At r% interest compounded annually, $400 earns $63.05 interest in 3 years. r = ?? Developing Greater "Interest" in Algebra...

SILLY RIDDLE OF THE WEEK
Why were the Romans so good at algebra?
You have to think outside the box and be in the mood for this groaner! Of course you've probably seen this elsewhere on the web...

It's been awhile since we've worked on financial math applications. Anyone recall those 3 mortgage investigations from last year? [Note: To see other mortgage/finance posts, click on the mortgage label/tag in the sidebar].
Considering the current economic situation, perhaps we should devote more attention in our math classes to the subtle trap of running up credit card debt. I'm working on that. There are strong mathematical similarities between loans, mortgages and investments and in this investigation students will focus on the investment problem in the title of this post.

The Problem in the Title of this Post:
At r% compounded annually, $400 earns $63.05 interest over 3 years. What is the value of r?
Let's agree, that r% has already been converted to a decimal so that we do not have to work with r/100 in the formulas below. That is, if r = 10% for example, we will work with r = 0.1.


OVERVIEW OF ACTIVITY
We will first consider a quick estimate of the interest rate by using simple interest to approximate compound interest. This develops sense about the formulas and could be helpful if a question like this appears as a multiple choice question on the SATs or other standardized tests. We will then apply standard compound interest formulas to validate our estimate. Students will be asked to use more than one method for this. Finally, there will be an extension for your students to try.
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KEY for this activity (not necessarily standard notation)
[Assume one interest period per year; no additional money deposited or withdrawn]

P = original amount invested (principal)
r = annual rate of interest (decimal form)
n = number of years
A_n = Amount original money is worth after n years
I_n = Interest earned during the nth year
T_n = Total Interest earned over n years
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Background for Simple vs. Compound Interest

Simple: Interest each year is constantly Pr so total interest for n years is T_n = Prn.
Example: If $400 is invested at 10% annually simple interest, then over 3 years one would earn (400)(0.1)(3) = $120 in interest.

Compound Interest
Example: Suppose $400 is compounded annually at 10%.
1st year: Interest earned = I₁ = (400)(0.1) = $40; money is now worth A₁ = $440.
2nd year: Interest earned = I₂ = (440)(0.1) = $44; A₂ = $484
In general:
A₁ = P + Pr = P(1+r)
A₂ = P(1+r) + rP(1+r) = P(1+r) (1+r) = P(1+r)²
(*) A_n = P(1+r)ⁿ

Beginning of Activity
I. Approximating the Rate using Simple Interest:
If the total interest over 3 years is about $63, show that r = 0.05 is a reasonable estimate for our problem using the simple interest formula above.

II. Using Compound Interest Formula
There are several approaches to solving the title problem:

Method I: Use the above compound interest formula (*) directly to solve for r.
Remember: The formula expresses A_n but it's the total interest, T_n that's given.

Method II: Derivation of Related Formulas

(a) Show that or explain why the total interest earned after n years can be expressed as
T_n = P[(1+r)ⁿ - 1].
(b) Use the formula in (a) to solve for r in our problem. Here you will be substituting the values for n, P and I_n first, then solve for r.
(c) Alternate Approach: Use the formula in (a) to derive a general formula for r in terms of n, P and I_n. Then use this formula to find the value for r in our problem. When do you think it makes more sense to use (b)? (c)?

Extension
In the above problem, we knew what the total interest was after 3 years and we needed to manipulate a formula to determine the rate. In other applications, we might want to determine the interest earned each year. This is usually done for us by our bank -- we certainly need this amount for federal and state income taxes. We will now derive the formula for I_n by two different methods:

(a) Derive a formula for I_n using the fact that I_n = A_n - A_n-1, for n = 1,2,3,...

(b) Derive a formula for I_n using the following pattern:
I₁ = rA₀ = rP = rP(1+r)⁰
I₂ = rA₁ = rP(1+r)¹
....
In general: I_n = ____________.
Note: This formula makes sense. Why? Can you show that the results in (a) and (b) are equivalent?

(c) For the original problem in the title of this post, complete the following table:

n................A_n....................I_n
0...............$400...............

1...............$400...............$40

2

3
.
.
.
10


Comments:

• The instructor may choose to use this activity to develop recursive functions. For example, A_n = (1+r)⋅A_n-1
  
• The chart above can be generated using the graphing calculator of course. More importantly, ask students to discover relationships among the columns.
• Much of the above is standard 'stuff' and not very challenging. However, the goal here is to help our students develop a feel for these formulas, rather than mechanically 'plugging in.' Considering that this topic is related to exponential functions, recursive thinking, and geometric sequences, there is unlimited potential for bringing more financial math into the algebra or precalculus classroom. And, yes, it's all standards-based...
  

Taking the Magic out of Mortgages Part I: Exponential Functions and Geometric Sequences to the Rescue

[Note: For an exceptionally clear and definitive exposition of all things financial, the best resource I have found is MoneyChimp. There are interactive calculators to thoroughly understand the concepts in this post and much much more. More importantly, for math nerds like me, the formulas are explained and, in some cases, derived. The mathematics is accurate and the analysis is excellent. Enjoy it!]

