 A portion went is the problem of a fair division of a discrete resource. But what does that really mean? What if we try to quantify unfairness? So suppose a state has population M and A representatives. Then the quotient M divided by A represents a number of persons represented by each congressman. And this suggests that one measure of fairness is the difference between the persons per congressman in each state. In a perfect apportionment there would be no difference. So the difference is a measure of the malapportionment. One way to measure the difference between two quantities, P and Q, is to find the absolute difference, the absolute value of the difference between the quantities. And this is usually what people mean when they say difference. For example, let's say we have two schools with 175 and 95 students respectively who are currently assigned five and two buses, respectively. What's the absolute difference in students per bus? So we find the number of students per bus at each school. At school one there's 175 students with five buses, and so there's 175 divided by five or 35 students per bus. Meanwhile school two with 95 students among two buses that has 47.5 students per bus. And so the absolute difference works out to be 12.5 students per bus. Now let's consider transferring a bus from one school to the other. The key question we want to ask is would the transfer reduce the malapportionment? So let's consider that. We'll compute the absolute difference under different scenarios. So in our current scenario with five and two buses, the absolute difference is 12.5 students per bus. So if we switch a bus to the larger school, school one now has six buses, so the number of students per bus will be, meanwhile school two now has one bus, so the number of students per bus will be, and we find the absolute difference in students per bus will be, what if we switch a bus to the smaller school? So now school one still has 175 students, but now they have four buses, and so their students per bus will be, the other school now has three buses, and so their students per bus will be, and so the absolute difference will be. And so now let's consider our three scenarios. Currently we have an absolute difference of 12.5. If we move a bus to the larger school, that actually increases the absolute difference and makes the malapportionment worse. But if we switch the bus to the smaller school, that decreases the absolute difference and makes the malapportionment less. And so if we switch the apportionment to four and three buses, the malapportionment would be reduced. Because switching one bus from the larger school to the smaller school would reduce the malapportionment, we should do it, or should we? And the problem is it depends on how we're measuring malapportionment. And the other way is we could look at the relative difference. This is the absolute difference divided by one of the quantities. So again, let's consider our scenario. And this time let's find the relative difference in students per bus. And we'll compare it to the smaller of the values. So we've already computed the students per bus and the absolute difference. And relative difference, we're going to compute relative to the smaller of the values, and that's this 35 students per bus. And so the smaller of the numbers is 35. And so our relative difference is going to be the absolute difference, 12.5 divided by the smaller of the numbers, 35, or? And now we'll ask the same question. If we could transfer buses, should we? So again, we'll compute the relative difference under different scenarios. So if we switch a bus to a larger school, we've already calculated the students per bus and the absolute difference. And the relative difference will be the absolute difference divided by the smaller of the quantities, 29.167. If we switch a bus to the smaller school, we've already computed the students per bus and the absolute difference. So the relative difference will be the absolute difference divided by the smaller of the quantities. And now we can make an informed decision. So in the current scenario, the relative difference is 0.357. And if we change the number of buses, that relative difference only goes up. And that means the current apportionment has the least malapportionment. And notice that if we use absolute difference as our measure of malapportionment, we should assign four buses to the school with 175 students and three to the school with 95. But if we use relative difference as our measure, then we should keep the current five to two assignment. And the important thing here is that mathematics gives us the objective facts on which to base our decision. But we have to make a choice of whether it's better to have a lower absolute difference or a lower relative difference. And that's not a question that mathematics can decide. What mathematics can do is address the real problem. Whichever method of malapportionment we decide to use, whether relative or absolute, we still have to solve the apportionment problem. Now, imagine we've apportioned all of our resources. The question you've got to ask yourself is, if we could switch a resource from one recipient to another, that would change the malapportionment. And so it seems that we want to apportion our resources so no reassignment would reduce the malapportionment. And as that happens, we have the best possible apportionment. But this seems to make it a lot more difficult. Do we have to consider every possible apportionment and select the best one? Fortunately, we don't, and we'll take a look at that next.