Sure, let's break down the given problem step-by-step.
### Step-by-Step Solution
Given Information:
- Total number of people surveyed is 80.
- Number of people who like only pineapple (P) is 60.
- Number of people who like both pineapple and jackfruit (BP) is 10.
- The number of people who like pineapple is 5 times the number of people who like jackfruit.
### Part a: State the cardinality of the universal set
The universal set (U) is the total number of people surveyed.
- [tex]\(|U| = 80\)[/tex]
### Part b: Venn diagram completion
To complete the Venn diagram, we need to find:
- Number of people who like only jackfruit (J)
- Number of people who didn't like any of these fruits (N)
Let’s denote:
- [tex]\( J \)[/tex] as the number of people who like only jackfruit
- BP = 10 (both pineapple and jackfruit)
- P = 60 (pineapple only)
Using the condition that the number of people who like pineapple is 5 times the number of people who like jackfruit (including both categories of jackfruit likers only and likers of both fruits):
[tex]\[
P + BP = 5(J + BP)
\][/tex]
Substitute the known values:
[tex]\[
60 + 10 = 5(J + 10)
\][/tex]
[tex]\[
70 = 5J + 50
\][/tex]
[tex]\[
20 = 5J
\][/tex]
[tex]\[
J = 4
\][/tex]
- Number of people who like only jackfruit [tex]\( J = 4 \)[/tex]
To find the number of people who did not like any of these fruits (N):
[tex]\[
N = \text{Total people} - \text{(P + J + BP)}
\][/tex]
[tex]\[
N = 80 - (60 + 4 + 10)
\][/tex]
[tex]\[
N = 80 - 74
\][/tex]
[tex]\[
N = 6
\][/tex]
- Number of people who didn't like any of these fruits [tex]\( N = 6 \)[/tex]
### Part c: Identify the number of people who like jackfruit
The number of people who like jackfruit is the sum of those who like only jackfruit and those who like both.
[tex]\[
\text{Jackfruit Total} = J + BP
\][/tex]
[tex]\[
\text{Jackfruit Total} = 4 + 10
\][/tex]
[tex]\[
\text{Jackfruit Total} = 14
\][/tex]
### Part d: Justify Rampyari’s statement
Rampyari stated that 40% of those who didn't like pineapple like the jackfruit.
First, we need to find the number of people who didn't like pineapple:
[tex]\[
\text{Not Pineapple} = \text{Total people} - P - BP
\][/tex]
[tex]\[
\text{Not Pineapple} = 80 - 70
\][/tex]
[tex]\[
\text{Not Pineapple} = 10
\][/tex]
Then, calculate 40% of these people:
[tex]\[
\text{40% of Not Pineapple} = 0.40 \times 10 = 4
\][/tex]
- Rampyari’s statement is correct because 4 people (40% of those who didn't like pineapple) like the jackfruit.
### Conclusion
- a. The cardinality of the universal set is [tex]\(|U| = 80\)[/tex].
- b. The number of people who like only jackfruit is 4, and 6 people didn't like any of these fruits.
- c. The total number of people who like jackfruit is 14.
- d. 40% (which is 4) of those who didn't like pineapple actually like jackfruit, justifying Rampyari’s statement.
