Chapter 3, Problem 34E

Exercises 33 and 34 refer to the following situation: Jackie, Karla, and Lori are planning to divide the half vegetarian—half meatball foot-long sub sandwich shown in Fig. 3-22 among themselves using the lone- divider method. Jackie likes the meatball and vegetarian parts equally well; Karla is a strict vegetarian and does not eat meat at all; Lori likes the meatball part twice as much as the vegetarian part. (Assume that when the sandwich is cut, the cuts are always made perpendicular to the length of the sandwich. You can describe different shares of the sandwich using the ruler and interval notation—for example, $\left[0,6\right]$ describes the vegetarian half. $\left[6,8\right]$ describes one-third of the meatball half, etc.).

Figure 3-22

Suppose that Lori ends up being the divider.

a. Describe how Lori should cut the sandwich into three shares. Label the three shares $s_1$ for the leftmost piece, $s_2$ for the middle piece, and $s_3$ for the rightmost piece. Use the ruler and interval notation to describe the three shares. (Assume that Lori knows nothing about Karla and Jackie’s likes and dislikes.)

b. Which of the three shares are fair shares to Jackie?

c. Which of the three shares are fair shares to Karla?

d. Suppose that Lori gets $s_3$. Describe how to proceed to find a fair division of the sandwich.