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 if they have 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 \(x\) and still have \(x\) be more than the votes of the other candidates). 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 we have the equation \(337=x+(4 - 1)(x - 1)\).

Step2: Solve the Equation for \(x\)

Expand the right - hand side of the equation:
\(337=x + 3(x - 1)=x+3x-3=4x-3\)

Add 3 to both sides of the equation:
\(337 + 3=4x\)
\(340 = 4x\)

Divide both sides by 4:
\(x=\frac{340}{4}=85\)

We need to check if this works. If the winning 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. Also, \(85>84\), so the candidate with 85 votes has more votes than the others.

Part (b)

Step1: Set up the Equation

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

Step2: Solve the Equation

Expand the right - hand side:
\(337=x+6(x - 1)=x + 6x-6=7x-6\)

Add 6 to both sides:
\(337+6=7x\)
\(343 = 7x\)

Divide both sides by 7:
\(x=\frac{343}{7}=49\)

Check: If the winning 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. And \(49>48\), so the candidate with 49 votes has more votes than the others.

Answer:

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

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