https://file.aiquickdraw.com/mathbot/file/1727109626154j1hoa4b7.jpg

The problem presented involves a Venn diagram representing the music preferences of 500 customers, listening to hip-hop, rock, and country music. The task asks for calculations regarding various regions in the diagram, based on the following dataThe problem presented involves a Venn diagram representing the music preferences of 500 customers, listening to hip-hop, rock, and country music. The task asks for calculations regarding various regions in the diagram, based on the following 

- 270 listen to hip-hop.
- 320 listen to rock.
- 160 listen to country.
- 140 listen to both rock and hip-hop.
- 120 listen to both rock and country.
- 80 listen to both hip-hop and country.
- 50 listen to all three genres.

Let's break down how to solve this:

### Steps to fill in the Venn diagram:
1. **Region V (All three genres):** 50 customers listen to all three genres (given).
2. **Regions II, IV, and VI (Two genres overlap excluding the third):**
   - Region II (Hip-hop and Rock but not Country): \( 140 - 50 = 90 \) customers.
   - Region IV (Rock and Country but not Hip-hop): \( 120 - 50 = 70 \) customers.
   - Region VI (Hip-hop and Country but not Rock): \( 80 - 50 = 30 \) customers.
3. **Regions I, III, and VII (Only one genre):**
   - Region I (Only Hip-hop): We know 270 people listen to hip-hop, and those in regions II, V, and VI also listen to hip-hop. Thus, \( 270 - (90 + 50 + 30) = 100 \) customers listen to only hip-hop.
   - Region III (Only Rock): 320 listen to rock, and those in regions II, IV, and V also listen to rock. Thus, \( 320 - (90 + 50 + 70) = 110 \) customers listen to only rock.
   - Region VII (Only Country): 160 listen to country, and those in regions IV, V, and VI also listen to country. Thus, \( 160 - (70 + 50 + 30) = 10 \) customers listen to only country.
4. **Region VIII (None of these genres):** The total number of customers is 500, so subtract the total number of people who listen to any of the genres:
   \[
   500 - (100 + 90 + 110 + 70 + 30 + 50 + 10) = 40
   \]
   So, 40 customers don't listen to any of the three genres.

Now, let’s address the specific questions:

### (a) Listen to only hip-hop:
This corresponds to **Region I**, so the answer is **100** customers.

### (b) Listen to rock and country but not hip-hop:
This corresponds to **Region IV**, so the answer is **70** customers.

### (c) Don’t listen to any of these three types of music:
This corresponds to **Region VIII**, so the answer is **40** customers.

### (d) Don’t listen to country music:
This group includes everyone outside of the country circle (i.e., Regions I, II, III, and VIII):
\[
100 + 90 + 110 + 40 = 340
\]
So, the answer is **340** customers.

Would you like further clarification on any of the steps? Here are five related questions to consider:

1. How would the solution change if more people listened to all three genres?
2. How would you calculate the number of people who listen to only rock?
3. What would happen if the total number of customers were different?
4. How would you express these set operations using formal set notation?
5. Can you visualize this data using other types of diagrams?

**Tip:** In Venn diagrams, always start by filling in the intersections with the most overlap and work your way outwards.

An online music service surveyed 500 customers and found that 270 listen to hip-hop music, 320 listen to rock, and 160 listen to country. In addition, 140 listen to both rock and hip-hop, 120 listen to rock and country, and 80 listen to hip-hop and country. Finally, 50 listen to all three. A Venn diagram is given, and the number of customers listening to different combinations is sought.

This problem uses a Venn diagram to represent the music preferences of 500 customers. Learn how to calculate the number of people listening to different combinations of hip-hop, rock, and country music. The problem involves key concepts of set theory, including intersections and exclusions, suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation