Answer :
Let's address each part of the question step by step.
### Part (i): Drawing the Venn Diagram
We are given:
- Total number of students: 150
- Students who passed in English: 65
- Students who passed in Mathematics: 75
- Students who passed in both subjects: 30
First, we need to find the number of students who passed only in English and only in Mathematics.
- Students who passed only in English = Total passed in English - Passed in both = 65 - 30 = 35
- Students who passed only in Mathematics = Total passed in Mathematics - Passed in both = 75 - 30 = 45
Now, we can draw the Venn Diagram:
1. Draw a rectangle to represent the universal set of 150 students.
2. Inside the rectangle, draw two intersecting circles.
3. Label one circle "English (65)" and the other "Mathematics (75)".
4. The intersection of the circles represents students passed in both subjects, which is 30.
5. In the "English" circle but outside the intersection, write 35 (students who passed only in English).
6. In the "Mathematics" circle but outside the intersection, write 45 (students who passed only in Mathematics).
7. Outside both circles, calculate the students who didn't pass in either subject (150 - (35 + 45 + 30) = 40).
### Part (ii): Number of Students Who Didn't Pass in Mathematics
To find the number of students who didn't pass in Mathematics:
- Total students = 150
- Students who passed in Mathematics = 75
- Therefore, students who didn't pass in Mathematics = Total students - Students who passed in Mathematics
- [tex]\( \text{Number of students who didn't pass in Mathematics} = 150 - 75 = 75 \)[/tex]
### Part (iii): Proof of [tex]\( n(U) = n(E) + n(M) - n(E \cap M) + n(\overline{E \cup M}) \)[/tex]
We need to show:
[tex]\[ n(U) = n(E) + n(M) - n(E \cap M) + n(\overline{E \cup M}) \][/tex]
Let's calculate each term:
1. [tex]\( n(U) \)[/tex]: Total number of students = 150
2. [tex]\( n(E) \)[/tex]: Students who passed in English = 65
3. [tex]\( n(M) \)[/tex]: Students who passed in Mathematics = 75
4. [tex]\( n(E \cap M) \)[/tex]: Students who passed in both subjects = 30
5. [tex]\( n(\overline{E \cup M}) \)[/tex]: Students who didn't pass in either subject = 40
Now, substituting these values into the equation:
[tex]\[ 150 = 65 + 75 - 30 + 40 \][/tex]
[tex]\[ 150 = 150 \][/tex]
Thus, the equation is proven to be correct.
### Part (iv): Ratio of Students Who Passed Mathematics Only and English Only
To find the ratio, calculate:
1. Students who passed only in Mathematics = 45 (from earlier calculation)
2. Students who passed only in English = 35 (from earlier calculation)
The ratio of students who passed Mathematics only to those who passed English only is:
[tex]\[ \text{Ratio} = \frac{\text{Students who passed Mathematics only}}{\text{Students who passed English only}} = \frac{45}{35} = \frac{9}{7} = 1.2857142857142858 \][/tex]
In conclusion:
1. Number of students who didn't pass in Mathematics: 75
2. Proof of the given equation: Verified.
3. Ratio of students who passed Mathematics only to English only: [tex]\( \approx 1.29 \)[/tex]
### Part (i): Drawing the Venn Diagram
We are given:
- Total number of students: 150
- Students who passed in English: 65
- Students who passed in Mathematics: 75
- Students who passed in both subjects: 30
First, we need to find the number of students who passed only in English and only in Mathematics.
- Students who passed only in English = Total passed in English - Passed in both = 65 - 30 = 35
- Students who passed only in Mathematics = Total passed in Mathematics - Passed in both = 75 - 30 = 45
Now, we can draw the Venn Diagram:
1. Draw a rectangle to represent the universal set of 150 students.
2. Inside the rectangle, draw two intersecting circles.
3. Label one circle "English (65)" and the other "Mathematics (75)".
4. The intersection of the circles represents students passed in both subjects, which is 30.
5. In the "English" circle but outside the intersection, write 35 (students who passed only in English).
6. In the "Mathematics" circle but outside the intersection, write 45 (students who passed only in Mathematics).
7. Outside both circles, calculate the students who didn't pass in either subject (150 - (35 + 45 + 30) = 40).
### Part (ii): Number of Students Who Didn't Pass in Mathematics
To find the number of students who didn't pass in Mathematics:
- Total students = 150
- Students who passed in Mathematics = 75
- Therefore, students who didn't pass in Mathematics = Total students - Students who passed in Mathematics
- [tex]\( \text{Number of students who didn't pass in Mathematics} = 150 - 75 = 75 \)[/tex]
### Part (iii): Proof of [tex]\( n(U) = n(E) + n(M) - n(E \cap M) + n(\overline{E \cup M}) \)[/tex]
We need to show:
[tex]\[ n(U) = n(E) + n(M) - n(E \cap M) + n(\overline{E \cup M}) \][/tex]
Let's calculate each term:
1. [tex]\( n(U) \)[/tex]: Total number of students = 150
2. [tex]\( n(E) \)[/tex]: Students who passed in English = 65
3. [tex]\( n(M) \)[/tex]: Students who passed in Mathematics = 75
4. [tex]\( n(E \cap M) \)[/tex]: Students who passed in both subjects = 30
5. [tex]\( n(\overline{E \cup M}) \)[/tex]: Students who didn't pass in either subject = 40
Now, substituting these values into the equation:
[tex]\[ 150 = 65 + 75 - 30 + 40 \][/tex]
[tex]\[ 150 = 150 \][/tex]
Thus, the equation is proven to be correct.
### Part (iv): Ratio of Students Who Passed Mathematics Only and English Only
To find the ratio, calculate:
1. Students who passed only in Mathematics = 45 (from earlier calculation)
2. Students who passed only in English = 35 (from earlier calculation)
The ratio of students who passed Mathematics only to those who passed English only is:
[tex]\[ \text{Ratio} = \frac{\text{Students who passed Mathematics only}}{\text{Students who passed English only}} = \frac{45}{35} = \frac{9}{7} = 1.2857142857142858 \][/tex]
In conclusion:
1. Number of students who didn't pass in Mathematics: 75
2. Proof of the given equation: Verified.
3. Ratio of students who passed Mathematics only to English only: [tex]\( \approx 1.29 \)[/tex]
