Respuesta :
Answer:
602 tourists visited only the LEGOLAND.
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 = 58[/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 1,107 tourists suveyed. This means that:
[tex]a + b + c + d + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 1,107[/tex]
We start finding the values from the intersection of three sets.
The problem states that:
36 tourists had visited all three theme parks. So:
[tex](A \cap B \cap C) = 36[/tex]
72 tourists had visited both LEGOLAND and Universal Studios. So:
[tex](A \cap B) + (A \cap B \cap C) = 72[/tex]
[tex](A \cap B) = 72 - 36[/tex]
[tex](A \cap B) = 36[/tex]
79 tourists had visited both the Magic Kingdom and Universal Studios
[tex](B \cap C) + (A \cap B \cap C) = 79[/tex]
[tex](B \cap C) = 79 - 36[/tex]
[tex](B \cap C) = 43[/tex]
68 tourists had visited both the Magic Kingdom and LEGOLAND
[tex](A \cap C) + (A \cap B \cap C) = 68[/tex]
[tex](A \cap C) = 68 - 36[/tex]
[tex](A \cap C) = 32[/tex]
258 tourists had visited Universal Studios:
[tex]B = 258[/tex]
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
[tex]258 = b + 43 + 36 + 36[/tex]
[tex]b = 143[/tex]
268 tourists had visited the Magic Kingdom:
[tex]C = 268[/tex]
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
[tex]268 = c + 32 + 43 + 36[/tex]
[tex]c = 157[/tex]
How many tourists only visited the LEGOLAND (of these three)?
We have to find the value of a, and we can do this by the following equation:
[tex]a + b + c + d + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 1,107[/tex]
[tex]a + 143 + 157 + 58 + 36 + 32 + 43 + 36 = 1,107[/tex]
[tex]a = 602[/tex]
602 tourists visited only the LEGOLAND.
