Absolutely, let me help you with that. Given a group of 200 students, we have some specific details about their preferences for two subjects: Men and Science. Here is how you can solve the problem step by step:
### Information and Problem Breakdown:
1. Total students (Est) = 200
2. Students who like both subjects = 77
3. The ratio of students who like only Men to only Science is 3:2.
### Step-by-Step Solution:
#### 1. Analyzing the Ratio and Finding Parts:
The ratio given is 3:2 for students who like only Men to those who like only Science.
To find the actual number of students in each category, we'll break this total into the parts of this ratio.
#### 2. Finding the Total Parts of the Ratio:
The ratio parts add up as follows:
[tex]\[ 3 + 2 = 5 \][/tex]
#### 3. Calculating Students Who Like Only Men:
Let's denote the students who like only Men as [tex]\( x \)[/tex] and those who like only Science as [tex]\( y \)[/tex].
Since [tex]\( \frac{x}{y} = \frac{3}{2} \)[/tex], we can say:
[tex]\[ x = 3k \][/tex]
[tex]\[ y = 2k \][/tex]
Where [tex]\( k \)[/tex] is a common multiplying factor.
#### 4. Total Students with Given Preferences:
The students who like only Men plus the students who like only Science plus the students who like both subjects equal the total number of students.
[tex]\[ x + y + \text{both subjects} = 200 \][/tex]
[tex]\[ 3k + 2k + 77 = 200 \][/tex]
Thus:
[tex]\[ 5k + 77 = 200 \][/tex]
#### 5. Solving for [tex]\( k \)[/tex]:
[tex]\[ 5k = 200 - 77 \][/tex]
[tex]\[ 5k = 123 \][/tex]
[tex]\[ k = \frac{123}{5} \][/tex]
[tex]\[ k = 24.6 \][/tex]
#### 6. Determining Number of Students for Each Category:
- Students who like only Men:
[tex]\[ x = 3k = 3 \times 24.6 = 73.8 \][/tex]
- Students who like only Science:
[tex]\[ y = 2k = 2 \times 24.6 = 49.2 \][/tex]
### Answers to the originally posed questions:
1. Number of students who like one subject (only Men or only Science):
[tex]\[ x + y = 73.8 + 49.2 = 123 \][/tex]
2. Number of students who like Science (both those who like only Science and both subjects):
[tex]\[ y + \text{students who like both subjects} = 49.2 + 77 = 126.2 \][/tex]
3. Number of students who like at least one subject:
[tex]\[ All students = 200 \][/tex]
### Venn Diagram Information:
To visualize this information in a Venn diagram:
- The circle representing Men includes 73.8 (only Men) plus 77 (both subjects).
- The circle representing Science includes 49.2 (only Science) plus 77 (both subjects).
Here is the visualization in terms of counts in each part:
- Only Men = 73.8
- Only Science = 49.2
- Both = 77
- Total = 200
That concludes our step-by-step solution.
