Can Algebra II Help Me With Finances?

If you have money in a savings account (or owe money on a loan or credit card), you're probably familiar with interest. Simple interest is calculated by multiplying the amount you have in the bank (principal) by the interest rate by the amount of time.

So if you have $100 in the bank at 5% interest per year, after one year you'll have $5 more. In banking, we don't usually worry about simple interest.

When we calculate interest, we need to take account of the interest your money has already earned. We also have to know how many times the interest is taken.

For example, suppose your principal, P, is still $100, and the interest rate, r, is still 5% per year. We refer to calculating interest as compounding interest. We let n equal the number of times interest is compounded in a year. If we compound interest once per year, then we have $105 after one year, like simple interest. But what if we compound twice? Then we would have $105.06 at the end of one year.

The formula (with the time in years equal to t) for compound interest is:

How does changing the number of times interest is compounded affect the amount of money we earn? To make things simple, we'll start with $1 and assume 100% interest per year. We'll look at the account for one year (t=1). This gives us a simple formula for A.

We'll consider compounding with the following frequency:

1) Yearly
2) Monthly

3) Daily

4) Hourly

5) Per Minute

6) Per Second

7) Per Tenth-of-Second

Though the amount we earn at the end of one year doesn't really change after compounding hourly $1, notice something about the numbers. They appear to be stabilizing. The larger number we use for n, the closer A appears to get to another number. What happens if we let n get really big? If we compound interest continuously (that is, we are ALWAYS calculating interest), then we can say that n goes to infinity. By taking a limit, we can see how A looks if we let n get really big.

The expression below is read as: the limit as n goes to infinity of the quantity one plus one over n quantity to the n power.

This is one of the original definitions for the constant e, which has a value of about 2.718. If we look at our table of values, we can see that as n gets bigger, A gets closer and closer to e.

What if the interest rate wasn't 100%? Would we still get e? As it turns out, we don't. We get something slightly different. If interest rate is a random number, a, then our limit doesn't yield e; it gives us e to the a, as seen below.

What happens if we continuously compound interest? We need to take the limit as n goes to infinity of the compound interest formula. Since P does not depend on n, we can bring it outside the limit.

Since t doesn't depend on n either, we can move the limit inside of the t-power. If we look at what we're taking the limit of now, it simplifies to e to the r, just like e to the a earlier. Using our rules for exponents, we know that raising an exponent to a power requires us to multiply the exponents, leaving us with e to the r t power, all multiplied by P.

So our formula for continuously compounded interest is:

This is a little surprising, as nothing about interest necessarily indicates the presence of e. The fact is, e has the habit of showing up whenever exponentials are involved, so keep an eye out for it.

Personally, I would never teach this in an Algebra II setting. I would teach the compound interest formula when learning exponential functions, but I would not delve into the e-problem. In my experience, Algebra II  students don't really care about interest, they don't know the definitions of e and they usually don't know limits.

That being said, this would be good enrichment for the advanced students in an Algebra II class, and this would be a valuable lesson for a Precalculus or Calculus class reviewing limits.

Anyway, I hope you got something out of this. If not, well, there's always next time.

If you have any questions, or if anything isn't clear, just let me know in the comments.

 

I first saw this definition for e from my Calculus Teacher Jan Haynes. I saw the table derivation for e in compound interest from my mentor teacher Jennifer Davis when I was completing my student teaching at Somerville High School and finished the rest of the derivation on my own.

The images used here were created by me using Noteshelf for iPad.