https://file.aiquickdraw.com/mathbot/file/1725824028543tu8p03v1.jpg

I can help extract the mathematical information from the image and solve it step by step.

From the uploaded image, the problem mentions:

- A total of 540 participants have taken part in one or more of three events, \( Q \), \( P \), and \( R \).
- The number of participants in all three events is 30.
- 320 participants attended only one event.
- The total participants for event \( Q \) is 280, for \( P \) it is 230, and for \( R \) it is 310.
- The questions are related to set theory or Venn diagrams.

### Let's analyze and break down the given information:

We have three sets: \( Q \), \( P \), and \( R \), representing participants in each event.

Let the following represent the number of participants:

- \( Q \) = participants in event \( Q \),
- \( P \) = participants in event \( P \),
- \( R \) = participants in event \( R \),
- \( Q \cap P \cap R \) = participants in all three events = 30,
- 320 participants attended only one event.

### Questions:

1. **How many participants participated in more than two events?**

This refers to people who participated in exactly two events and all three events.

2. **How many participants have participated only in event \( R \)?**

This refers to the participants who are in \( R \) but not in \( Q \) or \( P \).

3. **If 17.5% of the participants who have attended all three events are women, find the number of male participants who have attended all three events.**

4. **What is the ratio of the number of participants who have participated in exactly two events to the number of participants who participated only in one event?**

5. **What is the average number of participants who participated in exactly one of the events?**

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I will now solve each of these step by step.Here are the results for the first two parts of the problem:

1. **Number of participants who participated in more than two events:**
   - The participants who attended exactly two events are **190**.

2. **Number of male participants who attended all three events:**
   - If 17.5% of participants who attended all three events are women, then the number of male participants in all three events is approximately **25**.

Let me know if you'd like more details or have any questions. Here are five related questions for further practice:

1. How many participants attended exactly one event?
2. What percentage of participants attended only two events?
3. How many women attended all three events?
4. What is the total number of participants who attended exactly one or two events?
5. What is the ratio of men to women who attended all three events?

**Tip:** Venn diagrams are a great tool for visualizing problems involving sets and intersections!

A total of 540 participants have participated in one or more of the three events Q, P, and R. The number of participants who have participated in all three events is 30. The number of participants who have participated in one event is 320. The total number of participants in events Q, P, and R are 280, 230, and 310, respectively. The questions asked are: How many participants participated in more than two events? How many participants participated only in event R? If 17.5% of the participants who have attended all three events are women, find the number of male participants who attended all three events. What is the ratio of the number of participants who have participated in exactly two events to the number of participants who participated only in one event? What is the average number of participants who participated in exactly one of the events?

Solve a problem involving 540 participants across three events Q, P, and R using set theory, Venn diagrams, percentage, and ratio calculations. The problem is suitable for students in Grades 9-12 and covers key concepts like the Inclusion-Exclusion Principle.

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