When we borrow a certain sum of money over a period of time, we agree that we will pay it back, along with a fee, known as the interest owed. Similarly, when we invest a sum of money in a savings account, the account earns us interest. This lesson will show you how to calculate a certain type of interest called simple interest.

## What is Simple Interest?

**Simple interest** is a type of interest that is applied to the amount borrowed or invested for the entire duration of the loan, without taking any other factors into account, such as past interest (paid or charged) or any other financial considerations. Simple interest is generally applied to short-term loans, usually one year or less, that are administered by financial companies. The same applies to money invested for a similarly short period of time.The simple interest rate is a ratio and is typically expressed as a percentage.

It plays an important role in determining the amount of interest on a loan or investment. The amount of interest charged or earned depends on three important quantities that we will examine next.

## Simple Interest Formula

Sarah needs to borrow $2,000 in order to buy furniture. She’s approved for two different loans. Loan One allows her to borrow $2,000 now, provided that she pay off the loan by returning $2,200 exactly one year from the day that she borrows the money. Loan Two offers her $2,000 upfront as well, with a similar loan period of one year, at an annual interest rate of 7%.

Which is the better deal for Sarah?The amount borrowed or invested is called the **principal**. Using the example above, when Sarah borrows $2,000 to buy furniture, we say that the principal is $2,000.It’s customary for financial institutions to quote a quantity called the **interest rate** as a percentage. This interest rate represents a ratio of the principal borrowed or invested. Typically, this interest rate is given as a percentage per year, in which case it is called the **annual interest rate**. For example, if we borrow $100 at an annual rate of 5%, it means that we will be charged 5% of $100 at the end of the year, or $5.

The **loan period or duration** is the time that the principal amount is either borrowed or invested. It is usually given in years, but in some cases, it may be quoted in months or even days. If that is the case, we need to perform a conversion from a period given in months or days, into years.The simple interest formula allows us to calculate *I*, which is the interest earned or charged on a loan. According to this formula, the amount of interest is given by *I = Prt*, where *P* is the principal, *r* is the annual interest rate in decimal form, and *t* is the loan period expressed in years.

## Example

The second offer that Sarah has received is to borrow a principal amount *P* = $2,000, at an annual rate of 7%, over *t* = 1 year.

The rate *r* must be converted from a percentage into decimal form, which means that we divide the percentage value 7% by 100 to get *r* = 0.07.We now calculate the amount of interest Sarah would be charged if she accepts the loan offer just described:*I = Prt* = (2,000)(0.07)(1) = $140.Following our example, we determined that if Sarah accepts the second loan, the interest that she will owe the bank is $140. So, how much would Sarah have to pay the bank in order to pay off her debt? She would have to pay back the money she borrowed, or the principal, which is $2,000, and she would have to pay the bank the interest we calculated, in which *I* = $140.

Thus, she will owe the bank $2,000 + $140, which equals $2,140. We note that this is still less than the $2,200 Sarah would have to pay if she accepts Loan One. Obviously, Loan Two is the better choice.

## Future of Maturity Value

The total amount we would need to pay back when we take a loan is called the **future value** of the loan. Another name for future value is **maturity value**. The future value, *A*, of a loan is given by the equation *A = P + I*.

When we invest a principal amount (*P*), the future value (*A*) will represent the total amount we will have at the end of the loan period after simple interest is applied.Using the interest formula *I = Prt*, we can derive a formula for the future value, since *A = P + Prt*, or after factoring out *P* on the right hand side, *A = P(1 + rt)*.**Example 1**Lilya borrows $300 from her local bank at an annual interest rate of 3.

25% to be paid back in six months. How much interest will she pay at the end of the loan?**Solution**We are given the following values: the principal amount, *P* = 300, the annual interest rate, *r* = 3.25%, and the loan period *t* in years. The loan period is six months, so we have *t* = ½, calculated by dividing six months into 12 months, the number of months in a year.*t* = loan period in months / 12 months in a year = 6 / 12 = 1/2The value of *r* is determined by converting it from a percentage into decimal form:*r* = annual rate as a percentage / 100 = 3.25 / 100 = 0.

0325We now use the interest formula, *I = Prt*, to determine the interest to be paid at the end of the loan:*I = Prt* = (300) (0.0325) (1/2) = 4.875This value will be rounded to $4.88 by the bank, as it is more profitable to them. Therefore, Lilya will have to pay back $4.88 in interest.

**Example 2**Find the maturity value for a simple interest loan of $4,000 at an annual interest rate of 10.5% to be repaid in 105 days. It is common practice for banks to assume there are 360 days in a year.**Solution**We are given the principal amount, *P* = 4,000. The loan period, in years, is calculated by dividing 105 days into 360 days, which gives us *t* = 21/72.*t* = 105 / 360 = 21/72The annual interest rate is 10.

5%, or in decimal form, *r* = 0.105.The maturity value for the loan is given by the formula *A = P(1 + rt)*. Hence, we have:

Therefore, the future value of this loan is $4,122.50.**Example 3**Tom invests $3,000 in a savings account.

After one year, the account has earned $33.00 in interest. What is the simple annual interest rate?**Solution**We are given the principal amount, *P* = $3,000, the interest, *I* = 33.00, and the loan period in years is *t* = 1.

The interest rate is determined from the simple interest formula, *I = Prt*, solving for *r*:

Therefore, the annual simple interest rate is 1.1%.

## Lesson Summary

**Simple interest** is usually applied to short-term loans, where a sum of money, called the **principal amount**, is borrowed. At the end of the loan period, interest is applied to the principal amount, and the loan is paid off by paying back the principal amount borrowed, in addition to the interest incurred.

The amount of interest is proportional to the principal amount, the **annual interest rate**, which is given as a percentage per year, and the duration of the **loan period**, which is the time the principal amount is either borrowed or invested.The simple interest formula is *I = Prt*.The future value is determined by the formula *A = P(1 + rt)*.

## Key Terms

**Simple Interest**: amount of interest applied to a short-term loan applied to the loan amount for the duration of the loan**Principal Amount**: the amount borrowed or invested**Annual Interest Rate**: an interest rate given as a percentage per year**Loan Period**: the amount of time until the loan should be payed back**Future Value**: the amount to be paid back, including the principal amount and interest, at the end of the loan period

## Learning Outcomes

After you’ve finished, you should be able to:

- Explain what simple interest is
- Calculate simple interest and future value using the correct formulas