A committee has four members ​(Upper P 1​, Upper P 2​, Upper P 3​, and Upper P 4​). In this committee Upper P 1 has twice as many votes as Upper P 2​; Upper P 2 has twice as many votes as Upper P 3​; Upper P 3 and Upper P 4 have the same number of votes. The quota is qequals93. For each of the given definitions of the quota​, describe the committee using the notation​ [q: w1,​ w2, w3,​ w4]. ​(Hint​: Write the weighted voting system as ​[93​: ​4x, 2x,​ x, x], and then solve for​ x.) Complete parts​ (a) through​ (c) below.

Let's break down the problem step by step to describe the committee using the weighted voting system notation \([q: w_1, w_2, w_3, w_4]\).

### Step 1: Set up the equation
We know the following relationships between the weights (number of votes) of the members:

- \(w_1 = 4x\) (since \(P_1\) has twice as many votes as \(P_2\))
- \(w_2 = 2x\) (since \(P_2\) has twice as many votes as \(P_3\))
- \(w_3 = x\) (since \(P_3\) and \(P_4\) have the same number of votes)
- \(w_4 = x\) (since \(P_3\) and \(P_4\) have the same number of votes)

Thus, the weighted voting system is of the form:
\[
[93: 4x, 2x, x, x]
\]

### Step 2: Solve for \(x\)
We now need to consider the quota \(q = 93\) given in the problem to solve for \(x\).

(a) **When \(q = 93\)**

In this scenario, the equation remains:
\[
[93: 4x, 2x, x, x]
\]
But \(x\) is arbitrary in this context since \(q = 93\) is a fixed quota, and no specific total votes are provided. Thus, we don't solve for \(x\) directly as we don't have the total number of votes specified.

So, the notation for the committee remains:
\[
[93: 4x, 2x, x, x]
\]

(b) **When \(q = \frac{3}{4}\) of total votes**

Here, we need to find the total number of votes and set the quota \(q = \frac{3}{4}\) of the total votes. The total number of votes \(V\) is:
\[
V = 4x + 2x + x + x = 8x
\]

So, the quota \(q\) is:
\[
q = \frac{3}{4} \times 8x = 6x
\]

Thus, the weighted voting system becomes:
\[
[6x: 4x, 2x, x, x]
\]

(c) **When \(q = \frac{1}{2}\) of total votes**

Similarly, for this case, the quota \(q\) is:
\[
q = \frac{1}{2} \times 8x = 4x
\]

So, the weighted voting system is:
\[
[4x: 4x, 2x, x, x]
\]

### Summary of Answers:
- **(a) \(q = 93\):** The committee is described as \([93: 4x, 2x, x, x]\).
- **(b) \(q = \frac{3}{4}\) of total votes:** The committee is described as \([6x: 4x, 2x, x, x]\).
- **(c) \(q = \frac{1}{2}\) of total votes:** The committee is described as \([4x: 4x, 2x, x, x]\).

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### Related Questions:
1. How does the change in the quota affect the voting power of each committee member?
2. What happens if \(q\) is greater than the total number of votes?
3. How would the system change if \(P_1\) had three times as many votes as \(P_2\)?
4. Can \(P_3\) or \(P_4\) ever have veto power in these setups?
5. How can we calculate

A committee has four members (P1, P2, P3, and P4). In this committee, P1 has twice as many votes as P2; P2 has twice as many votes as P3; P3 and P4 have the same number of votes. The quota is q = 93. For each of the given definitions of the quota, describe the committee using the notation [q: w1, w2, w3, w4]. (Hint: Write the weighted voting system as [93: 4x, 2x, x, x], and then solve for x.) Complete parts (a) through (c) below.

Explore how to describe a committee's voting system using weighted voting notation. This problem involves understanding algebraic relationships and proportionality. Suitable for high school students in Grades 9-12.

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