# Math Matters: Why Do I Need To Know This?

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3 bottom, it would look like this, where you subtract the one in each case. And that looks a little neater in the formula I m getting ready to develop. (Figure 2) Figure 2, Segment 1 I m going to use this idea now to calculate the future value of that account. So we go back to this, where I ve got all the different accounts, all the different payments put in, and the compound interest that they, that they draw, I m going to add those things up. So each one of those has a one hundred there that I can factor out, k, and then here is my geometric series. And if you ll notice, that power of r is always one more than the highest power in the series. So this would be a four, which is actually equal to the number of payments I made, okay, and then it s just a matter of deciding what the monthly rate is. I do this with, uh, oh, there is a little simplification, one plus M minus one is just M. If I assume a 6% interest rate, that s 6% annual, divide that by twelve, then thats 1 % per month. So my monthly interest 2 rate would be And then, when I do the little calculation, it works out to be \$ So that means I gained \$3.01 from this process, other than just what I paid in. Not a big wad of money, I know. (Figure 3) But now lets think about, I d like to think about what happens if I try to do this over a long period of time. So, first let me say that the general formula, let me just go back, and I ll take the one-hundred, make that a P, n is the number of payments, okay? So here s the formula that I would use to calculate the value of the annuity after so many payments. Let s say for example that I pay a hundred dollars from each paycheck for thirty years. And I ll say still I m getting 6%, okay? So, taking those amounts, here s the 100, there s the 1% 2 interest, there s the 1 % interest. When I do that calculation, I end up with over \$100, Now, when you think about how much I paid in, I paid in \$100, 360 times. So thats \$36,000 3

4 Figure 3, Segment 1 that I paid in. I ve almost tripled my money by doing that over a long period of time, okay? You know, I m trying to show you the value of this kind of account. (Figure 4) The additional thing to kind of point out about this, is how it reacts to small changes. Now, lets say instead of \$100, I put in \$101. That means I m going to contribute an additional \$360, okay? So, how does that affect the calculation? Well, it raises it by a little over \$1,000, so again, I put in an extra \$360, I get out an additional \$1,000. I could also look at changes to the percentage rate. One percent of \$36,000 is \$360. However, compound interest works so that I draw interest on my interest, and that snowball effect, if I raise this up to 7%, as opposed to 6%, the end value, the future value of this account, is \$142,330, basically, thats a 41, or \$42,000 increase. That that s dramatic, okay? So it shows how affected that is by the interest rate. (Figure 5) I ve actually got a couple of summary pages that have those formulas on them, that will kind of summarize what we just talked about, and I ll get us set-up for the next segment. 4

5 Figure 4, Segment 1 Figure 5, Segment 1 5

6 Summary page 1, Segment 1 Summary page 2, Segment 1 6

8 Figure 1, Segment 2 Figure 2, Segment 2 8

9 Figure 3, Segment 2 share, that means they need to pay some back, and they end up paying the difference, which is \$29,000. That s paid into the estate, okay? Now the folks paid, they received less than their fair share, so we re going to make up that difference in cash, okay? So, Heir 2, when we subtract, they would receive \$48,000, paid from the estate. That s removed from the amount of cash we have on hand. And Heir 3 received the car I think. That difference in their fair share, what they consider their fair share, and the cars value, what they consider the cars value, is \$47,000, so we take that out of the estate, pay that to them. And when we re all done with this, we still have a little bit of money left over. There s nine thousand dollars left out of the cash. (Figure 4) And so the next step is easy. Divide 9,000 by three, give each of the people left, give each of them what s left, \$3,000 to each of the heirs. So what does that really give, what do they end up with, that s kind of the point of this. The first heir gets the house, which they value at \$85,000. They had to pay some cash, but they get \$3,000 of it back, for a grand total of, in their mind, \$59,000. Okay, now that s \$3,000 more than what they considered to be their fair share. The second heir got the, I think it was the boat, received some cash, and received the additional cash. That additional cash is over and above what they considered to be their fair share. And then the third heir, the same thing happens. This is the system, because everybody ended up getting more than what they considered to be fair. Everyone should be happy with this arrangement. Which is nice, that s a good system, that s a good way to kind of handle things. (Figure 5) Even beyond that, it s a good system in that it discourages folks from bidding artificially, you know, to make sure they get it. Everyone has to play honest, or they re going to get 9

