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In a survey of 80 people, it was found that 60 like pineapple only and 10 like both pineapple and jackfruit. The number of people who like pineapple is 5 times the number of people who like jackfruit.

a. In the given context, state the cardinality of the universal set.

b. Using a Venn diagram, find the number of people who like only jackfruit and those who did not like any of these fruits.

c. Identify how many people like only jackfruit.

d. If Rampyari stated that [tex]$40\%$[/tex] of those who did not like pineapple like jackfruit, justify this statement.



Answer :

GinnyAnswer
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.

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