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This chapter is about how to divide a set of objects into shares. Many objects, from pieces of hard candy to seats in the U.S. House of Representatives, cannot be distributed in fractional shares, and the apportionment problem occurs when the shares are rounded. The rounded shares may add up to more or less than the whole. We are interested in situations where it is imperative that the sum of the rounded shares is equal to the whole—even if the rounding of some shares must be adjusted to achieve that goal.

For example, we will consider the dilemma of a teacher who covers classes in geometry, precalculus, and calculus. She can teach five classes. There are 52 students enrolled in geometry, 33 in precalculus, and 15 in calculus. Do the math: With 100 students in all, the classes will average 20 students each. There ought to be 2.60 classes of geometry, 1.65 classes of precalculus, 0.75 classes of calculus—except it is ridiculous for her to teach fractional classes! How would you round these numbers to get a total of 5?

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The apportionment problem actually gets its name from Article 1, Section 2, of the U.S. Constitution, which, as modified by the Fourteenth Amendment, requires that “[representatives shall be apportioned among the several States according to their respective numbers, counting the whole number of persons in each State….” Representatives must be apportioned to states in whole numbers, which involves rounding. The framers of the Constitution gave no guidance in how to round. To be fair, an explicit method for apportionment must be chosen and agreed upon in advance by all parties. Many such methods have been proposed, and in the course of U.S. history, four methods (listed in Table 14.16 on page 597) have been employed to apportion seats in the House of Representatives. We will explore all of these methods.

You may wonder why we don’t just find the best apportionment method and dismiss the inferior ones. The answer is that there is no one “best” method. When apportioning classes to the teacher as in the situation we have alluded to, one would not choose the Hill-Huntington method—currently in use to apportion seats in Congress—because there would be no way to cancel a class with a very small enrollment. If only one student signed up for calculus, 60 students enrolled in geometry, and 39 students were in precalculus, the obvious solution would be 3 geometry classes, 2 precalculus classes, and no calculus class. The Hill-Huntington method would apportion two classes each for geometry and precalculus, and one class (with one student) for calculus.