a survey was conducted at an academy with 100 students, asking which industry they would like to work in. After the survey, 40 students expressed interest in the anime industry. Of these, 5 students wanted to work in both anime and video games, but not music and 10 wanted to word exclusively in the anime industry. Additionally , 15 students wanted to work in all 3 industries. How many students want to work both the anime and music industry?

To solve this problem, we will use the principle of inclusion-exclusion and some logical reasoning based on the given data.

### Step 1: Define Sets and Variables

Let:
- \( A \) = Students interested in the Anime industry.
- \( G \) = Students interested in the Video Games industry.
- \( M \) = Students interested in the Music industry.

Given values:
- Total students interested in Anime: \( |A| = 40 \).
- Students interested in both Anime and Video Games, but not Music: \( |A \cap G \cap M'| = 5 \).
- Students interested exclusively in Anime: \( |A|_{only} = 10 \).
- Students interested in all three industries: \( |A \cap G \cap M| = 15 \).

We want to find:
- The number of students who are interested in both Anime and Music.

### Step 2: Calculate Exclusive Sets

From the total Anime interest set \( |A| \), we need to calculate the number of students in various intersecting subsets.

1. Students interested only in Anime:
   \[
   |A|_{only} = 10
   \]

2. Students interested in both Anime and Video Games but not Music:
   \[
   |A \cap G \cap M'| = 5
   \]

3. Students interested in all three industries:
   \[
   |A \cap G \cap M| = 15
   \]

### Step 3: Find Students in Anime and Music but not Video Games

Let \( x \) be the number of students interested in both Anime and Music but not Video Games (\( |A \cap M \cap G'| \)).

Now, the total number of students interested in Anime is:
\[
|A| = |A|_{only} + |A \cap G \cap M'| + |A \cap G \cap M| + x
\]

Plug in the known values:
\[
40 = 10 + 5 + 15 + x
\]

Simplify:
\[
40 = 30 + x
\]

So:
\[
x = 10
\]

### Answer

The number of students who want to work in both the Anime and Music industries is:

\[
\boxed{10}
\]

Would you like further details or have any questions about this problem?

Here are some related questions to explore:

1. How many students want to work in the Music industry?
2. How many students are interested in Video Games exclusively?
3. What is the total number of students interested in the Video Games industry?
4. How many students are interested in only one industry?
5. How many students are not interested in the Anime industry at all?

**Tip:** When dealing with overlapping groups in surveys, use Venn diagrams to visualize and better understand the relationships among the sets!

A survey was conducted at an academy with 100 students, asking which industry they would like to work in. After the survey, 40 students expressed interest in the anime industry. Of these, 5 students wanted to work in both anime and video games, but not music, and 10 wanted to work exclusively in the anime industry. Additionally, 15 students wanted to work in all 3 industries. How many students want to work both in the anime and music industries?

Learn how to calculate the number of students interested in both the anime and music industries using the principle of inclusion-exclusion. This problem involves 100 students, with specific data points, including students interested in anime, video games, and music. Suitable for Grades 9-12.

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