In Algebra 2 and Precalculus (or whatever it may be entitled in your local schools), students often do compound interest problems. Typically, the author of the text and/or the instructor will derive the formula for what your original investment of P dollars will be worth in t years, if interest is compounded n times per year for t years at an annual rate given by r (a decimal for this discussion):
Compound Interest Formula: A(t) = P(1+r/n)^nt.

This is a nice practical application of exponential functions, exponential growth in particular. A similar, but more sophisticated, concept applies to annuities and amortization of a mortgage (paying off debt over time in n equal payments). In both an annuity and a mortgage, the original amount of money (whether it's the amount invested or the debt you owe) generally decreases over time. In an annuity, you receive a fixed amount at the end of each period, whereas, in a mortgage, you pay a fixed amount. In an annuity, your original investment is earning (accruing) interest (it may be possible to 'live off' the interest and not touch the principal), while you are receiving periodic equal payments that are deducted from your account. A central concept in both annuities and mortgages is that that interest is applied before receiving an annuity payment or before making a mortgage payment.

The following is the first part of an activity introducing students to the mysteries of mortgage calculations. The fact that the formulas for monthly payments or the decreasing amount of debt seem very intricate lead many to believe that this topic is too sophisticated for most secondary math students. Just give them the formulas, mention that it is related to exponential functions and let them plug it all into their graphing calculators. We know most adults, other than those in the business of lending, punch the numbers into the computer and read the results. Before calculators, bankers would look it up in those mortgage tables on some well-worn-out card. This activity may demystify a little bit of this. Students need good algebra skills, knowledge of exponential properties and functions in particular, a basic knowledge of compound interest and background in geometric sequences and series (later on). I am well-aware that sensitivity is needed here for students whose parents do not own a home, however, all students can benefit from these ideas since these principles apply to far more than a monthly mortgage payment.


STUDENT ACTIVITY
You can find many excellent web resources for mortgage calculations. You can also find the actual formulas for all of this either in your text or in other sources. Most of us would probably use the built-in applications typically found on a graphing calculator or more likely use those free mortgage calculators all over the web. In this activity you will take an active role in the process of borrowing and lending and see what lies behind those sophisticated formulas.

In actual practice, mortgages can range from less than a hundred thousand into the millions of dollars. Therefore, these loans are typically repaid over 5, 10, 15, 20, 25 or 30 years to make the monthly payments more manageable. In this activity you will be borrowing a small amount and considering an oversimplified form of repayment, leading up to more general considerations.

You will be borrowing $100 from a reputable lender, Stan, The Mortgage Man.
Stan is charging the going rate at the moment, which is 10% compounded annually.

(a) If you repay the loan in one year, explain why your single payment would be $110.
(b) If you agree to repay the loan at the end of two years, in one single payment, explain why that single payment would be $121?

Discussion: Parts (a) and (b) should remind you of the compound interest formula you've learned:
One year re-payment: 100(1+0.1)¹
Two year re-payment: 100(1+0.1)²

Surely, increasing the payment schedule to TWO payments over two years or one year, cannot be that much more difficult? Let's find out...

(c) This time you will make two equal payments over two years. Stan gives you the repayment schedule: Two equal payments of $60. He explains it as follows: The loan (your debt) of $100 is divided into two equal payments of $50 each. The interest charges are $10 (10% of the amount you borrowed) on each payment. Explain the mistake that Stan is making (or is he trying to take advantage of an unsuspecting borrower who didn't pay attention in algebra?). We're not asking you to correct Stan's error here - just explain why his calculation is either wrong or unfair.

(d) Now that you figured out that the two equal annual payments should not be $60, we will tell you what the actual payments would be according to mortgage formulas:
Each annual payment is $57.62 (rounded to the nearest penny).
Show why these two payments correctly repay the loan of $100 and the interest that is due on each payment. Show method clearly. Use calculator as needed.

(e) Do you think you could figure out an algebraic way to determine those equal payments of $57.62? You are about to...
Let A represent the equal annual payments you will make.
At the end of the first year, before you make your payment, you owe $100(1.1) = $110.00. Now the fun begins:
(i) Represent, in terms of A, the amount of debt (loan + interest) you will owe AFTER you make your first payment.
(ii) Represent, in terms of A, the interest you will be charged by the end of year 2 BEFORE you make your final payment.
(iii) Represent, in terms of A, your debt, AFTER making your final payment.
(iv) What should be the numerical value of your debt AFTER making your final payment? Now, write an equation and solve for A. You should come up with $57.62!

(f) Are you up to the challenge of solving for the general formula for A given an original loan of $P at an annual interest rate of i (expressed as a decimal)? Of course, you are! For now, you only need to do this for TWO payments, just as we did in (e). Of course, your formula for A should be in terms of P and i.

More to follow...