In A Recent Survey Of 100 Women, The Following Information W

1in A Recent Survey Of 100 Women The Following Information Was Gathe

1. In a recent survey of 100 women, the following information was gathered. 54 use shampoo A. 47 use shampoo B. 37 use shampoo C. 13 use shampoos A and B. 23 use shampoos A and C. 14 use shampoos B and C. 8 use all three. Use the figure to answer the question in the problem. How many are using shampoo A only (Region I)?

2. A poll was taken of 100 students at a commuter campus to find out how they got to campus. The results are below. How many are not using any of the three? 25 said they drove alone. 31 rode in a carpool. 31 rode public transportation. 2 used both carpools and public transportation. 6 used both a carpool and sometimes their own cars. 3 used buses as well as their own cars. 1 used all three methods.

3. At the time of George W. Bush's presidency, there have been 43 presidents of the United States. Draw a Venn diagram showing the following facts about where each served before assuming the presidency. How many presidents served as vice-president and in the Senate, but did not serve in a cabinet post? 14 were vice-president. 6 served in the Senate. 15 held a cabinet post. 2 were VP and served in the Senate. 8 were VP and a held a cabinet post. 4 served in the Senate and held a cabinet post. 1 was VP, in the Senate, and held a cabinet post.

4. Do the following. (a) What is the sum of the first 35 consecutive odd numbers? Tell whether it illustrates inductive or deductive reasoning. (b) What is the sum of the first 700 consecutive odd numbers? Tell whether it illustrates inductive or deductive reasoning.

Paper For Above instruction

The diverse range of problems involving sets, Venn diagrams, and number sequences exemplifies fundamental principles of mathematical reasoning and problem-solving strategies. These problems not only enhance comprehension of set theory and logical deduction but also demonstrate crucial techniques in quantitative analysis. This paper will analyze each problem, exploring methodologies such as Venn diagrams, inclusion-exclusion principles, and properties of odd numbers, and discussing how these methods underpin effective reasoning in mathematics.

Problem 1: Analyzing Shampoo Usage Among Women

The first problem involves a survey of 100 women and their shampoo usage habits. With data indicating the number of women using individual shampoos (A, B, and C) and their various combinations, the goal is to find the number of women using only shampoo A. This scenario is a classic example of applying the principles of set theory and Venn diagrams to decipher overlapping data.

Given data:

- Total women surveyed: 100

- Using shampoo A: 54

- Using shampoo B: 47

- Using shampoo C: 37

- Using both A and B: 13

- Using both A and C: 23

- Using both B and C: 14

- Using all three: 8

To determine the number of women using only shampoo A (Region I), we employ the inclusion-exclusion principle. First, calculate the number of women using A but not B or C:

• Women using A only = total using A - (A∩B) - (A∩C) + (A∩B∩C)

However, since the intersections are overlapping, the calculation is:

Women using A only = 54 - 13 - 23 + 8 = 54 - 36 + 8 = 26

Thus, 26 women are using only shampoo A, representing Region I in the Venn diagram. This problem illustrates the importance of systematic set analysis and the use of inclusion-exclusion principles to avoid double-counting.

Problem 2: Transportation Methods Among Students

The second problem involves a survey of 100 students and their modes of transportation to campus. The objective is to find how many students are not using any of the three transportation methods: driving alone, carpooling, or using public transportation.

Given data:

- Drove alone: 25

- Carpool: 31

- Public transportation: 31

- Both carpool and public transportation: 2

- Carpool and own cars: 6

- Buses and own cars: 3

- All three methods: 1

Applying inclusion-exclusion, the total number of students using at least one method can be calculated as:

Number using at least one method = D + C + P - (D∩C) - (D∩P) - (C∩P) + (D∩C∩P)

Where:

- D = drove alone

- C = carpool

- P = public transportation

- D∩C = students who carpooled and drove alone? Data indicates "used both carpools and own cars," which suggests overlap.

Assuming the data aligns, we sum:

Total = 25 + 31 + 31 - 2 - 6 - 3 + 1 = 86

Since total students are 100, the number not using any method is:

100 - 86 = 14

This calculation demonstrates the utility of inclusion-exclusion principles in determining the total count of students engaged in various overlapping choices and identifying those not participating in any of the options.

Problem 3: Presidential Service Patterns

The third problem involves analyzing the service history of U.S. Presidents before their presidency using a Venn diagram. The data indicates various overlapping roles, such as vice-president, senator, and cabinet member.

Given data:

- Vice-president: 14

- Senate: 6

- Cabinet: 15

- VP and Senate: 2

- VP and Cabinet: 8

- Senate and Cabinet: 4

- VP, Senate, and Cabinet: 1

The question asks how many presidents served as vice-president and in the Senate but did not serve in a cabinet post.

Calculating the number who served as VP and Senate: 2, but excluding those who also served in a cabinet, which is 1:

VP and Senate only = 2 - 1 = 1

Hence, only one president served as both vice-president and in the Senate but did not hold a cabinet position.

This problem highlights the effectiveness of Venn diagrams in visualizing complex overlapping roles and utilizing basic set algebra to extract specific subsets.

Problem 4: Summation of Odd Numbers and Mathematical Reasoning

The final problem involves summing the first 35 and 700 consecutive odd numbers and analyzing the reasoning type—inductive or deductive.

(a) The sum of the first 35 odd numbers:

Since the sum of the first n odd numbers equals n squared, the sum for n=35 is:
35^2 = 1225

This employs a well-known inductive reasoning principle or mathematical pattern recognition, where the pattern (sum equals n squared) is observed and generalized (inductive reasoning).

(b) For the first 700 odd numbers, similarly:

Sum = 700^2 = 490000

Again, this illustrates inductive reasoning, since the pattern (sum of first n odd numbers = n^2) is applied directly.

These examples demonstrate how recognizing mathematical patterns can lead to quick calculations without exhaustive addition, exemplifying inductive reasoning—observing and extending patterns based on initial data.

Conclusion

The array of problems analyzed emphasizes the centrality of set theory, Venn diagrams, inclusion-exclusion principles, and pattern recognition in mathematical thinking. These tools facilitate accurate data analysis, visualization of complex overlaps, and efficient computation. Understanding and applying these techniques are fundamental for students and professionals engaged in quantitative reasoning and problem-solving, empowering them to tackle diverse real-world scenarios with confidence and precision.

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