# How to Manually Calculate your Interest

## Finance Application – Interest Functions

Interest problems can be divided up into a couple of different categories. **It’s important to know what category of interest problem when you’re dealing with before you try to solve it.** I want to explain what each of these types of interests is and give the equation that you can use to solve for each one. The first type that we have is called simple interest.

**Simple interest is interest that is based only on the amount of the initial deposit.** It’s not affected by any interest that’s earned over the course of a time period you’re looking at. The function used to calculate simple interest is going to be f of t equals f naught, or the amount of the initial deposit, times 1 plus rt, where r is your interest rate and t is your time period. If you’re measuring timing years, which customarily you will, r should be your yearly interest rate.

**This is the equation for calculating simple interest.** Interest that is not simple is called compounded interest. We have compounded interest, but within compounded interest, we divide it up further into two categories. We have periodically compounded interest and we have continuously compounded interest.

**Periodically compounded interest is interest where the amount of interest earned is based not only on your initial deposit, but on the interest that’s been earned up to that point.** In fact, that applies to both periodically and continuously compound interest. With periodically compounded interest, interest is calculated or it is compounded at particular intervals of time or particular periods.

The function that we can use to calculate periodically compounded interest is f of t is equal to f naught times 1 plus r over n raised to the nt, where r and t are the interest rate and the time period and n is the number of periods, or the number of times that the interest is compounded per year. If you have monthly compounded interest, n would equal 12.

**If you have a semiannually compounded interest, n would equal 2.** That’s the equation for periodically compounded interest. Continuously compounded interest, essentially you can think of it as interest that’s compounded every fraction of a second, or so frequently that if we tried to use the periodic function to calculate it, the end in this equation would be so large that we wouldn’t be able to accurately calculate it.

What happens when n becomes incredibly large is that we end up with 1 plus some very small number here raised to a very large number. When you have that sort of situation, it becomes difficult to accurately calculate the amount of interest that you’ve earned. The equation that we use to model continuously compounded interest is just a form of the exponential growth equation. We have f of t equals f naught times e raised to the rt.

These are the three different equations you might have to use to solve an interest problem. I’ve already demonstrated how to solve an exponential growth equation on the biology video. I want to look primarily at the equation for periodically compound interest. The example I want to look at now is one where we’ve made an initial deposit of $1,000 into an account that earns 4% interest that is compounded quarterly (every three months).

Our initial deposit is 1,000. Our interest rate r is 4%, or written as the decimal is .04, and our n, because we have four periods in a year, is equal to 4. We can rewrite this equation as- What we’re looking for is the amount of time that it takes for us to earn $100 in interest, or when our amount of money that we have is equal to $1,100.

We can rewrite the equation now as 1,100 equals 1,000 times 1 plus 0.04 divided by four raised to the 4t. We have t, just one variable, and we need to solve for that. The first thing we want to do is divide both sides by 1,000. That will give us 1.1 equals 1 plus .04 divided by 4 is .01. 1 plus .01 raised to the 4t. We can rewrite this as just 1.01 raised to the 4t.

Now we have our t in an exponent, so what we need to do is take a logarithm of both sides. It doesn’t have to be the natural log, because we don’t have an e in our equation. Whatever base logarithm you use, it has to be the same on both sides. Let’s take the log base ten of both sides. That will give us a log of 1.1 is equal to a log of 1.01 to the 4t.

If you remember the rules for logarithms of numbers with exponents, this can also be written as 4t times the log of 1.01. We have log of 1.01 is equal to 4t times a log of 1.01. Now to solve for t, all we have to do is divide by 4 times the log of 1.01. Our t is equal to the log of 1.1 divided by 4 times the log of 1.01. At this point, you can plug these numbers into a calculator and what you get is 2.39 years.

It takes 2.39 years with 4% interest compounded every quarter for you to earn $100 in interest on a $1,000 deposit. If you’re curious, if you had had continuously compounding interest over that same period of time, you would earn maybe 50 cents more. If you had simple interest over that same period of time, you would earn about $4 less. This is the equation and the process for solving a periodically compounded interest problem.