Chapter 2, Problem 9E

A committee has four members $\left(P_1,P_2,P_3,\text{and }{P}_4\right)$. In this committee $P_1$ has twice as many votes as $P_2$; $P_2$ has twice as many votes as $P_3$; $P_3$ and $P_4$ have the same number of votes. The quota is $\text{q}=49$. For each of the given definitions of the quota, describe the committee using the notation $\left[q:w_1,w_2,w_3,w_4\right]$. (Hint: Write the weighted voting system as $\left[49:4x,2x,x,x\right]$, 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.