Respuesta :
Answer:
168 of these adults only watched TV 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, 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 525 adults suveyed. This means that:
[tex]a + b + c + D + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 525[/tex].
Solution
We build the sets from what the problem states:
96 did not do any of these three activities:
[tex]D = 96[/tex]
25 watched TV, hung out with friends, and ate pizza:
[/tex]A \cap B \cap C = 25[/tex]
32 hung out with friends and ate pizza, but did not watch TV:
[tex]B \cap C = 32[/tex]
50 watched TV and hung out with friends, but did not eat pizza:
[tex]A \cap B = 50[/tex]
26 watched TV and ate pizza, but did not hang out with friends:
[tex]A \cap C = 26[/tex]
152 ate pizza:
[tex]C = 152[/tex]
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
[tex]152 = c + 26 + 32 + 25[/tex]
[tex]c = 69[/tex]
166 hung out with friends
[tex]B = 166[/tex]
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
[tex]166 = b + 32 + 50 + 25[/tex]
[tex]b = 59[/tex]
How may 18-24 year olds (of these three activities) only watched TV last Friday night?
We can find the value of a from the following equation:
[tex]a + 59 + 69 + 96 + 50 + 26 + 32 + 25 = 525[/tex]
[tex]a = 525 - 357[/tex]
[tex]a = 168[/tex]
168 of these adults only watched TV last Friday night
