Question

There are 21 people on the basketball team and 45 people on the football team. If 100 people attend the athletic banquet …

a) What is the minimum and maximum number of people that are on both the basketball and the football teams?
b) What is the minimum and maximum number of people that are on neither of these teams?
Let x = number of individuals both in basketball and football
c) Draw a Venn diagram to illustrate this

Answer 1

a) min: 0; max: 21

b) min: 34; max: 55

Explanation:

In the attached Venn diagrams (part c), the overlap of the two circles represents the number on both teams.

a) In the top diagram, there is no overlap, so the number on both teams is 0.

In the bottom diagram, there is complete overlap between the teams, so the maximum number that play both sports is 21 (the minimum team size of the two teams).

both teams minimum: 0; maximum: 21

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b) The number on neither team is the number outside both circles. The minimum number will correspond to the case where there is no overlap between teams. In that case, there are 66 team players, so 34 people who are on neither team.

The maximum case is that where the teams completely overlap, so the number not on either team will be those not on the football team: 55 people.

neither team minimum: 34; maximum: 55