Problem Solver #2: Venn Diagram Problem

A group of 100 middle school students were surveyed and found out that all of them visited at least one of the following cities: Austin (TX), Boston (MA), and Chicago (IL). 

53 of the students visited Austin.
52 of the students visited Boston.
65 of the students visited Chicago.
19 of the students visited Austin and Boston.
28 of the students visited Boston and Chicago.
31 of the students visited Austin and Chicago. 
38 of the students visited exactly one of the cities: Austin, Boston and Chicago. 

Create a Venn Diagram that would display the overlap of the students who visited the different cities and then answer the following question.

How many students visited all 3 cities? 


To solve this problem, we'll begin by using a Venn diagram to visualize the data and break down the student counts based on the overlaps between the cities.

### Step 1: Define the variables
Let:
- \( A \) represent the set of students who visited Austin.
- \( B \) represent the set of students who visited Boston.
- \( C \) represent the set of students who visited Chicago.
- \( x \) represent the number of students who visited all three cities (Austin, Boston, and Chicago). This is what we need to find.

We also know:
- Total students = 100
- \( |A| = 53 \)
- \( |B| = 52 \)
- \( |C| = 65 \)
- \( |A \cap B| = 19 \)
- \( |B \cap C| = 28 \)
- \( |A \cap C| = 31 \)
- 38 students visited exactly one city.

### Step 2: Use the Venn diagram formula

The formula for the total number of elements in three sets is:

\[
|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |A \cap C| + |A \cap B \cap C|
\]

We are given that \( |A \cup B \cup C| = 100 \), so we can plug in the known values:

\[
100 = 53 + 52 + 65 - 19 - 28 - 31 + x
\]

Simplifying the right-hand side:

\[
100 = 170 - 78 + x
\]

\[
100 = 92 + x
\]

Solving for \( x \):

\[
x = 100 - 92 = 8
\]

### Step 3: Confirm the result
Now that we know \( x = 8 \), we can check if the remaining information matches.

- The number of students who visited **only Austin** is \( |A| - (|A \cap B| + |A \cap C| - |A \cap B \cap C|) = 53 - (19 + 31 - 8) = 11 \).
- The number of students who visited **only Boston** is \( |B| - (|A \cap B| + |B \cap C| - |A \cap B \cap C|) = 52 - (19 + 28 - 8) = 13 \).
- The number of students who visited **only Chicago** is \( |C| - (|A \cap C| + |B \cap C| - |A \cap B \cap C|) = 65 - (31 + 28 - 8) = 14 \).

So, the number of students who visited exactly one city is:

\[
11 \ (\text{only Austin}) + 13 \ (\text{only Boston}) + 14 \ (\text{only Chicago}) = 38
\]

This matches the given data, so the solution is correct.

### Final Answer:
The number of students who visited all three cities (Austin, Boston, and Chicago) is **8**.

---

Would you like further details, or do you have any questions?

Here are 5 related questions:
1. How many students visited only Austin and Chicago?
2. How many students visited only Boston and Chicago?
3. How many students visited exactly two cities?
4. What is the number of students who visited Austin but not Chicago?
5. Can you find how many students did not visit Boston?

**Tip:** Using the principle of inclusion-exclusion helps to solve complex set problems systematically.

A group of 100 middle school students were surveyed and found out that all of them visited at least one of the following cities: Austin (TX), Boston (MA), and Chicago (IL). 53 of the students visited Austin. 52 of the students visited Boston. 65 of the students visited Chicago. 19 of the students visited Austin and Boston. 28 of the students visited Boston and Chicago. 31 of the students visited Austin and Chicago. 38 of the students visited exactly one of the cities: Austin, Boston, and Chicago. How many students visited all 3 cities?

Explore the Venn Diagram problem where 100 students visited Austin, Boston, or Chicago. Learn how to apply the inclusion-exclusion principle to calculate how many students visited all three cities with detailed steps. Perfect for Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation