Chapter 2, Problem 10E

A committee has six members $\left(P_1,P_2,P_3,P_4,P_5,\text{and }{P}_6\right)$. In this committee $P_1$ has twice as many votes as $P_2$; $P_2$ and $P_3$ each has twice as many votes as $P_4$; $P_4$ has twice as many votes as $P_5$; $P_5$ and $P_6$ have the same number of votes. The quota is $\text{q}=121$. For each of the given definitions of the quota, describe the committee using the notation $\left[q:w_1,w_2,w_3,w_4,w_5,w_6\right]$. (Hint: Write the weighted voting system as $\left[121:8\text{x},4\text{x},4\text{x},2\text{x},\text{x},\text{x}\right]$, and then solve for $x$.)

a. The quota is defined as a simple majority of the votes.

b. The quota is defined as more than two-thirds of the votes.

c. The quota is defined as more than three-fourths of the votes.