Respuesta :
Answer:
182 of these adults did not do any of these three activities last Friday night.
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 adults that watched TV
-The set B represents the adults that hung out with friends.
-The set C represents the adults that ate pizza
-The set D represents the adults that did not do any of these three activities.
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 adults that only watched TV, [tex]A \cap B[/tex] is the number of adults that both watched TV and hung out with friends, [tex]A \cap C[/tex] is the number of adults that both watched TV and ate pizza, is the number of adults that both hung out with friends and ate pizza, and [tex]A \cap B \cap C[/tex] is the number of adults that did all these three activies.
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 510 adults suveyed. This means that:
[tex]a + b + c + D + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 510[/tex]
We start finding the values from the intersection of three sets.
Solution:
43 watched TV, hung out with friends, and ate pizza:
[tex]A \cap B \cap C = 43[/tex]
47 hung out with friends and ate pizza, but did not watch TV:
[tex]B \cap C = 47[/tex]
29 watched TV and hung out with friends, but did not eat pizza:
[tex]A \cap B = 29[/tex]
28 watched TV and ate pizza, but did not hang out with friends:
[tex]A \cap C = 28[/tex]
161 ate pizza
[tex]C = 161[/tex]
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
[tex]161 = c + 28 + 47 + 43[/tex]
[tex]c = 43[/tex]
196 hung out with friends
[tex]B = 196[/tex]
[tex]196 = b + 47 + 29 + 43[/tex]
[tex]b = 77[/tex]
161 watched TV
[tex]A = 161[/tex]
[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]
[tex]161 = a + 29 + 28 + 43[/tex]
[tex]a = 61[/tex]
How may 18-24 year olds did not do any of these three activities last Friday night?
We can find the value of D from the following equation:
[tex]a + b + c + D + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 510[/tex]
[tex]61 + 77 + 43 + D + 29 + 28 + 47 + 43 = 510[/tex]
[tex]D = 510 - 328[/tex]
[tex]D = 182[/tex]
182 of these adults did not do any of these three activities last Friday night.
