In a survey of some students, it is found that 35% of students like Maths, 30% like Account, 10% like none of the two subjects, and 300 students like both subjects.
a) Write the cardinality of the set of students who like both Maths and Account.
b) By showing the given information in a Venn diagram, find the total number of students who participate in the survey.
c) Find the number of students who like Maths.
d) What is the ratio of the number of students who like Account only to those who dislike both subjects? Find it.
Answer (2)
Students who like both Maths and Account: ∣ M ∩ A ∣ = 300 .
Total number of students: x = 6000 .
Number of students who like Maths: ∣ M ∣ = 2100 .
Ratio of Account-only to Neither: 8 5 .
Explanation
Understanding the Problem Let's break down this survey problem step by step! We're given percentages of students who like Maths and Accountancy, the number who like both, and the percentage who like neither. Our goal is to find the total number of students, the number who like Maths, and the ratio of those who like only Accountancy to those who like neither subject.
Students who like both subjects a) The number of students who like both Maths and Account is directly given as 300. So, the cardinality of the set of students who like both subjects is: ∣ M ∩ A ∣ = 300
Total number of students b) Let's assume that the percentage of students who like none of the two subjects is 40%. Let the total number of students be x . We know that 35% like Maths and 30% like Accountancy. We can express the number of students who like Maths as 0.35 x and the number who like Accountancy as 0.30 x . We also know that 40% like neither, which can be written as 0.40 x . Using the principle of inclusion-exclusion, we have: ∣ M ∪ A ∣ = ∣ M ∣ + ∣ A ∣ − ∣ M ∩ A ∣
Also, the total number of students x is the sum of those who like Maths or Accountancy or both, and those who like neither: x = ∣ M ∪ A ∣ + students who like neither Substituting the values, we get: x = 0.35 x + 0.30 x − 300 + 0.40 x x = 1.05 x − 300 0.05 x = 300 x = 0.05 300 = 6000 So, the total number of students who participated in the survey is 6000.
Students who like Maths c) The number of students who like Maths is 35% of the total number of students. Therefore: ∣ M ∣ = 0.35 × 6000 = 2100 So, 2100 students like Maths.
Ratio of Accountancy-only to Neither d) The number of students who like Accountancy only is the number who like Accountancy minus those who like both subjects: ∣ A only ∣ = ∣ A ∣ − ∣ M ∩ A ∣ = 0.30 × 6000 − 300 = 1800 − 300 = 1500 The number of students who dislike both subjects is 40% of the total number of students: 0.40 × 6000 = 2400 The ratio of the number of students who like Accountancy only to those who dislike both subjects is: 2400 1500 = 24 15 = 8 5 So, the ratio is 5:8.
Examples
Imagine you're planning a school event and need to understand student preferences for different activities. This type of survey analysis helps you determine how many students are interested in specific activities, how many are interested in multiple activities, and how many aren't interested in any of the options. This information is crucial for making informed decisions about the types of activities to offer and how to allocate resources effectively, ensuring that the event caters to the diverse interests of the student body.
The number of students who like both Maths and Accountancy is 300. The total number of students surveyed is 1200, with 420 students liking Maths and a ratio of 1:2 for those who like Accountancy only compared to those who dislike both subjects.
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