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Chapter 15: Problem 24

In a survey among 80 people, 50 people like arrange marriage and 70 people like love marriage. What is the minimum and maximum number of people like both the marriages respectively? (a) 40,45 (b) 40,50 (c) 30,40 (d) can't be determined

Short Answer

Expert verified
Answer: (b) 0,50

Step by step solution

01

1. Analyzing the given information

First, we are given a total of 80 people. 50 people like arranged marriage, and 70 people like love marriage. We need to determine the minimum and maximum number of people who like both types of marriages.
02

2. Calculate minimum overlap

To find the minimum possible overlap between the two preferences, we take into consideration that there could be people who don't like either type of marriage. In this case, we calculate the number of people who fall into this category by subtracting the total number of people who like either marriage from the total population: Minimum overlap = Total people - (People like arranged marriage + People like love marriage) Minimum overlap = 80 - (50 + 70) = 80 - 120 = -40 Since we cannot have a negative overlap, the minimum overlap is 0, but we also need to double-check if there are only 80 people in total. By checking the total, we have: 50 arranged marriage like + 0 both marriage like + (70 - 0) love marriage like = 80 So the minimum overlap is 0.
03

3. Calculate maximum overlap

To find the maximum possible overlap between the two preferences, we consider the scenario where all the people who like arranged marriage also like love marriage. In this case, the overlap between the two preferences is equal to the number of people who like arranged marriage: Maximum overlap = People like arranged marriage = 50
04

4. Choose the correct option

Comparing our calculated minimum and maximum overlaps with the given options, we find that our answer matches with option (b). The minimum and maximum number of people who like both types of marriages are 0 and 50, respectively. So, the correct answer is (b) 0,50.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Set Theory
Set theory is a fundamental branch of mathematics that deals with the study of sets, which are well-defined collections of objects. In our scenario, we used set theory to analyze groups of people with different preferences. By representing preferences as sets, we can solve problems related to membership and relationships between these sets.
The key components include:
  • **Set:** A collection of distinct objects, considered as an object in its own right. In our example, one set is people who like arranged marriages, and another set is people who like love marriages.
  • **Subset:** A set whose elements are all contained in another set. For example, the set of people who like both marriages is a subset of each of the marriage preference sets.
  • **Union:** The set containing all the elements of two sets. Here, it represents everyone who likes either of the two types of marriages or both.
By utilizing these concepts, we determine potential overlaps and divergences, helping us find the minimum and maximum number of people who like both marriage types.
Overlapping Sets
Overlapping sets occur when there is some commonality between different sets. This means there are elements that belong to multiple sets simultaneously. In dealing with survey problems, overlapping sets help us identify individuals who share multiple preferences.
  • **Intersection:** The set containing all elements that are common to both sets. For this problem, it represents the people who enjoy both arranged and love marriages.
  • **Calculation of Overlaps:** In practice, we calculated the overlap by checking how many people fit into both preferences. The solutions showed us how the minimum overlap could theoretically be zero if no one liked both, and the maximum overlap could match the smaller preference set if everyone shared both interests.
Understanding overlaps allows us to properly distribute the survey data without exceeding the total number surveyed, ensuring all possibilities are accounted for.
Survey Analysis
Survey analysis involves the interpretation and organization of collected data to provide insights into preferences, trends, and overall outcomes. When applying to our problem, it focused on analyzing responses from a population about their marriage preferences.
Key components of survey analysis include:
  • **Data Collection:** Obtaining responses from a specified group鈥攊n this case, 80 people providing preference information for two marriage types.
  • **Data Interpretation:** Finding meaningful patterns and overlaps such as the people who liked both marriage types.
  • **Validation:** Ensuring that the total number of responses align with the imposed constraints (e.g., total must be less than or equal to the total surveyed population).
By creating Venn diagrams or similar tools, survey analysis allows for visual representation of the data, enhancing understanding of relationships and overlaps and helping decide the best-fitted solution.

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