Simple interest is when the interest on a loan or investment is calculated only on the amount initially invested or loaned. This is different from compound interest, where interest is calculated on on the initial amount and on any interest earned. As you will see in the examples below, the simple interest formula can be used to calculate the interest earned, the total amount, and other values depending on the problem.

## Examples of finding the interest earned with the simple interest formula

In many simple interest problems, you will be finding the total interest earned over a set period, which is represented as \(I\). The formula for this is:

Let’s use an example to see how this formula works. Remember that in the formula, the principal \(P\) is the initial amount invested.

### Example

A 2-year loan of $500 is made with 4% simple interest. