A survey of 1,107 tourists visiting Orlando was taken. Of those surveyed: 268 tourists had visited the Magic Kingdom 258 tourists had visited Universal Studios 68 tourists had visited both the Magic Kingdom - QAmmunity.org

A survey of 1,107 tourists visiting Orlando was taken. Of those surveyed: 268 tourists had visited the Magic Kingdom 258 tourists had visited Universal Studios 68 tourists had visited both the Magic Kingdom

asked May 24, 2020 229k views

1 Answer

Answer:

602 tourists visited only the LEGOLAND.

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:
d = 58

We have that:

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

In which a is the number of tourists that only visited LEGOLAND,
$A \cap B$ is the number of tourists that visited both LEGOLAND and Universal Studies,
$A \cap C$ is the number of tourists that visited both LEGOLAND and the Magic Kingdom. and
$A \cap B \cap C$ is the number of students that visited all these parks.

By the same logic, we have:

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

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

This diagram has the following subsets:

$a,b,c,d,(A \cap B), (A \cap C), (B \cap C), (A \cap B \cap C)$

There were 1,107 tourists suveyed. This means that:

$a + b + c + d + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 1,107$

We start finding the values from the intersection of three sets.

The problem states that:

36 tourists had visited all three theme parks. So:

$(A \cap B \cap C) = 36$

72 tourists had visited both LEGOLAND and Universal Studios. So:

$(A \cap B) + (A \cap B \cap C) = 72$

$(A \cap B) = 72 - 36$

$(A \cap B) = 36$

79 tourists had visited both the Magic Kingdom and Universal Studios

$(B \cap C) + (A \cap B \cap C) = 79$

$(B \cap C) = 79 - 36$

$(B \cap C) = 43$

68 tourists had visited both the Magic Kingdom and LEGOLAND

$(A \cap C) + (A \cap B \cap C) = 68$

$(A \cap C) = 68 - 36$

$(A \cap C) = 32$

258 tourists had visited Universal Studios:

B = 258

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

258 = b + 43 + 36 + 36

b = 143

268 tourists had visited the Magic Kingdom:

C = 268

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

268 = c + 32 + 43 + 36

c = 157

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:

$a + b + c + d + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 1,107$

a + 143 + 157 + 58 + 36 + 32 + 43 + 36 = 1,107

a = 602

602 tourists visited only the LEGOLAND.

Gabriel Gartz answered May 28, 2020

by Gabriel Gartz

4.9k points

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