# Finding Compound Interest With a Calculator

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Compound interest is a type of interest many of us will encounter in our lives. This lesson will show formulas for different types of compound interest, and explain how to break the formulas down so we can use a calculator to find compound interest.

## Compound Interest

Everyone has to deal with money at some point in their lives, and when it comes to investing money, it is a good idea to be familiar with different ways to earn interest. For instance, suppose you want to invest \$5,000. You go to a bank, and the banker says that you have three options. The first is an account that pays 1.7% interest compounded annually. The second is an account that pays 1.9% compounded monthly. The third is an account that pays 1.8% compounded continuously.

Each of these accounts are an example of compound interest. With compound interest, interest is compounded, or added to the account balance, on regular intervals. Then, the account earns interest on the new balance, rather than just on the original investment. I'd say that sounds like a pretty good deal!

As our example illustrates, there are three main ways in which interest is compounded; annually, a certain number of times per year, and continuously. After the banker tells you your options, you need to figure out which account will yield the most interest in a two-year period. Thankfully, you have your calculator with you, so let's take a look at how to use formulas and your calculator to calculate compound interest in each instance.

## Interest Compounded Annually

When interest is compounded annually, we just need to know the interest rate, r (in decimal form), the amount of the original investments, P, and the length of time the money will be in the account, t (in years). If we know these things, we can use the following formula to calculate the amount in the account at the end of a given time period.

It's important to note that the rate has to be in decimal form. To convert a given percentage rate to decimal form, you simply move the decimal point two places to the left.

In our example, when interest is compounded annually, we have an interest rate of 1.7%. To convert to decimal form, we move the decimal two places to the left to get 0.017. Thus, r = 0.017. You are investing \$5000, so P = 5000. Lastly, you are leaving the money in the account for two years, so t = 2. We plug these into our formula.

And now we use our calculator. You want to work from the inside out using your calculator to find each value. First, add what's in the parentheses to get 1+0.017 = 1.017. Now, raise 1.017 to the power of 2 to get (1.017) 2 = 1.034289. Lastly, you multiply 1.034289 by 5000 to get 5000*1.034289 = 5171.45.

We see at the end of two years, you would have \$5,171.45 in the account, so you would gain \$171.45 in interest if you go with this account.

## Interest Compounded a Set Number of Times Per Year

The second option of our example says the interest is compounded monthly, so it is compounded 12 times in a year. When an account compounds interest a set number of times per year, we need to know the number of times the interest is compounded, n, the amount of the original investment, P, the interest rate, r (in decimal form), and the time the money will be in the account, t (in years). When we know this, we use the following formula.

In our example, we have n = 12, P = 5000, r = 0.019, and t = 2. We plug these values into our formula.

Once again, we can use our calculator to work from the inside out. Start by dividing 0.019 by 12 to get 0.019/12 = 0.001583 (rounded to 6 decimal places). Next, we add that to 1 to get 1 + 0.001583 = 1.001583. The next thing we want to do is raise 1.001583 to the power of 12*2 = 24 to get (1.001583) 24 = 1.038692 (rounded to 6 decimal places). Last step is to multiply 5000 by 1.038692 to get 5000*1.038692 = 5193.46.

We see that with this option, you would end up with \$5,193.46 at the end of two years, and you would have made \$193.46 in interest.

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