The problem asks about the weighted voting system $[q: 7, 5, 3, 1, 1]$. Let’s break it down step-by-step to answer the questions.

Understanding Weighted Voting System

In this system:

• $q$ is the quota — the minimum number of votes required to pass a motion.
• The weights of the five voters are $7, 5, 3, 1, 1$.
• The total number of votes (or sum of the weights) is: $7 + 5 + 3 + 1 + 1 = 17 \, \text{votes}.$

Now, let’s address each part of the question.

(a) What is the smallest value that the quota $q$ can take?

• The smallest quota must be more than half of the total votes to prevent a tie (simple majority system).
• Half of 17 is: $\frac{17}{2} = 8.5$
• Since the quota must be an integer greater than 8.5, the smallest possible value of $q$ is: $q_{\text{min}} = 9$

(b) What is the largest value that the quota $q$ can take?

• The largest quota must be at most the total number of votes (since a motion requires votes to pass). A quota higher than 17 would make it impossible to pass a motion.

Thus, the maximum value of $q$ is: $q_{\text{max}} = 17$

(c) What is the value of the quota if at least two-thirds of the votes are required to pass a motion?

• Two-thirds of the total votes is: $\frac{2}{3} \times 17 = 11.333$
• Since the quota must be an integer, it must be at least 12 to meet the two-thirds requirement.

Thus, the value of the quota for two-thirds is: $q = 12$

Summary of Answers:

• (a) The smallest value of the quota $q_{\text{min}}$ is 9.
• (b) The largest value of the quota $q_{\text{max}}$ is 17.
• (c) If two-thirds of the votes are required, the quota $q$ is 12.

Would you like further clarification on any part of this solution?

Here are 5 related questions for practice:

1. What is the minimum number of voters needed to pass a motion with quota 9?
2. If a voter with weight 7 leaves, what is the new minimum quota?
3. What is the majority quota for a total of 21 votes?
4. How many coalitions are possible with a total weight of exactly 12 votes?
5. What happens if the weights of two voters are doubled?

Tip: In weighted voting systems, quotas are typically designed to balance fairness and prevent small groups from dominating decisions.