In Example 4.9, three brothers used the method of points to divide a collection of chore...

In Example 4.9, three brothers used the method of points to divide a collection of chores. The chores, mowing the grass, weeding the garden, and vacuuming the house, were considered to be discrete, indivisible items. The result of the division was not proportional, but it was the best under the circumstances. Consider chores you do every day. a. Make a list of as many chores as you can think of that must be treated as discrete and chores that can be treated as continuous. b. Reconsider Example 4.9. Suppose Jeremy is sick and cannot do any of the chores, so Kenny and Luke must divide the chores between the two of them. Use the points provided by Kenny and Luke in Example 4.9, treat the chores as continuous, and apply the adjusted-winner procedure. Which boy will do each chore, and which chore(s) must be shared? c. In your opinion, is the division obtained by the adjusted-winner procedure in part (b) sensible? Explain.

Example 4.9:

It is Saturday and three brothers, Jeremy, Kenny, and Luke, have chores to do before they can do anything fun. Their mother says that they must mow the grass, weed the garden, and vacuum the house, but she leaves it up to them to decide who does what chore. The boys each assign points to the chores based on which one they would rather do if they must do one (Table 4.22).

Using the method of points, determine which task each brother should do.

SOLUTION

STEP 1: Assign points.

The results of step 1 are shown in Table 4.22. It appears that none of the boys prefers to vacuum the house.

STEP 2: List possible arrangements.

Table 4.23 lists all six possible assignments of chores and the associated points.

STEP 3: Consider the smallest numbers of assigned points.

The smallest point values appear in column 4. In this case, exactly two arrangements have a maximum smallest point value, 25. We keep the first and third arrangements, discard the other arrangements, and move on to step 4.

STEP 4: Consider the middle numbers of assigned points.

We record the middle number in the two remaining arrangements, as shown in Table 4.24.