Answer :
Based on the results provided in the survey:
Here is a tabulated summary:
[tex]\[ \begin{array}{|l|c|c|c|c|} \hline & 8 \text{ voters} & 6 \text{ voters} & 3 \text{ voters} & 3 \text{ voters} \\ \hline \text{First choice} & \text{Tacos} & \text{Pizza} & \text{Sandwiches} & \text{Pizza} \\ \hline \text{Second choice} & \text{Pizza} & \text{Tacos} & \text{Tacos} & \text{Sandwiches} \\ \hline \text{Third choice} & \text{Sandwiches} & \text{Sandwiches} & \text{Pizza} & \text{Tacos} \\ \hline \end{array} \][/tex]
Step-by-Step Solution for Instant Runoff Voting:
1. First Choice Count:
- Tacos: 8 voters
- Pizza: 6 voters + 3 voters = 9 voters
- Sandwiches: 3 voters
2. Determine if there is a majority winner:
- Total voters = 8 + 6 + 3 + 3 = 20.
- A majority requires more than half of the total votes, so more than 10 votes are needed to win outright.
3. Check for a majority:
- Tacos: 8 votes
- Pizza: 9 votes
- Sandwiches: 3 votes
No single candidate has more than 10 votes, so we proceed to eliminate the candidate with the fewest votes.
4. Eliminate the candidate with the fewest votes:
- Sandwiches have the fewest votes (3 votes), so they are eliminated.
5. Redistribute votes based on the next choice for those who voted for Sandwiches:
- The 3 voters who initially voted for Sandwiches will have their votes transferred to their second choice:
- Second choice of Sandwiches voters: All 3 voters prefer Tacos as their second choice.
6. Recount with redistributed votes:
- Tacos: 8 original + 3 redistributed = 11 votes
- Pizza: 9 votes
7. Determine if there is a majority winner after redistribution:
- Tacos: 11 votes
- Pizza: 9 votes
With Tacos securing 11 votes, which is more than 10 and thus a majority, Tacos is the winner according to the instant runoff method.
Therefore, the number of first-place votes needed to be the winner using the instant runoff method, given the provided voting results, is at least: 11 votes.
And based on the process, Tacos wins.
Here is a tabulated summary:
[tex]\[ \begin{array}{|l|c|c|c|c|} \hline & 8 \text{ voters} & 6 \text{ voters} & 3 \text{ voters} & 3 \text{ voters} \\ \hline \text{First choice} & \text{Tacos} & \text{Pizza} & \text{Sandwiches} & \text{Pizza} \\ \hline \text{Second choice} & \text{Pizza} & \text{Tacos} & \text{Tacos} & \text{Sandwiches} \\ \hline \text{Third choice} & \text{Sandwiches} & \text{Sandwiches} & \text{Pizza} & \text{Tacos} \\ \hline \end{array} \][/tex]
Step-by-Step Solution for Instant Runoff Voting:
1. First Choice Count:
- Tacos: 8 voters
- Pizza: 6 voters + 3 voters = 9 voters
- Sandwiches: 3 voters
2. Determine if there is a majority winner:
- Total voters = 8 + 6 + 3 + 3 = 20.
- A majority requires more than half of the total votes, so more than 10 votes are needed to win outright.
3. Check for a majority:
- Tacos: 8 votes
- Pizza: 9 votes
- Sandwiches: 3 votes
No single candidate has more than 10 votes, so we proceed to eliminate the candidate with the fewest votes.
4. Eliminate the candidate with the fewest votes:
- Sandwiches have the fewest votes (3 votes), so they are eliminated.
5. Redistribute votes based on the next choice for those who voted for Sandwiches:
- The 3 voters who initially voted for Sandwiches will have their votes transferred to their second choice:
- Second choice of Sandwiches voters: All 3 voters prefer Tacos as their second choice.
6. Recount with redistributed votes:
- Tacos: 8 original + 3 redistributed = 11 votes
- Pizza: 9 votes
7. Determine if there is a majority winner after redistribution:
- Tacos: 11 votes
- Pizza: 9 votes
With Tacos securing 11 votes, which is more than 10 and thus a majority, Tacos is the winner according to the instant runoff method.
Therefore, the number of first-place votes needed to be the winner using the instant runoff method, given the provided voting results, is at least: 11 votes.
And based on the process, Tacos wins.
