Respuesta :
Answer:
16 students take only English or only Math.
Step-by-step explanation:
We can solve this problem by treating these values as sets, and building the Venn Diagram.
I am going to say that:
A is the number of students who take Math.
B is the number of students who take English.
C is the number of students who take History.
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 take Math, [tex]A \cap B[/tex] is the number of students who take both Math and English, [tex]A \cap C[/tex] is the number of students that take both Math and History, and [tex]A \cap B \cap C[/tex] is the number of students that take all these classes.
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]
There were 42 students suveyed. This means that:
[tex]a + b + c + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 42[/tex]
We start finding the values from the intersection of three sets.
12 students said they take math, English, and history. This means that:
[tex]A \cap B \cap C = 12[/tex]
18 students said they take English and history. This also takes into account those who take math, english and history. So:
[tex](B \cap C) + (A \cap B \cap C) = 18[/tex]
[tex]B \cap C = 6[/tex]
17 students said they take math and English.
[tex](A \cap B) + (A \cap B \cap C) = 17[/tex]
[tex]A \cap B = 5[/tex]
15 students said they take math and history
[tex](A \cap C) + (A \cap B \cap C) = 15[/tex]
[tex]A \cap C = 3[/tex]
2 students said they only take history.
[tex]c = 2[/tex]
How many students take only English or only math?
This is a + b, that we can find by the following formula:
[tex]a + b + c + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 42[/tex]
[tex]a + b + 2 + 5 + 3 + 6 + 12 = 42[/tex]
[tex]a + b = 16[/tex]
16 students take only English or only Math.
