EXAMPLE 10 Sharing Diamonds by the Jefferson Method

Winnie, Louise, and Tim have bought 18, 4, and 1 lottery tickets, respectively, and have won 100 diamonds. The Jefferson method can be used to share the diamonds. The house size is 100 diamonds, and the total population is the 23 lottery tickets. The standard divisor is 0.23. Table 14.11 shows what happens with a few trial divisors, starting with 0.22, which turns out to be too small, as it apportions a total of 103 diamonds. We gradually increase the divisor until 100 are apportioned. For each trial divisor, there are two columns: the apportionment quotients on the left, and the same quotients rounded down on the right. The sums of the rounded apportionment quotients are shown, and we see that the divisor 0.227 gets the correct house size, 100. (There is no need to add the apportionment quotients, as their sums play no role in determining the apportionment.)

587

The apportionment is what Winnie wanted: 79 diamonds for her, 17 for Louise, and 4 for Tim.