Question

question 3 consider an election with 337 votes a) if there are 4 candidates, what is the smallest number of first - place votes a candidate could win with under the plurality method? votes b) if there are 7 candidates, what is the smallest number of first - place votes a candidate could win with under the plurality method? votes question help: video

Explanation:

Part (a)

Step1: Understand Plurality Method

In the Plurality method, a candidate wins by having more first - place votes than any other candidate. To find the smallest number of votes a candidate can win with when there are \(n\) candidates, we assume the votes are distributed as evenly as possible among the candidates. Let the number of votes the winning candidate gets be \(x\), and the remaining \(n - 1\) candidates get \(x- 1\) votes (since we want to minimize \(x\), we assume the other candidates get as many votes as possible without exceeding \(x\)). The total number of votes \(V\) is the sum of the votes of all candidates. So, \(V=x+(n - 1)(x - 1)\).

For part (a), \(V = 337\) and \(n=4\). So the equation becomes \(337=x + 3(x - 1)\).

Step2: Solve the Equation

First, expand the right - hand side of the equation:
\(337=x+3x - 3\)
Combine like terms:
\(337 = 4x-3\)
Add 3 to both sides of the equation:
\(337 + 3=4x\)
\(340 = 4x\)
Divide both sides by 4:
\(x=\frac{340}{4}=85\)

But we need to check if this is correct. Let's verify the distribution. If one candidate has 85 votes, the other 3 candidates have \(85 - 1=84\) votes each. The total number of votes is \(85+3\times84=85 + 252=337\), which matches the total number of votes. However, we need to make sure that when we distribute the votes, we can't have a smaller \(x\). Suppose \(x = 84\), then the other 3 candidates would have at most 83 votes each. The total number of votes would be \(84+3\times83=84 + 249=333\), which is less than 337. We have 337 - 333 = 4 extra votes. We can distribute these 4 votes among the candidates. But if we give these extra votes to the other candidates, we might end up with a candidate having more than 84 votes. So, the minimum number of votes is 85.

Part (b)

Step1: Set up the Equation

For \(n = 7\) candidates and \(V=337\) votes, using the same logic as in part (a), the equation is \(337=x+6(x - 1)\).

Step2: Solve the Equation

Expand the right - hand side:
\(337=x + 6x-6\)
Combine like terms:
\(337=7x - 6\)
Add 6 to both sides:
\(337 + 6=7x\)
\(343=7x\)
Divide both sides by 7:
\(x = 49\)

Let's verify. If one candidate has 49 votes, the other 6 candidates have \(49 - 1 = 48\) votes each. The total number of votes is \(49+6\times48=49+288 = 337\), which is correct. If we try \(x = 48\), then the other 6 candidates have at most 47 votes each. The total number of votes is \(48+6\times47=48 + 282=330\), which is less than 337. We have 337 - 330 = 7 extra votes. If we distribute these 7 votes, we would end up with a candidate having more than 48 votes. So, the minimum number of votes is 49.

Answer:

a) \(\boldsymbol{85}\) votes

b) \(\boldsymbol{49}\) votes