Mathematics in Contemporary Society: A Work in Progress

Interest is the term used for the fee charged to borrow money in a loan or the return on a savings account or other investment. Simple interest is calculated as a percentage of the starting amount of the loan or investment.

For example, suppose a college student borrowed $2,000 from their sibling to pay for tuition. They will pay it back after one year with 3% interest. The amount of interest charged will be 3% of $2,000, or \(0.03\cdot2\,000=$60\text{.}\) At the end of the year, the student should pay their sibling the $2,000 borrowed plus $60 in interest.

Interest rates are most-commonly given as annual rates. So, if this student were to repay the money after 3 years, they would owe \(3\cdot60=$180\) in interest. If they were to repay the money after 6 months, which is half of a year, they would owe \(\frac{1}{2}\cdot60=$30\) in interest.

We can generalize to obtain the simple interest formula:

With our financial formulas we measure time in years, but the time period for a loan or investment may be given in months or days. In this case, we can still use the simple interest formula, but we need to convert the time period to years before substituting into the formula. For example, since there are 12 months in a year, a time period of 1 month would be \(t=\frac{1}{12}\text{,}\) and a time period of 6 months would be \(t=\frac{6}{12}=\frac{1}{2}\text{.}\) Similarly, since there are 365 days in a year, a time period of 1 day would be \(t=\frac{1}{365}\) and a time period of 31 days would be \(t=\frac{31}{365}\text{.}\)

We use the term future value to refer to the total value of an account at maturity or the total amount repaid at the end of a simple interest loan. It is the sum of the principal (or present value) and the interest. We typically denote it with the variable \(A\) for amount.

Which simple interest future value formula should I use? For the most part, you may use any of the three formulas. However, there are some situations where you can make your work easier by selecting the appropriate formula:

An amortized loan is a loan that is paid off in equal payments over a specified period of time. In each payment, a portion goes toward the interest and the rest goes toward paying down the principal. The interest portion of each payment is calculated using the simple interest formula, and the rest of the payment goes toward reducing the principal or amount owed.

Amortized loans are commonly used for car loans, mortgages, and student loans. The payment amount is calculated so that the loan will be completely paid off at the end of the loan term. This means that the payment amount is based on the original loan amount, the interest rate, and the length of the loan term. We will cover how to calculate the payment amount for an amortized loan in Section 3.5, but for now we will focus on how to calculate the interest and principal portions of each payment and the outstanding balance after each payment.

For each subsequent payment, the interest is calculated on the remaining balance. This means that the interest portion of each payment will decrease over time as the outstanding balance decreases, while the principal portion will increase over time. By the end of the loan term, the final payment will be mostly principal with very little interest.

An amortization table is a useful tool for keeping track of the interest and principal portions of each payment and the outstanding balance after each payment. It can help you see how much of each payment goes toward interest and how much goes toward paying down the principal over the life of the loan.

A spreadsheet program can be used to create an amortization table for the entire loan term, which can be very helpful for keeping track of the payments and the outstanding balance over time. Once the first few rows of the table are completed, you can use the fill-down feature to automatically fill in the rest of the table for the remaining payments. A sample amortization table

¹

docs.google.com/spreadsheets/d/1KWOdAlZzCc-_GcjmdUnZBuJDNvoFuJrVmib_M9hOQ60/edit?usp=sharing

is provided as an example.

In the sample amortization table, the ending balance after 30 years was $2.69 rather than $0 because of rounding. The interest and principal portions are rounded to the nerest cent for each payment. In practice, the final payment would be adjusted to account for this rounding error so that the loan would be completely paid off at the end of the loan term.