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Chapter 5: Problem 2

Ahmed and Tiana buy a cake for \(\$ 14\) that is half chocolate and half vanilla. They cut the cake into 8 slices. If Ahmed likes chocolate four times as much as vanilla, what is the value of a slice that is half chocolate, half vanilla?

Short Answer

Expert verified
The value of a slice that is half chocolate, half vanilla, is $1.75.

Step by step solution

01

Understanding the Problem

Ahmed and Tiana bought a cake for $14, and they cut it into 8 slices. Each slice is worth $14 divided by 8, or $1.75. The cake is half chocolate and half vanilla.
02

Calculate Relative Liking

Ahmed likes chocolate four times as much as vanilla. If we denote his liking for vanilla as 1 unit, then his liking for chocolate is 4 units, making a total of 5 units (1 vanilla + 4 chocolate).
03

Distribute Liking Points to the Cake

Since the cake is half chocolate and half vanilla, the value contributed by Ahmed's liking for the chocolate part is 4 units, and the vanilla part is 1 unit. This makes a total of 5 units of liking-value for the entire slice.
04

Calculate Value of Each Part

Distribute the $1.75 equally to the 5 liking units determined. Each unit of liking represents $0.35 (since $1.75 divided by 5 is $0.35). Thus, the chocolate part, with 4 units, is valued at $1.40, and the vanilla part, with 1 unit, is valued at $0.35.
05

Double-check Understanding

For a slice that is half chocolate and half vanilla, Ahmed's enlarged preference for chocolate results in that component effectively comprising most of its subjective 'value' given his preferences. The total computed price matches the actual slice price of $1.75.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Relative Liking
In problems involving subjective preferences, like the one between Ahmed and Tiana, understanding relative liking is crucial. When we say "relative liking," we refer to how much more one person prefers one thing over another. In this exercise, Ahmed's liking ratio shows that he values chocolate four times more than vanilla. This kind of preferences are often expressed in ratios or multipliers.
For instance, having Ahmed like chocolate four times more than vanilla sets up a scenario where his total liking is split into five 'liking units': one for vanilla and four for chocolate. Thus, relative liking allows us to quantify and balance subjective preferences in practical, mathematical terms when making decisions, such as dividing a cake fairly based on likes.
Value Calculation
Calculating the value of an object based on preferences involves understanding how these preferences impact the perceived worth. In this case, the cake was initially divided into slices each valued at $1.75, calculated by dividing the total cake cost by the number of slices.
Then, because Ahmed's preference is for chocolate, we reassigned value according to how much more he likes chocolate than vanilla. By converting preferences into units (1 for vanilla, 4 for chocolate), each preference unit represents $0.35. Hence, the slice is split into a chocolate part valued at $1.40 and a vanilla part valued at $0.35. The concept of value calculation here helps translate subjective preferences into monetary equivalents, ensuring that everyone feels their preferences are represented in proportion.
Cake Slice Division
Division of the cake slices isn't just about physical cuts; it's about an equitable distribution considering personal preferences. With Ahmed and Tiana, they needed to ensure that the division of the cake (half chocolate and half vanilla) recognizes Ahmed's particular fondness for chocolate.
By calculating each slice's worth in terms of their liking, the slices don't only get a physical dimension but also a preference-weighted one. This approach allows for more nuanced distributions, wherein each part of something can be evaluated based on its appeal to different people, not just its physical size or cost.
Distribution of Preferences
The distribution of preferences involves allocating parts of something鈥攊n this case, a cake鈥攂ased on how much different individuals value each part. In Ahmed's situation, his preferences significantly skew the value of each slice of cake toward the chocolate half.
This exercise shows that when preferences are distributed, even simple things like cake slices can vary greatly in their perceived value. By understanding and calculating these preferences, individuals can fairly share resources in a way that reflects their unique tastes, ensuring satisfaction and fairness. This method is vital whenever distribution involves varying degrees of desire or utility among participants.

