Not sure which "assumption" you are referring to. In fact, I am just taking the information from the question and re-organising it, so not really any assumptions are made.
Let me try using including a model to help demonstrate how we start this question. The first picture shows the two IF situations (Cases) where the grey shows the $38 spent by both people. But they do not spend the $38 at the same time. They spend it in two different Cases - this point becomes important.
The second picture introduces a "midpoint" which is the actual amount they both have. (It's important to note that both cases say IF which means that it doesn't really happen thus the midpoint reflects the actual amount they have.) When using the Total Changed concept, we need a "starting" and "ending" ratio / situation. But we do not have one for this question ... mmm ... However, the question states two Cases, so we can create a "starting" position using Case 1 and an "Ending" position using Case 2 (because these Cases do not happen at the same time).
Using a similar example: Imagine that you and I have $10 each. Case 1) IF you spend $2, you have $8 and I have $10. Case 2) IF I spend $2, you have $10 and I have $8. Written as ratios: We have a ratio of 10:10. Case 1) the ratio becomes 8:10. Case 2) the ratio becomes 10:8. Now, if I want to show the change between the two Cases, like following a path from Case 1 to Case 2 you would have: Case 1) 8: 10 --> Case 2) 10:8, so the change is +2 : -2. So even though both Cases are spending $2 when we following the change from Case 1 to Case 2 it is actually +2 : -2. Yet from the midpoint, the actual amount we have, it is -2:0 for Case 1 and 0:-2 for Case 2.
and "move toward" the other end. I have arbitrarily chosen where Keith spent $38 (case 1). From this point it appears as though Keith is + $38 to get to the mid point and then -$38 for what Sean spends (after case 2).
