Calculating Interest on Auto LoansDate: 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 formula will be helpful. 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 you the answer... 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 Plug this into your calculator and get the answer: 17.33 years 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/ |
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