
From the information given, we know that the number of students who offer Biology is equal to those who offer Physics only. So, the number of students who offer Physics only is the same as the number of students who offer Biology:
Number of students who offer Physics = Number of students who offer Biology = 40
(V) Chemistry:
From the information given, we know that the sum of the number of students who offer Biology and those who offer Chemistry is equal to the number of students who offer Physics. So, the number of students who offer Chemistry is the same as the number of students who offer Physics:
Number of students who offer Chemistry = Number of students who offer Physics = 40
(VI) Chemistry and Physics only:
We can subtract the number of students who offer Chemistry and Biology (10) from the total number of students who offer Chemistry (40) to find the number of students who offer Chemistry and Physics only:
Number of students who offer Chemistry and Physics only = 40 - 10 = 30
To solve this problem, we will start by illustrating the information on a Venn diagram.
(a) Venn Diagram:
Let's label the three circles as Biology (B), Chemistry (C), and Physics (P).
According to the information given:
- 40 students offer Biology
- 10 students offer Chemistry and Biology
- 10 students offer Physics and Chemistry
- 30 students offer Physics and Biology
We need to find the overlapping regions and place the respective numbers:
1. Biology (B) circle: This circle represents students who offer Biology. We know that 40 students offer Biology, which includes 10 students who also offer Chemistry (B ∩ C = 10) and 30 students who also offer Physics (B ∩ P = 30).
2. Chemistry (C) circle: This circle represents students who offer Chemistry. We have 10 students who offer Chemistry and Biology (B ∩ C = 10) and 10 students who offer Physics and Chemistry (C ∩ P = 10).
3. Physics (P) circle: This circle represents students who offer Physics. We know that 30 students offer Physics and Biology (B ∩ P = 30) and 10 students offer Physics and Chemistry (C ∩ P = 10).
Now, we can determine the remaining regions of the diagram:
4. Only Biology (B, not C and not P): This region represents the students who offer Biology only. To find this number, we subtract the students who offer Biology and additional subjects (i.e., Biology and Chemistry, and Biology and Physics) from the total number of students who offer Biology:
Students who offer Biology only = Number of Students offering Biology (B) - Students offering Biology and Chemistry (B ∩ C) - Students offering Biology and Physics (B ∩ P)
= 40 - 10 - 30
= 0
5. Only Chemistry (C, not B and not P): This region represents the students who offer Chemistry only. To find this number, we subtract the students who offer Chemistry and additional subjects (i.e., Chemistry and Biology, and Chemistry and Physics) from the total number of students who offer Chemistry:
Students who offer Chemistry only = Number of Students offering Chemistry (C) - Students offering Chemistry and Biology (B ∩ C) - Students offering Chemistry and Physics (C ∩ P)
= 0 - 10 - 10
= -20
6. Only Physics (P, not B and not C): This region represents the students who offer Physics only. To find this number, we subtract the students who offer Physics and additional subjects (i.e., Physics and Biology, and Physics and Chemistry) from the total number of students who offer Physics:
Students who offer Physics only = Number of Students offering Physics (P) - Students offering Biology and Physics (B ∩ P) - Students offering Chemistry and Physics (C ∩ P)
= 0 - 30 - 10
= -40
From the Venn diagram, we can see that there are no students who offer Chemistry only, Physics only, or Biology and Physics only. It means that the information given may have a discrepancy or inconsistency.
Please note that due to the given information, we cannot determine the number of students who offer Chemistry only, Physics only, or Biology and Physics only.