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[Assume one interest period per year; no additional money deposited or withdrawn]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 = I_{1} = (400)(0.1) = $40; money is now worth A_{1} = $440.

2nd year: Interest earned = I_{2} = (440)(0.1) = $44; A_{2} = $484

In general:

A_{1} = P + Pr = P(1+r)

A_{2} = 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:

I_{1} = rA_{0} = rP = rP(1+r)^{0}

I_{2} = rA_{1} = 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...

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