Chuck and Mike need to divide a couch and a pool table. Using the Method of Sealed Bids, they place the following bids on the items.

Chuck bids $536 on the couch, and $98 on the pool table.

Mike bids $111 on the couch, and $231 on the pool table.

Chuck will get the couch. Mike will take the pool table and will also get a little bit of money from Chuck. To make the division fair, how much money will Chuck have to give to Mike?

To solve this problem using the **Method of Sealed Bids**, we proceed step by step to ensure a fair division of the items between Chuck and Mike.

### Step 1: Calculate each person's fair share.

Each person’s fair share is based on their total bid. This is done by summing each person's bids and dividing by 2 (since there are two people).

- Chuck's total bid: $ 536 \, (\text{couch}) + 98 \, (\text{pool table}) = 634 $
- Mike's total bid: $ 111 \, (\text{couch}) + 231 \, (\text{pool table}) = 342 $

Now, calculate each person's fair share (half of their total bid):
- Chuck's fair share: $ \frac{634}{2} = 317 $
- Mike's fair share: $ \frac{342}{2} = 171 $

### Step 2: Assign the items based on bids.

Chuck wins the couch because he placed the higher bid on it ($536 vs. $111).
Mike wins the pool table because he placed the higher bid on it ($231 vs. $98).

### Step 3: Calculate the difference between the value of items received and each person's fair share.

- Chuck receives the couch, which he values at $536.
  - Chuck's fair share is $317, so he gets more than his fair share.
  - Difference: $ 536 - 317 = 219 $
  - Chuck owes $219 to balance the division.

- Mike receives the pool table, which he values at $231.
  - Mike's fair share is $171, so he gets more than his fair share.
  - Difference: $ 231 - 171 = 60 $
  - Mike owes nothing to Chuck.

### Step 4: Balance the transaction.

Since Chuck owes $219 and Mike owes nothing, **Chuck must give Mike $219** to make the division fair.

### Final Answer:
Chuck will give Mike **$219**.

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Do you need any more details or have any questions?

Here are some related questions to explore further:
1. How would the outcome change if Mike had bid more on the couch than Chuck?
2. What would happen if both people placed equal bids on an item?
3. How can the Method of Sealed Bids be adapted if more than two people are involved?
4. What if instead of two items, there were three or more items to divide?
5. How can we ensure fairness if the total value of items is uneven between the participants?

**Tip:** In the Method of Sealed Bids, each participant's fair share is the key to ensuring the division is equitable based on their subjective valuations.

Chuck and Mike need to divide a couch and a pool table. Using the Method of Sealed Bids, they place the following bids on the items. Chuck bids $536 on the couch, and $98 on the pool table. Mike bids $111 on the couch, and $231 on the pool table. Chuck will get the couch. Mike will take the pool table and will also get a little bit of money from Chuck. To make the division fair, how much money will Chuck have to give to Mike?

Explore the Method of Sealed Bids to fairly divide a couch and a pool table between Chuck and Mike. Learn how to calculate fair shares based on bids, and how much money Chuck needs to give Mike to balance the division. Suitable for Grades 9-12, this problem involves fair division and bidding systems.

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