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Henry, Tom, and Fred are dividing among themselves a set of common assets equally owned
Chapter 3, Problem 1(choose chapter or problem)
Henry, Tom, and Fred are dividing among themselves a set of common assets equally owned by the three of them. The assets are divided into three shares \(\left(s_{1}, s_{2}, \text { and } s_{3}\right)\). Table 12 shows the values of the shares to each player expressed as a percent of the total value of the assets.
(a) Which of the shares are fair shares to Henry?
(b) Which of the shares are fair shares to Tom?
(c) Which of the shares are fair shares to Fred?
(d) Find all possible fair divisions of the assets using \(s_{1}, s_{2}\), and \(s_{3}\) as shares.
(e) Of the fair divisions found in (d), which one is the best?
\(\begin{array}{|l|c|c|c|} \hline & \boldsymbol{s}_{1} & \boldsymbol{s}_{\mathbf{2}} & \boldsymbol{s}_{\mathbf{3}} \\ \hline \text { Henry } & 25 \% & 40 \% & 35 \% \\ \hline \text { Tom } & 28 \% & 35 \% & 37 \% \\ \hline \text { Fred } & 33 \frac{1}{3} \% & 33 \frac{1}{3} \% & 33 \frac{1}{3} \% \\ \hline \end{array}\)
Table 12
Questions & Answers
QUESTION:
Henry, Tom, and Fred are dividing among themselves a set of common assets equally owned by the three of them. The assets are divided into three shares \(\left(s_{1}, s_{2}, \text { and } s_{3}\right)\). Table 12 shows the values of the shares to each player expressed as a percent of the total value of the assets.
(a) Which of the shares are fair shares to Henry?
(b) Which of the shares are fair shares to Tom?
(c) Which of the shares are fair shares to Fred?
(d) Find all possible fair divisions of the assets using \(s_{1}, s_{2}\), and \(s_{3}\) as shares.
(e) Of the fair divisions found in (d), which one is the best?
\(\begin{array}{|l|c|c|c|} \hline & \boldsymbol{s}_{1} & \boldsymbol{s}_{\mathbf{2}} & \boldsymbol{s}_{\mathbf{3}} \\ \hline \text { Henry } & 25 \% & 40 \% & 35 \% \\ \hline \text { Tom } & 28 \% & 35 \% & 37 \% \\ \hline \text { Fred } & 33 \frac{1}{3} \% & 33 \frac{1}{3} \% & 33 \frac{1}{3} \% \\ \hline \end{array}\)
Table 12
ANSWER:
Step 1 of 6
The table showing the values of assets is:
\(\begin{array}{|l|c|c|c|} \hline & \boldsymbol{s}_{1} & \boldsymbol{s}_{\mathbf{2}} & \boldsymbol{s}_{\mathbf{3}} \\ \hline \text { Henry } & 25 \% & 40 \% & 35 \% \\ \hline \text { Tom } & 28 \% & 35 \% & 37 \% \\ \hline \text { Fred } & 33 \frac{1}{3} \% & 33 \frac{1}{3} \% & 33 \frac{1}{3} \% \\ \hline \end{array}\)
Let the total value of assets be \(V=100\).
Let the value of \(s_1=v_1,\ s_2=v_2\text{ and }s_3=v_3\).
Thus the mean value of the assets will be:
\(\begin{array}{l} v_{m}=\frac{100}{3} \\ v_{m}=33 \frac{1}{3} \end{array}\)