Angela, Boris, and Carlos are dividing the​ vanilla-strawberry cake shown using the​ lone-chooser method. The table to the right shows the value that each player places on each part of the cake. Suppose that Angela and Boris are dividers and Carlos is the chooser. In the first​ division, Angela cuts the cake into two​ shares, s 1s1 and s 2s2. Boris picks the share he likes​ best, and Angela gets the other share. In the second​ division, Angela subdivides her share of cake into three pieces and Boris subdivides his share of cake into three pieces. Answer parts ​(a) through ​(d) below.
Strawberry
Vanilla
 
Angela
​$2727
​$99
Boris
​$4545
​$55
Carlos
​$1212
​$2424

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Part 1
​(a) Describe which share ​(s 1s1 or s 2s2​) Boris picks and how he might subdivide it. The shares are shown. Share s 1s1 is a 120 degrees120° ​strawberry-only piece and share s 2s2 is a 60 degrees60° strawberrydash–180 degrees180° vanilla piece. Which share does Boris​ pick?
s 2
s 2s2
s 1
s 1s1

To solve this, we need to analyze how Boris evaluates the two shares of the cake based on his preferences.

The two shares are:

- **Share s₁:** A 120° strawberry-only piece.
- **Share s₂:** A 60° strawberry + 180° vanilla piece.

According to the table, here are the values that Boris places on each flavor:

- **Strawberry:** \$45
- **Vanilla:** \$5

### Step 1: Calculate the value Boris assigns to each share.

- **For Share s₁** (120° of strawberry):
  - Since Boris values strawberry at \$45, we need to find what fraction of the whole strawberry this is:
  - The total cake is 360°, and 120° represents one-third of the whole cake. So, the value of s₁ to Boris would be:
    \[
    \text{Value of s₁ to Boris} = \frac{120^\circ}{360^\circ} \times 45 =  \frac{1}{3} \times 45 = 15
    \]

- **For Share s₂** (60° of strawberry and 180° of vanilla):
  - **Strawberry part**: The 60° strawberry part represents one-sixth of the whole cake:
    \[
    \text{Value of 60° of strawberry} = \frac{60^\circ}{360^\circ} \times 45 = \frac{1}{6} \times 45 = 7.5
    \]
  - **Vanilla part**: The 180° vanilla part represents half of the whole cake:
    \[
    \text{Value of 180° of vanilla} = \frac{180^\circ}{360^\circ} \times 5 = \frac{1}{2} \times 5 = 2.5
    \]
  - **Total value of s₂ for Boris**:
    \[
    \text{Total value of s₂ to Boris} = 7.5 \ (\text{strawberry}) + 2.5 \ (\text{vanilla}) = 10
    \]

### Step 2: Determine which share Boris picks.

- **Value of s₁ to Boris** = 15
- **Value of s₂ to Boris** = 10

Since Boris values **s₁** more highly (\$15) than **s₂** (\$10), he will pick **s₁** (the 120° strawberry-only piece).

### Step 3: How Boris might subdivide his share.

Since Boris values strawberry highly (\$45), he would want to divide **s₁** into three pieces that maintain this value distribution. He might divide **s₁** based on how he values the strawberry flavor, likely attempting to create three relatively equal shares in terms of the strawberry portion.

### Final Answer:

- Boris picks **share s₁** (the 120° strawberry-only piece).
- He subdivides it into three parts based on the value he places on strawberry to ensure fairness.

Would you like further details on Boris’s subdivision strategy or how Angela might proceed? 

### Related Questions:
1. What method would Angela use to divide her share to ensure fairness?
2. How does Carlos determine which subdivision piece to pick?
3. What are some potential challenges in dividing mixed-flavor cakes using the lone-chooser method?
4. Could Boris choose s₂ under different value assumptions, and how would that affect the process?
5. How does the lone-chooser method ensure that no player feels envious?

### Tip:
Always calculate the value of each part relative to the total, especially when dealing with unequal divisions like this one.

Angela, Boris, and Carlos are dividing the vanilla-strawberry cake shown using the lone-chooser method. The table to the right shows the value that each player places on each part of the cake. Suppose that Angela and Boris are dividers and Carlos is the chooser. In the first division, Angela cuts the cake into two shares, s1 and s2. Boris picks the share he likes best, and Angela gets the other share. In the second division, Angela subdivides her share of cake into three pieces and Boris subdivides his share of cake into three pieces. Answer parts (a) through (d) below.

Explore how to fairly divide a vanilla-strawberry cake using the lone-chooser method. This detailed solution involves evaluating the value each participant places on different portions of the cake, using fractions and the lone-chooser method to ensure fairness. Suitable for high school students (Grades 9-12).

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