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Sports club has a membership of 800 and operates 26 baseball fields and 24 soccer fields. Before deciding whether to accept new members, the club council would like to know how many members regularly use each field. A survey of the membership indicates that 62% use the baseball field regularly and 64% regularly use the soccer field, and 3% do not regularly use either of these fields. What percentage of the 800 use both the soccer and baseball fields?​

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The percentage of the 800 use both the soccer and baseball fields is 34 %​.

Let S = set of people that use soccer field and B = set of people that use base ball field and B U S represent the universal set.

Since 62% use the baseball field regularly, P(B) = 64 %.

Also, 64% regularly use the soccer field, P(S) = 64 % and 3% do not regularly use either of these fields. Since this is a complement of the universal set, (B U S)'. So, P(B ∪ S)' = 3 %.

Since P(B ∪ S) + P(B ∪ S)' = 100 %

P(B ∪ S) = 100 % - P(B ∪ S)'

= 100 % - 3 %

= 97 %

Now, from set theory,

P(B ∪ S) = P(B) + P(S) - P(B ∩ S)

where P(B ∩ S) = percentage that use both the soccer and baseball fields.

So, P(B ∩ S) = P(B) + P(S) - P(B ∪ S)

Substituting the values of the variables into the equation, we have

P(B ∩ S) = P(B) + P(S) - P(B ∪ S)

P(B ∩ S) = 64 % + 62 % - 97 %

P(B ∩ S) = 126 % - 97 %

P(B ∩ S) = 34 %

So, the percentage of the 800 use both the soccer and baseball fields is 34 %​.

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