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Compound Interest (Finite Number of Calculations) - Problem 1

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

We’re now going to look at a compounded interest question. So basically what we have is a set up where we invest $3000 in an account that earns 4% interest, how much will you have after 4 years if interest is compounded in a number of different ways?

For this what we need to remember is our equation. Let’s go write that over here and we end up with A, the amount we end up with is equal to P, our initial amount, 1 plus r over n to the nt. Going through this problem what we need to figure out is what we’re given. We know that we are investing $3000. That’s going to be our initial amount, so that’s just going to be our P. We’re asked to figure out how much we end up with, so that’s going to be A for a number of different scenarios. Our rate is 4%, we need to put it in as a decimal so that goes in as .04. Let’s write that a little bit bigger so to make sure you see it, this goes in as .04. N is the number of times per year that we calculate interest, that’s going to vary depending on which part we look at, if we look the a, b, or c portions. So we’re going to hold off on that and t is the time we invest, which the problem tells us is 4 years. This is going to be, let’s use a different color, this is going to be 4. I didn’t use a different color at all.

We have our general set up and now we’re just going to go and plug these numbers in for the different cases. Biannually, our 3000 is still there. Our 1 plus our rate is still the same but now our n is the number of times per year. Biannually, that means twice a year so the number of times per year is 2, and then it’s to the nt, n is the number of times per year so that’s just going to be 2 as well and then length is 4. I forgot a zero in my 3000, didn’t I? Let’s plug that back in, make sure it’s actually 3000. There we go. I’ll hold off on calculating this until we get all the actual equations up.

Monthly, so we still have the same initial investment, we still have the same rate but now we’re dealing with monthly. That’s going to affect the n, which instead of being 2 for biannually it’s now going to be 12 and then it’s to the nt so this is just going to be 12 times, our time hasn’t changed, it’s still 4 years.

Daily, same exact formula except instead of having n being 2 or 12, we’re now talking about daily which is just going to be 365. This is now 3000, 1 plus our rate hasn’t changed over 365 to the 365 times 4.

We can go ahead and plug this into out calculator which I’ve already done, I hope you don’t mind, and what we end up getting is, this is 3514.98. This is 3519.60 and this is 3520.50.

It doesn’t really matter how many times we invest, they’re all roughly the same, the more you invest slightly more interest you get.

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