Respuesta :
Answer:
624 tourists only visited the Magic Kindgom.
Step-by-step explanation:
To solve this problem, we must build the Venn's Diagram of this set.
I am going to say that:
-The set A represents the tourists that visited LEGOLAND
-The set B represents the tourists that visited Universal Studios
-The set C represents the tourists that visited Magic Kingdown.
-The value d is the number of tourists that did not visit any of these parks, so: [tex]d = 74[/tex]
We have that:
[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]
In which a is the number of tourists that only visited LEGOLAND, [tex]A \cap B[/tex] is the number of tourists that visited both LEGOLAND and Universal Studies, [tex]A \cap C[/tex] is the number of tourists that visited both LEGOLAND and the Magic Kingdom. and [tex]A \cap B \cap C[/tex] is the number of students that visited all these parks.
By the same logic, we have:
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
This diagram has the following subsets:
[tex]a,b,c,d,(A \cap B), (A \cap C), (B \cap C), (A \cap B \cap C)[/tex]
There were 1,168 tourists suveyed. This means that:
[tex]a + b + c + d + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 1,168[/tex]
We start finding the values from the intersection of three sets.
The problem states that:
16 tourists had visited all three theme parks. So:
[tex]A \cap B \cap C = 16[/tex]
91 tourists had visited both LEGOLAND and Universal Studios. So:
[tex](A \cap B) + (A \cap B \cap C) = 91[/tex]
[tex](A \cap B) = 91-16[/tex]
[tex](A \cap B) = 75[/tex]
68 tourists had visited both the Magic Kingdom and Universal Studios. So
[tex](B \cap C) + (A \cap B \cap C) = 68[/tex]
[tex](B \cap C) = 68-16[/tex]
[tex](B \cap C) = 52[/tex]
87 tourists had visited both the Magic Kingdom and LEGOLAND
[tex](A \cap C) + (A \cap B \cap C) = 87[/tex]
[tex](A \cap C) = 87-16[/tex]
[tex](A \cap C) = 71[/tex]
295 tourists had visited Universal Studios
[tex]B = 295[/tex]
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
[tex]295 = b + 52 + 75 + 16[/tex]
[tex]b + 143 = 295[/tex]
[tex]b = 152[/tex]
266 tourists had visited LEGOLAND
[tex]A = 266[/tex]
[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]
[tex]266 = a + 75 + 71 + 16[/tex]
[tex]a + 162 = 266[/tex]
[tex]a = 104[/tex]
How many tourists only visited the Magic Kingdom (of these three)?
This is the value of c, the we can find in the following equation:
[tex]a + b + c + d + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 1,168[/tex]
[tex]104 + 152 + c + 74 + 75 + 71 + 52 + 16 = 1,168[/tex]
[tex]c + 544 = 1,168[/tex]
[tex]c = 624[/tex]
624 tourists only visited the Magic Kindgom.
