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Sagot :
Sure, let's break down the problem step by step:
### Given Information:
- Total number of students: 150
- 25% of students dislike both subjects
- Ratio of students who like Maths to those who like Science is 4:6
### Part (a): Cardinality of the Sets
To find the number of students who like Maths and the number of students who like Science:
1. Calculate the percentage that dislike both subjects:
[tex]\[ \text{Number of students who dislike both subjects} = 150 \times \frac{25}{100} = 37.5 \][/tex]
2. Given the ratio 4:6 (or 2:3) of students who like Maths to Science means that for every 10 students, 4 like Maths and 6 like Science. We need to scale this ratio to the total number of students who like either subject.
3. Total number of students who like at least one subject is:
[tex]\[ \text{Number of students who like at least one subject} = 150 - 37.5 = 112.5 \][/tex]
4. To split these 112.5 students into Maths and Science according to the 4:6 ratio:
- Ratio Maths: [tex]\( \frac{4}{4 + 6} = \frac{4}{10} = 0.4 \)[/tex]
- Ratio Science: [tex]\( \frac{6}{4 + 6} = \frac{6}{10} = 0.6 \)[/tex]
5. Number of students who like Maths:
[tex]\[ \text{Number who like Maths} = 112.5 \times 0.4 = 45 \][/tex]
6. Number of students who like Science:
[tex]\[ \text{Number who like Science} = 112.5 \times 0.6 = 67.5 \][/tex]
So, the cardinality of the sets is:
- Number of students who like Maths: 45
- Number of students who like Science: 67.5
### Part (b): Venn Diagram
To represent the information in a Venn diagram:
- Let's denote the set of students who like Maths as [tex]\( M \)[/tex] and those who like Science as [tex]\( S \)[/tex].
- Total students: [tex]\( 150 \)[/tex]
- Students who dislike both subjects (outside both circles): [tex]\( 37.5 \)[/tex]
- [tex]\( |M| = 60 \)[/tex], [tex]\( |S| = 90 \)[/tex]
- Both sets will overlap representing students who like both subjects if there were any.
### Part (c): Number of Students Who Liked At Least One Subject
We have already determined this:
- Number of students who liked at least one subject = 112.5
### Part (d): Compare the Number of Students Who Liked At Least One Subject to Those Who Did Not Like Both Subjects
1. Students who liked at least one subject: [tex]\( 112.5 \)[/tex]
2. Students who did not like both subjects: [tex]\( 37.5 \)[/tex]
[tex]\[ \text{Comparison}: \text{More students liked at least one subject (112.5)} \text{ than those who disliked both (37.5)}. \][/tex]
This methodical approach ensures that every part of the problem is addressed correctly based on the given ratios and percentages.
### Given Information:
- Total number of students: 150
- 25% of students dislike both subjects
- Ratio of students who like Maths to those who like Science is 4:6
### Part (a): Cardinality of the Sets
To find the number of students who like Maths and the number of students who like Science:
1. Calculate the percentage that dislike both subjects:
[tex]\[ \text{Number of students who dislike both subjects} = 150 \times \frac{25}{100} = 37.5 \][/tex]
2. Given the ratio 4:6 (or 2:3) of students who like Maths to Science means that for every 10 students, 4 like Maths and 6 like Science. We need to scale this ratio to the total number of students who like either subject.
3. Total number of students who like at least one subject is:
[tex]\[ \text{Number of students who like at least one subject} = 150 - 37.5 = 112.5 \][/tex]
4. To split these 112.5 students into Maths and Science according to the 4:6 ratio:
- Ratio Maths: [tex]\( \frac{4}{4 + 6} = \frac{4}{10} = 0.4 \)[/tex]
- Ratio Science: [tex]\( \frac{6}{4 + 6} = \frac{6}{10} = 0.6 \)[/tex]
5. Number of students who like Maths:
[tex]\[ \text{Number who like Maths} = 112.5 \times 0.4 = 45 \][/tex]
6. Number of students who like Science:
[tex]\[ \text{Number who like Science} = 112.5 \times 0.6 = 67.5 \][/tex]
So, the cardinality of the sets is:
- Number of students who like Maths: 45
- Number of students who like Science: 67.5
### Part (b): Venn Diagram
To represent the information in a Venn diagram:
- Let's denote the set of students who like Maths as [tex]\( M \)[/tex] and those who like Science as [tex]\( S \)[/tex].
- Total students: [tex]\( 150 \)[/tex]
- Students who dislike both subjects (outside both circles): [tex]\( 37.5 \)[/tex]
- [tex]\( |M| = 60 \)[/tex], [tex]\( |S| = 90 \)[/tex]
- Both sets will overlap representing students who like both subjects if there were any.
### Part (c): Number of Students Who Liked At Least One Subject
We have already determined this:
- Number of students who liked at least one subject = 112.5
### Part (d): Compare the Number of Students Who Liked At Least One Subject to Those Who Did Not Like Both Subjects
1. Students who liked at least one subject: [tex]\( 112.5 \)[/tex]
2. Students who did not like both subjects: [tex]\( 37.5 \)[/tex]
[tex]\[ \text{Comparison}: \text{More students liked at least one subject (112.5)} \text{ than those who disliked both (37.5)}. \][/tex]
This methodical approach ensures that every part of the problem is addressed correctly based on the given ratios and percentages.
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