The Franklin Residence
Philadelphia
Pennsylvania
The Twenty-Fourth of September, Seventeen Hundred and Fifty

In Reply to: Your letter of the fifth of September
SPECIAL DELIVERY
Mr. Myron Sopher

My Dear Mr. Sopher,

I feel honored that you have entrusted me to this important task, I assure you that I shall do my best to deliver you from your predicament. Just so you understand how I approached the situation, this is what I understand your problem to be. Your mentor left you a sum of money upon his death and left it in a CD to earn interest while you grow up. It was a fixed rate CD and upon your coming of age, you would receive the original amount that he put in, and the object of your affection, Miss. Philomena Willoby, would receive the interest that it earned. Now you sent the bank statements to her so you wouldn't have any more sad reminders of your benefactor. Now you don't know how much money is owed to Philomena, and how much is yours. You did keep a note letting yourself know the expected final sum but it did not agree with the check that you received from the bank. Of course, you want everything to work out properly. It is, I know very well, necessary to keep track of finances carefully. As I always say, a penny saved is a penny earned. 

As I've mentioned, I was honored to receive your letter. However, I found myself perplexed as to why you would have chosen myself to solve this particular problem. As you know, I have served my country in several capacities; most know me as a diplomat, a revolutionary. You must have heard that I, as a typical renaissance man, am also a scientist and inventor. I'm not sure how you found out, but my most recent invention is a time travel machine, man's greatest aspirations can finally be fulfilled. I decided to use my machine to solve the problem in your letter. 

You see, I tried very hard to discover the solution to your letter myself but found that I would need outside help. This is where my invention proved to be so useful. You may have heard of the college that I founded along with a colleague, his name escapes me right now. I decided to travel to the year 2003, to my college, and ask a student for help. Her name was Beth; I went to see her because she is a math tutor for the other students at the school. We met on a lovely Monday afternoon, the fifteenth. This is when I discovered the true complexity of your problem. You see, neither of us could come to an answer after working on it for over an hour. So, I decided to find the most powerful mathematician that I could.   

I didn't have very far to go. At my institution, there is, of course, a department devoted to demystifying the problems of mathematics. The head of this department, Dr. Annalisa Crannell, is a professor, and was perfectly suited to clarifying this problem for me. I wanted to be sure to thank and acknowledge Dr. Crannell and Beth for their time, effort, and talent.

Now, it rests with me to complete this task; you need to know the answer to your question and understand how to find it. What if you're talking with Philomena and she asks you a question? You need to be able to explain all of this to her. Therefore, I will try to take you through the process that I went through (with help) to find the solutions to your questions.

First of all, I tried to figure out exactly what information you needed. You asked me, first and foremost, how much money Philomena is entitled to. Remember, she is supposed to receive all of the interest that accrued over the ten years that the principal amount was in the bank. The principal amount is the number of dollars that Mr. Gusterson originally put into the CD for you. You also asked me to find out how much of the final $11,406.14 was yours; you are entitled to the principal amount that Mr. Gusterson put in.

Just to ease your mind, I'll let you know the answers to your questions right away.  The amount of money that Mr. Gusterson put into the bank was about $3,455.45; this is the amount that you are entitled to. The interest that this money earned over a period of ten years is about $7950.69, this is how much money Philomena receives.  So, now that you can rest assured that your queries have been answered, wouldn't you like to know how I came to these conclusions, with the help of the aforementioned Beth and Dr. Crannell?

The second part to solving a problem, after figuring out what it is, is to decide what information you have that might be useful. I took a careful look at your letter. You told me that Mr. Gusterson passed away when you were eleven years old. I'm sorry about that loss, it must have been very difficult for you. You also told me that you would be allowed to take out the money in the bank when you are 21 years old. This means that the money sat in the CD for ten years. I know this because before you were 11, there was no money in the bank. It was only there between the ages of 11 and 21. Therefore, I subtracted 11 years from 21 years:

You also explained to me that this money was placed in a fixed rate CD, where the money earned (the interest) is at a constant rate. That means that every year, you will not earn the same amount of money. It means that, based on how much is in the bank, the same percentage will be earned every year. So, to explain this tricky concept, I'll give you a little example. Let's say I put one dollar into a fixed rate CD, and the fixed rate of interest will be .5, or 50% earned per year. After the first year, I will have the principal amount I put in, the dollar, plus 50 cents, which is the interest that it earned. So, after the first year, I would have $1.50. So, now, the second year begins with $1.50, not $1 as before. At the end of the second year, I would have $1.50 plus half of that, which is 75 cents. So at the end of the second year, I would have $2.25. If I earned the same amount of money each year, I wouldn't have that much. Luckily for you, your fixed rate CD allows you to earn the same percentage each year rather than the same dollar amount. Mr. Gusterson must have really liked you.

You also gave me some other information. You told me that after the first year, the principal, (the amount Mr. Gusterson put in) earned $438.30 in interest. Now, you thought that the principal would earn this amount every year. Remember that I explained that this isn't the case. So, your reasoning that Philomena's share would be $4383.00 wasn't quite right. You did give me the hint, though, that this amount is more than the principal amount (your share) and that will be helpful later.

The final piece of information I found particularly useful was the amount of money made out to you in the check from the bank. Since you had incorrectly calculated the final amount earlier, I assumed that this figure was an accurate reflection of the principal amount plus the interest that it earned over ten years. So, the total amount to be split between you and Philomena is $11,406.14. You should be pleased, this is a nice sum. It could almost pay for a third of the cost of going to my college for one year!