10 Figure 4, Segment 2 Figure 5, Segment 2 10

11 caught, and I ll show you how that works. Let s say that the first heir really, really, really wanted the house. So they gave this exorbitant price, this exorbitant bid on the house. Even though they really thought it was valued at \$85,000, they bid \$200,000 on it, okay? Now watch how this effects their totals. Their fair share now is going to go up, right when I change this to \$200,000, the value goes up here, and then when you divide by three, it goes up. So, their fair share is inflated as well, okay, the rest of them stay the same. (Figure 6) Figure 6, Segment 2 They were going to get the house anyway, and so they still get the house. What changes now though, is how much is paid to the estate. If this is two-hundred thousand, this is now a pretty big sum, right? This is way over and above what they considered their fair share. So they have to pay quite a bit back to the estate, and what that does is that puts a lot more cash on hand for the estate, okay? (Figure 7 and 8) Everything else is as it was, but what changes is, instead of taking the amount \$9,000, dividing by three, and getting \$3,000, now when you divide by three you get \$28,556. (Figure 9) So that s paid out, and watch what happens to the value here with each of the heirs, okay? And what I ve done is, instead of the \$200,000, I used the actual value, what they thought it was actually valued at. They end up with a lot less than the others. And what they ve done is they cheated themselves out of, they did get the item they wanted, but they cheated themselves out of actual value of the estate. (Figure 10) This is a remarkable little system for doing this. Chances are at some point in your life, you might encounter a situation like this. Now you kind of have an idea how to handle the mathmatics of it. I ve summarized the algorithm for this in a couple of summary pages, and I ll get set up for the last little segment. (Figure 10) 11

12 Figure 7, Segment 2 Figure 8, Segment 2 12

13 Figure 9, Segment 2 Figure 10, Segment 2 13

14 Summary page 1, Segment 2 Summary page 2, Segment 2 14

15 3 Wrap-up The last thing I d like to do with you today, since it s our last episode, is I would like to kind of wrap things up, and end with some encouraging words. Now I realize, at this stage of the game, you re probably getting ready for finals, you may be getting ready for finals, and you may even be a little sick of mathmatics at this point if you re finishing up the semester. But I would like to encourage you to follow up, and take some more mathmatics. And I have a couple of reasons for telling you this. We ve already demonstrated that we could handle some very tough situations with the mathmatics that we find in the general math courses. We ve looked, we spent a semester looking at things like that. It s also true though, that you can get into some fairly routine situations that have tough mathmatical answers, and that s the kind of thing that we handle in the upper level courses. For example, here s a very easy problem. If I say okay, what are all the solutions to this equation, well, you ve probably realized by now that that s a line. It s a linear equation, so the graph of that is a line, and the solutions are all the points on that line. That s pretty easy, a graphing calculator can do that for us. (Figure 1) However, if I ask the question this way, I say, what are the integer solutions to that equation? That is a lot less exacting. And, a lot of the solutions here will not work. They have to fall along an integer lattice here, like these points do. This one actually occurs at (1, 1), this one occurs at ( 2, 3). Those are integer solutions, they re not always, not everything on the line though will work for this answer. And so we re getting into a field of mathematics called discrete mathematics, and that requires some further study, although that s a very, that s a very simple kind of question. (Figure 2) Figure 1, Segment 3 15

17 Figure 3, Segment 3 Figure 4, Segment 3 17

18 these things, and I can numerically isolate that point, and there are other techniques for doing that. But exact solutions are very difficult for non-linear systems. So, these are the kind of things that can pop up, and, and in day-to-day applications, I mean not everything is linear, thats just the sad fact of things. (Figure 6) Figure 5, Segment 3 Here s another example of a very tough problem. It sounds very simple. We have a bakery that makes five different types of fudge you know it sounds like something you would ask out of just being naive how many different ways can we fill a package with twelve pieces of fudge? I ve got five, I can make them all one type, I can make them all the other types, make half,.... I mean, you start thinking about all the ways to do this, and there are a lot of ways to do this. That is a very difficult question to answer. That s also something we d handle in the field of mathematics called discrete mathematics. We have counting techniques for doing things like that. But it s a little difficult. If I go back and say, well okay, let me put a restriction on that. Let s make sure we have one of each type in the package, then it gets just that much tougher to kind of figure out, okay, well, how many ways can you do this? You certainly don t want to try and list them all out by process of elimination, because the answer could be up in the thousands, ten-thousands, whatever it is, okay? So this is some fairly difficult mathmatics to deal with, although it s very simply stated, okay? (Figure 7) The other kind of approach, the other kind of thing I would say as far as why I would encourage you to continue to study math is, you know, I think we jump the gun sometimes when we re studying, and we make this assumption that because we can t immediately find some applications for things, that it s not really very useful, okay? I couldn t disagree with that attitude more, okay? The problem sometimes is not in the mathematics, it s in our 18

19 Figure 6, Segment 3 Figure 7, Segment 3 19

21 Summary page 1, Segment 3 Summary page 2, Segment 3 21

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