Anne​, Bess​, and Cindy are roommates planning to move out of their apartment. They identify five major chores that need to be done before moving out and decide to use the method of sealed bids to reverse auction the chores. The accompanying table shows the bids that each roommate made for each chore. Describe the final outcome of the division​ (which chores are done by each roommate and how much each roommate pays or gets​ paid.)
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Part 1
Describe Anne​'s part of the agreement. Select the correct choice below and fill in the answer​ box(es) within your choice.
​(Round to the nearest cent as​ needed.)
A.
Anne does​ chore(s) 
  
enter your response here and pays ​$
  
enter your response here.
​(Use a comma to separate answers as​ needed.)
B.
Anne does no chores and pays ​$
  
enter your response here.
C.
Anne does​ chore(s) 
  
enter your response here and gets ​$
  
enter your response here.
​(Use a comma to separate answers as​ needed.)
D.
Anne does no chores and gets ​$
  
enter your response here.
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Bid Table
Anne
Bess
Cindy
 Chore 1
​$33
​$23
​$43
 Chore 2
​$13
​$53
​$25
 Chore 3
​$23
​$33
​$18
 Chore 4
​$23
​$33
​$13
 Chore 5
​$43
​$23
​$18

### Method of Sealed Bids (Reverse Auction)

In a reverse auction using the sealed bids method, the following steps are taken to divide the chores:

1. **Each person submits a bid** for how much they are willing to be paid to do each chore.
2. **Chores are assigned to the person with the lowest bid**, as the goal is to minimize the total cost.
3. **Calculate the fair share**: Each person's total bid for all the chores is summed, and the average of these total bids is used to determine how much each should "receive" to balance out the costs.
4. **Compensation or payment adjustments**: People who perform chores will either be paid or owe money based on how their work compares to their fair share.

### Step-by-Step Solution for Anne:

#### Step 1: Assign the Chores Based on the Bids
Let's assign the chores to the person with the lowest bid:

- **Chore 1**: Anne ($33), Bess ($23), Cindy ($43) → **Bess** wins with $23.
- **Chore 2**: Anne ($13), Bess ($53), Cindy ($25) → **Anne** wins with $13.
- **Chore 3**: Anne ($23), Bess ($33), Cindy ($18) → **Cindy** wins with $18.
- **Chore 4**: Anne ($23), Bess ($33), Cindy ($13) → **Cindy** wins with $13.
- **Chore 5**: Anne ($43), Bess ($23), Cindy ($18) → **Cindy** wins with $18.

So, the assignments are:
- **Bess** does Chore 1.
- **Anne** does Chore 2.
- **Cindy** does Chores 3, 4, and 5.

#### Step 2: Calculate Fair Share
Each roommate's fair share is the average of the total bids they made for all the chores.

- **Anne's total bid**: $33 + $13 + $23 + $23 + $43 = $135
- **Bess's total bid**: $23 + $53 + $33 + $33 + $23 = $165
- **Cindy's total bid**: $43 + $25 + $18 + $13 + $18 = $117

The total bid for all chores = $135 + $165 + $117 = $417.

The fair share for each roommate is the total divided by 3:
- Fair share = $417 / 3 = **$139**.

#### Step 3: Calculate Anne's Payment/Compensation
- Anne's **fair share** is $139.
- Anne only does Chore 2, for which she bid **$13**.

Since Anne does less work than her fair share, she needs to **pay the difference**:
- Anne needs to pay = $139 - $13 = **$126**.

### Final Outcome for Anne:
- **Anne does Chore 2**.
- **Anne pays $126**.

### Answer:
Option A: Anne does Chore 2 and pays $126.

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Would you like more details or have any questions about the calculations? Here are 5 related questions to think about:

1. How is the total bid for each person used in calculating the fair share?
2. How does the method of sealed bids ensure a fair distribution of work and payments?
3. What would happen if one person did all the chores?
4. How would the result change if one person had a very low bid for every chore?
5. What are other methods for dividing chores fairly?

**Tip**: When calculating fair shares, make sure to check the total sum of bids first, as this ensures each person's contribution is balanced based on their bids.

Anne​, Bess​, and Cindy are roommates planning to move out of their apartment. They identify five major chores that need to be done before moving out and decide to use the method of sealed bids to reverse auction the chores. The accompanying table shows the bids that each roommate made for each chore. Describe the final outcome of the division​ (which chores are done by each roommate and how much each roommate pays or gets​ paid).

Explore how Anne, Bess, and Cindy use the sealed bids reverse auction method to fairly divide five major chores before moving out. This solution demonstrates key concepts in game theory and fair division, including calculating fair shares and determining payments or compensation for each participant. Ideal for Grades 10-12 students.

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