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### Calculating Interest on Auto Loans

```
Date: 7/10/96 at 20:9:17
From:  Sampath Oks
Subject: Calculate Interest

Hello Dr.Math,

I would appreciate any information on how the lien holders for
auto loans etc. calculate interest for the principal. Any useful

Thanks,
Sampath
```

```
Date: 7/11/96 at 9:16:41
From: Doctor Paul
Subject: Re: Sampath

Good question!  The idea here is that the rate at which your money
grows is proportional to how much you've got.  Makes sense, right?  If
you have more money than someone else, you're going to accumulate more
interest than that person.

There is a simple differential equation to model this:

dP
-- = k*P
dt

P is your principle; k is an arbitrary constant.

In words:  The rate at which the Principle changes over time is a
constant times how much Principle you've currently got.

I'm not going to show you how to solve this Diff Eq.  If you're
interested, write us back and we'll solve it for you.  I'll just tell

P(t) = P0 * e^(k*t)

P0 denotes how much money is made in your initial deposit; k is the
interest rate.

Let's do an example:

If you deposit \$5000 in the bank at 4% interest, how long will it
take to become \$10,000?

Well, let's set it up..

We know that P0 (our initial deposit) is \$5000 and our interest rate
(k) is .04.

Let's plug those in:

P(t) = 5000 * e^(.04*t)

We have two unknowns left: t and P(t).  The problem says find how long
it will take for the money to become \$10,000.  In other words, solve
for 't';

10000 = 5000 * e(.04*t)     plug in 10,000 for P(t)...
2 = e^(.04*t)               divide both sides by 5000
ln (2) = ln (e^(.04*t))     take the natural log of both sides

Note: ln is the inverse of 'e', so  ln (e^(garbage)) = garbage
ln (2) = .04 * t

divide both sides by .04
t = ln(2)/.04

One final note: This problem was done assuming that the interest was
compounded (or computed) continuously. Most banks don't actually
compute interest continuously (that would be a great drain on their
computers). They usually just compound it quarterly or monthly.

I hope this answers your question.  If you have any more questions
feel free to contact Dr. Math again.

-Doctor Paul,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Interest

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