Given the voting system {q:8,7,7,5,4,1}
The quota (q) of the system must be over half and must not be more than the total weight
a) The smallest value that the quota q can take is given below;
[tex]\begin{gathered} \text{Total weights}\Rightarrow8+7+7+5+4+1=32 \\ \text{Half of the total =}\frac{32}{2}=16 \\ Thus,\text{ the smallest value falls in the range} \\ 16b) The largest value the quota can take must not be more than the total weight, thus the largest value is 32c) If at least three fourth of the vote are required to pass, the quota will be
[tex]\frac{3}{4}\times32=8\times3=24[/tex]The quota for at least three-fourth to pass the votes will be 24.
d) If at most three fourth of the vote are required to pass, the quota will be
[tex]\begin{gathered} \text{The value will fall in the range} \\ \frac{3}{4}\times32