Answer :
Certainly! Let's delve into the survey problem step-by-step.
### Part (a): Venn Diagram Representation
We have two sets in our Venn diagram:
- Set A: People who like modern songs.
- Set B: People who like folk songs.
Given Information:
- 50 people like both modern and folk songs.
- 40 people like only folk songs.
- 80 people like neither modern nor folk songs.
- The ratio of people who like modern songs to those who like folk songs is 8:9.
To represent this information in a Venn diagram:
- Let the circle for modern songs be labeled as A.
- Let the circle for folk songs be labeled as B.
In the intersection of A and B (both sets), we place "50" (those who like both songs).
In the part of B excluding the intersection, we place "40" (those who like only folk songs).
Outside both circles, we place "80" (those who like neither type of song).
### Part (b): Find the Number of People Who Participated in the Survey
To find the total number of people who participated in the survey, we need to add up all the individual groups:
1. People who like both songs: 50
2. People who like only folk songs: 40
3. People who like neither type of song: 80
We need to find the number of people who like only modern songs. Given the ratio of modern song lovers to folk song lovers is 8:9:
Let's denote the total number of people who like folk songs (both those who like only folk and those who like both) as [tex]\(x\)[/tex]. We know:
[tex]\[ x = 50 \text{ (both)} + 40 \text{ (only folk)} = 90 \][/tex]
Since the ratio of modern to folk song lovers is 8:9,
The total number of modern song lovers can be found using this ratio:
Let the number of modern song lovers be [tex]\( \frac{8}{9} \times x = \frac{8}{9} \times 90 \)[/tex].
Once we find this, we subtract the number of people who like both (50) to find those who like only modern songs.
[tex]\[ \text{Only modern song lovers} = \frac{8}{9} \times 90 - 50 = 80 - 50 = 30 \][/tex]
Thus, the total number of participants can be calculated by:
[tex]\[ \text{Total participants} = (\text{Both}) + (\text{Only Folk}) + (\text{Neither}) + (\text{Only Modern}) \][/tex]
[tex]\[ = 50 + 40 + 80 + 30 \][/tex]
[tex]\[ = 126.25 \][/tex]
### Part (c): Find the Number of People Who Like Folk Songs
People who like folk songs include both those who like only folk songs and those who like both types of songs.
Number of people who like folk songs:
[tex]\[ \text{Total folk song lovers} = (\text{Only folk}) + (\text{Both}) \][/tex]
[tex]\[ = 40 + 50 \][/tex]
[tex]\[ = 90 \][/tex]
### Part (d): Find the Number of People Who Like Only One Song
We need to find the sum of people who like only folk songs and those who like only modern songs.
Number of people who like only one song:
[tex]\[ \text{Only one song lovers} = (\text{Only folk}) + (\text{Only modern}) \][/tex]
[tex]\[ = 40 + 30 \][/tex]
[tex]\[ = 46.25 \][/tex]
### Summary of Results
- Total participants: 126.25
- People who like folk songs: 90
- People who like only one song: 46.25
These steps explain the complete solution to the problem as requested.
### Part (a): Venn Diagram Representation
We have two sets in our Venn diagram:
- Set A: People who like modern songs.
- Set B: People who like folk songs.
Given Information:
- 50 people like both modern and folk songs.
- 40 people like only folk songs.
- 80 people like neither modern nor folk songs.
- The ratio of people who like modern songs to those who like folk songs is 8:9.
To represent this information in a Venn diagram:
- Let the circle for modern songs be labeled as A.
- Let the circle for folk songs be labeled as B.
In the intersection of A and B (both sets), we place "50" (those who like both songs).
In the part of B excluding the intersection, we place "40" (those who like only folk songs).
Outside both circles, we place "80" (those who like neither type of song).
### Part (b): Find the Number of People Who Participated in the Survey
To find the total number of people who participated in the survey, we need to add up all the individual groups:
1. People who like both songs: 50
2. People who like only folk songs: 40
3. People who like neither type of song: 80
We need to find the number of people who like only modern songs. Given the ratio of modern song lovers to folk song lovers is 8:9:
Let's denote the total number of people who like folk songs (both those who like only folk and those who like both) as [tex]\(x\)[/tex]. We know:
[tex]\[ x = 50 \text{ (both)} + 40 \text{ (only folk)} = 90 \][/tex]
Since the ratio of modern to folk song lovers is 8:9,
The total number of modern song lovers can be found using this ratio:
Let the number of modern song lovers be [tex]\( \frac{8}{9} \times x = \frac{8}{9} \times 90 \)[/tex].
Once we find this, we subtract the number of people who like both (50) to find those who like only modern songs.
[tex]\[ \text{Only modern song lovers} = \frac{8}{9} \times 90 - 50 = 80 - 50 = 30 \][/tex]
Thus, the total number of participants can be calculated by:
[tex]\[ \text{Total participants} = (\text{Both}) + (\text{Only Folk}) + (\text{Neither}) + (\text{Only Modern}) \][/tex]
[tex]\[ = 50 + 40 + 80 + 30 \][/tex]
[tex]\[ = 126.25 \][/tex]
### Part (c): Find the Number of People Who Like Folk Songs
People who like folk songs include both those who like only folk songs and those who like both types of songs.
Number of people who like folk songs:
[tex]\[ \text{Total folk song lovers} = (\text{Only folk}) + (\text{Both}) \][/tex]
[tex]\[ = 40 + 50 \][/tex]
[tex]\[ = 90 \][/tex]
### Part (d): Find the Number of People Who Like Only One Song
We need to find the sum of people who like only folk songs and those who like only modern songs.
Number of people who like only one song:
[tex]\[ \text{Only one song lovers} = (\text{Only folk}) + (\text{Only modern}) \][/tex]
[tex]\[ = 40 + 30 \][/tex]
[tex]\[ = 46.25 \][/tex]
### Summary of Results
- Total participants: 126.25
- People who like folk songs: 90
- People who like only one song: 46.25
These steps explain the complete solution to the problem as requested.
