Ms. Teller surveyed the 38 students in her class about their families. 28 students have at least a brother or a sister, 15 students have a brother, and 8 students have both a brother and a sister. How many students have only a sister?

To solve the problem, we can use the principle of inclusion-exclusion.

Let's define the following:
- \( B \): the set of students who have a brother.
- \( S \): the set of students who have a sister.

We are given:
- \( |B \cup S| = 28 \) (the number of students who have at least one brother or sister)
- \( |B| = 15 \) (the number of students who have a brother)
- \( |B \cap S| = 8 \) (the number of students who have both a brother and a sister)

We need to find the number of students who have only a sister, denoted as \( |S \setminus B| \).

### Step 1: Find the number of students who have at least a sister
Using the principle of inclusion-exclusion:
\[
|B \cup S| = |B| + |S| - |B \cap S|
\]
Plugging in the values we know:
\[
28 = 15 + |S| - 8
\]
\[
28 = 7 + |S|
\]
\[
|S| = 21
\]
So, 21 students have a sister.

### Step 2: Find the number of students who have only a sister
The number of students who have only a sister is the number of students who have a sister but do not have a brother:
\[
|S \setminus B| = |S| - |B \cap S|
\]
\[
|S \setminus B| = 21 - 8 = 13
\]

So, **13 students have only a sister**.

Would you like more details or have any questions?

Here are five related questions you might consider:

1. What is the principle of inclusion-exclusion?
2. How can Venn diagrams help visualize problems like this?
3. What if there were 12 students who had only a brother? How would that affect the answer?
4. Can you determine the number of students who have only a brother?
5. What is the total number of students who do not have any siblings?

**Tip:** When solving problems involving sets, carefully define each set and use formulas like inclusion-exclusion to handle overlaps.

Learn how to analyze survey data about students' families using sets and the inclusion-exclusion principle. Discover the solution to finding the number of students who have only a sister, based on given data. Suitable for Grades 7-9.

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