 In the United States, the number of congresspersons assigned to a state is based on the state's population. Since you can't have a fraction of a person, this is an apportionment problem. Alexander Hamilton suggested the following approach. We'll compute the standard quota. Then round down, but keep track of the fractional parts. This will always assign fewer items than are available, and so the remaining items will be assigned to the recipients with the largest fractional parts. So let's use Hamilton's method to apportion 5 computers to classrooms of size 31, 72, and 43 students. So first we find there are 146 students, and since there are 5 computers to apportion, there should be 29.2 students per computer. This is the standard divisor, which we'll use in the Hamilton apportionment. So using a divisor of 29.2, we can then compute the standard quotas. And so we can divide each of our class sizes by 29.2. And we round these numbers down to get the standard quota, but we do want to record the fractional parts. Now at this point, we've only assigned 1, plus 2, plus 1, 4 computers, and since we have 5 available, we can give out one more. And so we look at our fractional parts, and since the class of size 43 has the largest fractional part, it gets the last computer. And so our final apportionment will be 1, 2, 2. Thomas Jefferson argued against Hamilton's method, calling it too complicated. However, Congress passed an apportionment based on Hamilton's method. But Washington vetoed it, the first ever veto in American history. But the veto meant that a new solution to the apportionment problem had to be found. To handle the apportionment problem, Thomas Jefferson suggested we choose a divisor and compute the corresponding quotas, and again use the rounded down values as the assignment. And if some resources haven't been allocated, use a new divisor and repeat. So let's see what happens if we apply Jefferson's method. So if we use our standard quota, 29.2, as our divisor, then we only allocated 1, plus 2, plus 1, that's 4 computers. And since we have 5 to allocate, we'll use a smaller divisor. Now the easiest way to do this is to use trial and error. We'll pick a divisor and see what our apportionment is. In the 18th century, they would have used a computer. Of course, in the 18th century, a computer was actually a person trained to do computations. So if we use the divisor of D equals 20, we find our quotas by dividing the class sizes by 20 and rounding down. And this would allocate 1, 3, and 2 computers. But this requires 6 computers, and we only have 5. So our divisor is too small, so we'll choose another. If we use a divisor of 25, we find our quotas by dividing the class sizes by 25 and rounding down. And this would have us assigning 1, plus 2, plus 1, 4 computers, which means that we have one more computer left to assign. And so our divisor is now too big, so we'll choose another. And finally, if we use a divisor of 24, we find our quotas and round them down. And now we've assigned our 1, plus 3, plus 1, 5 computers, which is the amount we have available. And so our final apportionment will be 1, 3, 1. It's worth noting that the two apportionment methods actually give us different apportionments, which means it is important to decide how we want to proceed. And this choice was left to George Washington. Thomas Jefferson and George Washington both objected to Hamilton's approach. Jefferson even said that it depended on a difficult and in-obvious doctrine of fractions. In other words, they thought it was too complicated. Now remember Washington was a surveyor, and Jefferson read widely on mathematics, so this probably wasn't the real reason. And in fact, Jefferson's apportionment gave Virginia more influence in Congress than Hamilton's plan did, and Jefferson and Washington were both Virginians. Jefferson's method would be used for apportionments from 1792 until 1842.