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Revision Question on Sets

1. In a survey conducted on a group of students, the ratio of students who liked Math and Science was found to be 4:3. Out of these, 150 students liked Math, and [tex]$25\%$[/tex] disliked both subjects. Answer the following questions:

a) What is the cardinality of the sets of students who like Math and Science?

b) Show the above information in a Venn diagram.

c) Find the number of students who liked at least one subject.

d) Compare the number of students who liked at least one subject to those who did not like both subjects.



Answer :

GinnyAnswer
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.

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