Respuesta :
Answer:
There were 49 students in the class
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 students that watched TV on Monday
-The set B represents the student that watched TV on Tuesday.
-The set C represents the students that watched TV on Wednesday.
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 students that only watched TV on Monday, [tex]A \cap B[/tex] is the number of adults that watched TV both on Monday and Tuesday, [tex]A \cap C[/tex] is the number of students that watched TV both on Monday and Wednesday, and [tex]A \cap B \cap C[/tex] is the number of students that watched TV on every day.
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,(A \cap B), (A \cap C), (B \cap C), (A \cap B \cap C)[/tex]
The sums of all of this values is the number of student that were there in the class. This means that we want to find the value of T:
[tex]a + b + c + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = T[/tex]
We start finding the values from the intersection of three sets.
Solution:
12 students watched TV on all three days:
[tex]A \cap B \cap C = 12[/tex]
14 students watched TV on both Monday and Tuesday
[tex]A \cap B + A \cap B \cap C = 14[/tex]
[tex]A \cap B = 14 - 12[/tex]
[tex]A \cap B = 2[/tex]
Of those who watched TV on only one of these days, 13 choose Monday, 9 chose Tuesday, and 10 chose Wednesday.
[tex]a = 13, b = 9, c = 10[/tex]
29 students watched television on Monday:
[tex]A = 29[/tex]
[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]
[tex]29 = 13 + 2 + (A \cap C) + 12[/tex]
[tex]A \cap C = 29 - 27[/tex]
[tex]A \cap C = 2[/tex]
24 on Tuesday
[tex]B = 24[/tex]
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
[tex]24 = 9 + (B \cap C) + 2 + 12[/tex]
[tex]B \cap C = 24 - 23[/tex]
[tex]B \cap C = 1[/tex]
Now we have every value needed to find T:
[tex]T = a + b + c + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
[tex]T = 13 + 9 + 10 + 2 + 2 + 1 + 12[/tex]
[tex]T = 49[/tex]
There were 49 students in the class
