Question

A survey was conducted of 42 students who take math, Eng lish, or history. In the survey, 15 students said they take math and history. 2 students said they only take history. 17 students said they take math and English. 12 students said they take math, English, and history. 18 students said they take English and history. How many students take only English or only math? •

Answer 1

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:

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

In which a is the number of students that only take Math, $A \cap B$ is the number of students who take both Math and English, $A \cap C$ is the number of students that take both Math and History, and $A \cap B \cap C$ is the number of students that take all these classes.

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:

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

There were 42 students suveyed. This means that:

$a + b + c + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 42$

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

12 students said they take math, English, and history. This means that:

$A \cap B \cap C = 12$

18 students said they take English and history. This also takes into account those who take math, english and history. So:

$(B \cap C) + (A \cap B \cap C) = 18$

$B \cap C = 6$

17 students said they take math and English.

$(A \cap B) + (A \cap B \cap C) = 17$

$A \cap B = 5$

15 students said they take math and history

$(A \cap C) + (A \cap B \cap C) = 15$

$A \cap C = 3$

2 students said they only take history.

$c = 2$

How many students take only English or only math?

This is a + b, that we can find by the following formula:

$a + b + c + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 42$

$a + b + 2 + 5 + 3 + 6 + 12 = 42$

$a + b = 16$

16 students take only English or only Math.