Question

A survey on fruit preference found that 40% of all students liked bananas while 70% of all students liked apples. Suppose x% of the students have opposite preferences for bananas and apples. What are the maximum and the minimum possible values of x?

Answer 1

Let A represent the set of students who like apples, B represent the set of students who like bananas, and C represent the set of students who have opposite preferences for bananas and apples. Then we have:

|A| = 70% of all students
|B| = 40% of all students
|A ∩ B| = the number of students who like both apples and bananas, which is not given
|C| = x% of all students
We can use the inclusion-exclusion principle to find an expression for |A ∪ B|, the number of students who like either apples or bananas (or both):

|A ∪ B| = |A| + |B| - |A ∩ B|

Substituting the values we have:

|A ∪ B| = 70% + 40% - |A ∩ B|

|A ∪ B| = 110% - |A ∩ B|

Since the total percentage of students cannot exceed 100%, we know that:

|A ∪ B| ≤ 100%

Combining the above equations, we get:

110% - |A ∩ B| ≤ 100%

Solving for |A ∩ B|, we get:

|A ∩ B| ≥ 10%

This means that at least 10% of the students like both apples and bananas. Therefore, the maximum possible value of x is 30% (when all students who do not like apples or bananas have opposite preferences) and the minimum possible value of x is 0% (when all students who like bananas also like apples, and vice versa).