Question

Thirty-eight cities were researched to determine whether they had a professional sports​ team, a​ symphony, or a​ children's museum. of these cities, 21 had a professional sports team, 20 had a​ symphony, 17 had a​ children's museum, 14 had a professional sports team and a​ symphony, 10 had a professional sports team and a​ children's museum, 10 had a symphony and a​ children's museum, and 6 had all three activities. How many of the cities surveyed had only a professional sports​ team?

Answer 1

Answer:

3 of the cities surveyed had only a professional sports​ team.

Step-by-step explanation:

We solve this question treating these as Venn sets.

I am going to say that:

Set A: professional sports team.

Set B: Symphony.

Set C: Children's museum.

We start building from the intersection of these three.

6 had all three activities.

This means that $A \cap B \cap C = 6$

10 had a symphony and a​ children's museum,

This means that:

$(B \cap C) + (A \cap B \cap C) = 10$

$(B \cap C) + 6 = 10$

$(B \cap C) = 4$

That is, 4 have only the symphony and children's museum, while 6 have all three.

10 had a professional sports team and a​ children's museum

$(A \cap C) + (A \cap B \cap C) = 10$

$(A \cap C) + 6 = 10$

$(A \cap C) = 4$

14 had a professional sports team and a​ symphony

$(A \cap B) + (A \cap B \cap C) = 14$

$(A \cap B) + 6 = 14$

$(A \cap B) = 8$

21 had a professional sports team

This means that:

$(A - B - C) + (A \cap B) + (A \cap C) + (A \cap B \cap C) = 21$

In which (A - B - C) represents the cities which had only a sports team. So

$(A - B - C) + 8 + 4 + 6 = 21$

$(A - B - C) = 3$

3 of the cities surveyed had only a professional sports​ team.