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Most popular questions from this chapter

Three heirs \((\mathrm{A}, \mathrm{B}, \mathrm{C})\) must fairly divide an estate consisting of three items - a house, a car, and a coin collection - using the method of sealed bids. The players' bids (in dollars) are: \begin{tabular}{|c|r|r|r|} \hline & \multicolumn{1}{|c|} {\(\mathbf{A}\)} & \multicolumn{1}{|c|} {\(\mathbf{B}\)} & \multicolumn{1}{|c|} {\(\mathbf{C}\)} \\ \hline House & 180,000 & 210,000 & 220,000 \\ \hline Car & 12,000 & 10,000 & 8,000 \\ \hline Coins & 3,000 & 6,000 & 2,000 \\ \hline \end{tabular} a. What is A's fair share? b. Find the initial allocation. c. Find the final allocation.

All the problems we have looked at in this chapter have assumed that all participants receive an equal share of what is being divided. Often, this does not occur in real life. Suppose Fred and Maria are going to divide a cake using the divider-chooser method. However, Fred is only entitled to \(30 \%\) of the cake, and Maria is entitled to \(70 \%\) of the cake (maybe it was a \(\$ 10\) cake, and Fred put in \(\$ 3\) and Maria put in \(\$ 7\) ). Adapt the divider- choose method to allow them to divide the cake fairly. Assume (as we have throughout this chapter) that different parts of the cake may have different values to Fred and Maria, and that they don't communicate their preferences/values with each other. You goal is to come up with a method of fair division, meaning that although the participants may not receive equal shares, they should be guaranteed their fair share. Your method needs to be designed so that each person will always be guaranteed a share that they value as being worth at least as much as they're entitled to.

This question explores how bidding dishonestly can end up hurting the cheater. Four partners are dividing a million-dollar property using the lone-divider method. Using a map, Danny divides the property into four parcels \(\mathrm{s}_{1}, \mathrm{~s}_{2}, \mathrm{~s}_{3},\) and \(\mathrm{s}_{4} .\) The following table shows the value of the four parcels in the eyes of each partner (in thousands of dollars): \begin{tabular}{|c|c|c|c|c|} \hline & s1 & s2 & s3 & s 4 \\ \hline Danny & \(\$ 250\) & \(\$ 250\) & \(\$ 250\) & \(\$ 250\) \\ \hline Brianna & \(\$ 460\) & \(\$ 180\) & \(\$ 200\) & \(\$ 160\) \\ \hline Carlos & \(\$ 260\) & \(\$ 310\) & \(\$ 220\) & \(\$ 210\) \\ \hline Greedy & \(\$ 330\) & \(\$ 300\) & \(\$ 270\) & \(\$ 100\) \\ \hline \end{tabular} a. Assuming all players bid honestly, which piece will Greedy receive? b. Assume Brianna and Carlos bid honestly, but Greedy decides to bid only for s1, figuring that doing so will get him \(\mathrm{s} 1\). In this case there is a standoff between Brianna and Greedy. Since Danny and Carlos are not part of the standoff, they can receive their fair shares. Suppose Danny gets \(s 3\) and Carlos gets \(s 2,\) and the remaining pieces are put back together and Brianna and Greedy will split them using the basic divider-chooser method. If Greedy gets selected to be the divider, what will be the value of the piece he receives? c. Extension: Create a Sealed Bids scenario that shows that sometimes a player can successfully cheat and increase the value they receive by increasing their bid on an item, but if they increase it too much, they could end up receiving less than their fair share.

Explain why divider-chooser method with two players will always result in an envyfree division.

In \(1974,\) the United States and Panama negotiated over US involvement and interests in the Panama Canal. Suppose that these were the issues and point values assigned by each side \(^{2}\). Apply the Adjusted Winner method. $$ \begin{array}{|l|l|l|} \hline & \text { United States } & \text { Panama } \\ \hline \text { US defense rights } & 22 & 9 \\ \hline \text { Use rights } & 22 & 15 \\ \hline \text { Land and water } & 15 & 15 \\ \hline \text { Expansion rights } & 14 & 3 \\ \hline \text { Duration } & 11 & 15 \\ \hline \text { Expansion routes } & 6 & 5 \\ \hline \text { Jurisdiction } & 2 & 7 \\ \hline \text { US military rights } & 2 & 7 \\ \hline \text { Defense role of Panama } & 2 & 13 \\ \hline \end{array} $$
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