1. For each of the following weighted voting systems, determine the total number of players, the total
weight, the quota, and the weight of P4.
(a) [65 : 30, 28, 22, 15, 7, 6] 

Let's break down the given weighted voting system \([65 : 30, 28, 22, 15, 7, 6]\) and analyze its components.

### Step 1: Identify the Elements
The format \([q : w_1, w_2, w_3, \dots]\) represents a weighted voting system where:
- \(q\) is the **quota** (minimum total weight required to pass a motion),
- \(w_1, w_2, w_3, \dots\) are the weights of the individual players.

Here:
- Quota (\(q\)) = 65
- Player 1's weight (\(w_1\)) = 30
- Player 2's weight (\(w_2\)) = 28
- Player 3's weight (\(w_3\)) = 22
- Player 4's weight (\(w_4\)) = 15
- Player 5's weight (\(w_5\)) = 7
- Player 6's weight (\(w_6\)) = 6

### Step 2: Determine the Total Number of Players
The total number of players is simply the number of weights given. There are 6 players in total.

- **Total number of players** = 6

### Step 3: Calculate the Total Weight
The total weight is the sum of all the players' weights:
\[
\text{Total Weight} = w_1 + w_2 + w_3 + w_4 + w_5 + w_6 = 30 + 28 + 22 + 15 + 7 + 6 = 108
\]

- **Total weight** = 108

### Step 4: Identify the Quota
The quota is explicitly given as 65.

- **Quota** = 65

### Step 5: Determine the Weight of Player 4 (P4)
From the weights given:
- The weight of Player 4 (\(w_4\)) = 15

### Final Summary:
- Total number of players = **6**
- Total weight = **108**
- Quota = **65**
- Weight of Player 4 = **15**

Would you like more details on how to interpret these numbers?

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Here are five follow-up questions:
1. What is the minimum number of players needed to meet the quota in this system?
2. What is a weighted voting system used for in practical terms?
3. How does the weight distribution affect the voting power of each player?
4. Could the system be manipulated by forming coalitions? If so, how?
5. What happens if the quota is set higher or lower than 65?

**Tip:** In weighted voting systems, players with larger weights often have more influence, but coalitions of smaller players can still win if they combine their weights.

For each of the following weighted voting systems, determine the total number of players, the total weight, the quota, and the weight of P4. (a) [65 : 30, 28, 22, 15, 7, 6]

Explore the analysis of a weighted voting system [65 : 30, 28, 22, 15, 7, 6]. Learn how to determine the total number of players, the total weight, the quota, and the weight of Player 4. This problem covers key concepts in weighted voting systems, suitable for Grades 9-12.

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