For the following preference​ schedule, answer questions a through g.

First

Upper DD

Upper CC

Upper EE
 
Second

Upper EE

Upper AA

Upper BB

Third

Upper BB

Upper DD

Upper CC

Fourth

Upper CC

Upper BB

Upper AA

Fifth

Upper AA

Upper EE

Upper DD

Number of voters

4545

3434

1717

Question content area bottom

Part 1

a. How many votes were​ cast?

A total of 

9696 

votes were cast.

Part 2

b. Find the plurality winner. Did the plurality winner also receive a​ majority? Explain.

A.

Candidate 

Upper EE 

is the plurality​ winner, but not by a majority because 

Upper EE 

received less than​ 50% of the votes.

B.

Candidate 

Upper CC 

is the plurality winner by a majority because 

Upper CC 

received more than​ 50% of the votes.

C.

Candidate 

Upper DD 

is the plurality​ winner, but not by a majority because 

Upper DD 

received less than​ 50% of the votes.

Your answer is correct.

D.

Candidate 

Upper AA 

is the plurality​ winner, but not by a majority because 

Upper AA 

received less than​ 50% of the votes.

Part 3

c. Find the winner by a runoff of the top two candidates. Choose the correct answer below.

A.

Candidate 

Upper DD 

is the winner of the​ top-two runoff.

B.

Candidate 

Upper AA 

is the winner of the​ top-two runoff.

C.

Candidate 

Upper EE 

is the winner of the​ top-two runoff.

D.

Candidate 

Upper CC 

is the winner of the​ top-two runoff.

Your answer is correct.

Part 4

d. Find the winner by a sequential runoff. Choose the correct answer below.

A.

The winner by sequential runoff is 

Upper DD.

B.

The sequential runoff method gives a tie among 

Upper DD 

and 

Upper CC.

C.

The winner by sequential runoff is 

Upper EE.

D.

The winner by sequential runoff is 

Upper CC.

Let's go through the solutions for each part based on the preference schedule provided:

### Part 1
**a. How many votes were cast?**

There are 96 voters in total:
- 45 from the first group
- 34 from the second group
- 17 from the third group

So, the total number of votes is:

\[
45 + 34 + 17 = 96 \text{ votes.}
\]

### Part 2
**b. Find the plurality winner. Did the plurality winner also receive a majority?**

The plurality winner is the candidate with the most first-place votes. Let’s count the first-place votes for each candidate:

- **DD**: 45 votes from the first group
- **CC**: 34 votes from the second group
- **EE**: 17 votes from the third group

The candidate with the most votes is **DD** with 45 votes. However, for a majority, a candidate needs more than 50% of the votes. Since 50% of 96 is 48, **DD** did not receive the majority (only 45 votes).

Answer: **C. Candidate DD is the plurality winner, but not by a majority because DD received less than 50% of the votes.**

### Part 3
**c. Find the winner by a runoff of the top two candidates.**

In a runoff, the top two candidates (those with the most first-place votes) compete against each other. The top two are:

- **DD** (45 votes)
- **CC** (34 votes)

To conduct the runoff, we need to examine how the supporters of **EE** (the third candidate) would vote between **DD** and **CC**. Based on the preferences:

- **EE** voters (17 votes) prefer **DD** as their second choice (according to the given schedule).

Thus, **DD** would receive the 17 votes from **EE** voters in addition to its original 45 votes, giving **DD** a total of 62 votes.

**CC** remains with its original 34 votes.

Therefore, the winner of the top-two runoff is **DD**.

Answer: **A. Candidate DD is the winner of the top-two runoff.**

### Part 4
**d. Find the winner by a sequential runoff.**

In a sequential runoff, the candidate with the fewest first-place votes is eliminated in each round, and their votes are redistributed based on voter preferences until one candidate remains. Let’s follow this process:

1. **First round**: The first-place votes are:
   - **DD**: 45 votes
   - **CC**: 34 votes
   - **EE**: 17 votes

   Since **EE** has the fewest votes (17), they are eliminated. The 17 votes for **EE** will be redistributed to their second choice, which is **DD**.

2. **Second round**: After redistributing **EE**'s votes, we now have:
   - **DD**: 45 + 17 = 62 votes
   - **CC**: 34 votes

   **DD** has the majority with 62 votes, so **DD** wins the sequential runoff.

Answer: **A. The winner by sequential runoff is DD.**

---

Do you need further details or have any questions? Here are 5 related questions to expand on the topic:

1. What is the difference between a plurality and a majority in voting systems?
2. How does a sequential runoff differ from an instant runoff?
3. Can a candidate with fewer first-round votes still win in a sequential runoff?
4. What are some potential advantages of using a runoff system over a plurality system?
5. How might voter preferences affect the outcome of different runoff methods?

**Tip**: In sequential runoffs, the candidate eliminated first can significantly impact the final outcome, especially if their voters’ second preferences are highly polarized.

For the following preference schedule, answer questions a through g: First: DD, CC, EE; Second: EE, AA, BB; Third: BB, DD, CC; Fourth: CC, BB, AA; Fifth: AA, EE, DD. Number of voters: 45, 34, 17. Part 1: How many votes were cast? Part 2: Find the plurality winner. Did the plurality winner also receive a majority? Explain. Part 3: Find the winner by a runoff of the top two candidates. Part 4: Find the winner by a sequential runoff.

Analyze a preference schedule with 96 voters to determine the plurality winner and find the winner using runoff methods. Learn about plurality rules and how sequential runoffs work in voting theory. Suitable for Grades 10-12.

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