Problem 18

Explanation

Given:

• Total teenagers ($S$) = $300$
• Sports activities ($O$) = $135$
• Nature lover activities ($C$) = $255$

We are asked to find the difference between the maximum value ($B$) and the minimum value ($A$) of the intersection of these two sets ($O \cap C$).

Finding Minimum Value A

The minimum intersection occurs when the union of the two sets is maximized, which equals the total universe ($300$). This means everyone participates in at least one activity.

Finding Maximum Value B

The maximum intersection occurs when the smaller set is a subset of the larger set. In this case, all sports participants also participate in nature lover activities.

Calculating the Difference

Thus, the difference between $B$ and $A$ is $45$ people.