At this point, I had figured out what your questions were and what information you gave me. Then, I needed to actually solve your problem. Well, I have to admit, this is where things got a bit tricky. I started out with a good equation but it got a little too complicated. This was when I decided to go see Beth. We worked with this equation for a while but it proved to be too difficult, even with our combined efforts. So, I had to go to the best. Dr. Crannell was able to help me find a better equation. She also guided me through all of the steps that I will be showing you. The equation that we needed would be one that could relate to the information you provided. She thought that the best equation to use would be one that we had discussed before:

Right now, this means nothing to you. It confused me at first as well. If I explain this general equation to you now, then it will be easier to understand the equations we use later, when we put in the information from your letter. The P represents the final amount of money. This could be for one year, or all of the years that the money is in the account. The Po is the principal amount. I've explained this one before; it's the amount that Mr. Gusterson put in the account, and how much you will receive. A good way to remember this is that the o in Po can stand for original, as in the original amount of money put in the account. The t in this equation represents the amount of time that the money has been in the account, or how many years the account has accrued interest. The slightly tricky part of this equation is the (1 + r). The r is the rate at which the interest is earned by the money in the CD. A one is added to it so that the principal will not be multiplied by a number less than one, which would make the amount of money in the account decrease! This equation results in an exponential function, which you can read about if you are interested, in the book, Calculus, pages 11-14, by Hughes-Hallet et al. (this is assuming that you don't live in the 18th century, like me) If you want more information as to how to derive this equation, you can always consult Dr. Crannell, who is very helpful with things like that. We knew to use this equation because it has variables that are the same things that you need us to figure out.

We put information into the equation that you provided in your letter, to get

The 11406.14 is the amount that was in the account after ten years passed by. This equation is very important, it will be used later and usually referred to as the "big equation."

So, once we knew what your question was, what information we had, and what formula we had to use, we were all set. Now, we knew that after ten years, the final amount was $11,406.14 so we had two variables to fill into our equation: P and t. Unfortunately, if you look at the equation, you'll see that there are still two unknown values. If you want to solve for a variable, you can only have one unknown for every equation that you have. So, we needed to figure out how we could end up with one unknown in our equation. You told us that in the first year, the principal amount earned $438.30 so we used this figure. The interest that a sum of money earns is equal to the principal amount multiplied by the rate at which the interest is earned. You can see this same expression mathematically:

Using this equation, one can solve for either the principal amount or the rate. The following equation has been solved for r:

and the next one has been solved for Po:

The two preceding equations will now be referred to as the "small equations" and keep in mind that the first was solved for r and the second was solved for Po. Now, it looks like we have the same problem with having two things to solve for within each equation. It works out very well though because we still have the big equation from before. Instead of putting a number into the big equation for Po or r, we are going to substitute in one of these small equations. Then, there will be only one variable to solve for in the big equation. Isn't that wonderful! Let's see what would happen if we plugged in the small equation that is equal to r:

We could also substitute in the equation that was solved for Po. This would yield:

So now each of these equations has one variable to solve for, we can use either one. Let's work with the first one, the one where you need to solve for Po. It might just be me, but I think that this equation looks very complicated. Instead of trying to solve for Po with algebra, lets make a graph. We can make a graph of the final amount (P) as a function of the principal amount (Po). This means that the amount of money you end up with depends on how much you put in the bank in the beginning. We can put a range of the possible principle amounts on the x-axis and the range of possible final amounts on the y-axis. Our graph can be set up like this...

You may notice that the differences in the amounts on the x-axis and on the y-axis are large. Remember, the values on the x-axis represent the possible original amounts that Mr. Gusterson could have put into the account. The values on the y-axis represent the final amount in the account, (the principal plus the interest that it earned over ten years). We know that the actual final amount is $11,406.14, so we need to find the value on the axis that forms a point with that y-value. There happen to be two such points. One has a principal value of $3,455.45 and one has a principal value of $4,524.60. I could eliminate the point that tells us the higher original amount because you told me in your letter that the principal was less that $4,383.00. Therefore, the principal amount that yields $11,406.14 after 10 years is $3,455.45. Now, this is an approximation, but it is very close. Since we were able to solve for the principle amount using a graph, we could also go through the same process to solve for the rate at which interest was earned. To do this, just substitute the other small equation (the one that solves for r) into the original big equation. I won' t go through this tedious process again with you but I will tell you, that when I did it, I found that the rate at which your interest was earned was approximately .127, or 12.7%. That's a nice high rate!

After I figured out the principal amount that Mr. Gusterson placed in the CD for you (your share of the final total), I needed to find out how much total interest was earned. (Philomena's share.) This was much easier than what we already went through. Since I knew the final amount, $11406.14 and your share, $3455.45, I could just subtract your share from the total. Philomena's share of the money is

So, now that you have the answers to your questions and the knowledge of how they were found, you should feel happy and confident during your meeting with Philomena. Best of luck to you! Perhaps you and Philomena should consider putting your combined money in a CD and taking it out on your 50th wedding anniversary…

Cordially,

Mr. Benjamin Franklin

Diplomat, Scientist, Inventor,

Founder-Franklin College

cc: Dr. Annalisa Crannell, Mathematics Department Chair,

Franklin and Marshall College