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To determine the chore assignments and calculate the surplus cash from the first settlement, we’ll analyze the bids given for each chore and assign them to the roommate who bid the lowest amount.

Here's the breakdown:

### Bids Table
| Chore | Roommate A | Roommate B | Roommate C |
|-------|------------|------------|------------|
| 1     | $85        | $75        | $70        |
| 2     | $125       | $115       | $120       |
| 3     | $80        | $75        | $65        |
| 4     | $100       | $95        | $111       |

### Step 1: Assigning Chores
1. **Chore 1**: Lowest bid is $70 by **Roommate C**.
2. **Chore 2**: Lowest bid is $115 by **Roommate B**.
3. **Chore 3**: Lowest bid is $65 by **Roommate C**.
4. **Chore 4**: Lowest bid is $95 by **Roommate B**.

So, the assignments are:
- **Chore 1**: Roommate C
- **Chore 2**: Roommate B
- **Chore 3**: Roommate C
- **Chore 4**: Roommate B

### Step 2: Calculating Surplus Cash
The surplus cash is the difference between the total cost of all chores (using the highest bids) and the actual amount paid (using the lowest bids).

1. **Total Cost (Using Bids)**:
   - **Highest bids** for each chore:
     - Chore 1: $85
     - Chore 2: $125
     - Chore 3: $80
     - Chore 4: $111
   - Total cost: $85 + 125 + 80 + 111 = 401$

2. **Actual Payment (Lowest Bids)**:
   - **Lowest bids** assigned:
     - Chore 1 (C): $70
     - Chore 2 (B): $115
     - Chore 3 (C): $65
     - Chore 4 (B): $95
   - Total payment: $70 + 115 + 65 + 95 = 345$

3. **Surplus Calculation**:
   - Surplus cash = Total cost - Actual payment
   - Surplus cash = $401 - 345 = 56$

### Final Answer
The first settlement results in a surplus of **$56**.

Would you like more details, or do you have any questions?

Here are some related questions for further understanding:
1. How does the reverse auction method benefit each roommate?
2. What would happen if two roommates bid the same amount on a chore?
3. Can the surplus be divided among roommates? If so, how?
4. How does this method ensure fairness in chore assignments?
5. What other methods could be used to assign chores equitably?

**Tip:** In reverse auctions, assigning tasks to the lowest bidder often results in cost savings or surplus, as demonstrated here.

Three roommates are planning to move out of their apartment. They identify four 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 table shows the bids that each roommate made for each chore. Describe the final outcome of the division. Determine which roommate receives each chore from the division and calculate the surplus cash from the first settlement.

Explore a step-by-step solution for dividing chores among roommates using a reverse auction method with sealed bids. Learn how to assign chores based on the lowest bids and calculate the surplus cash from the settlement, resulting in a $56 surplus. Ideal for high school students studying practical applications of cost calculation and auctions.

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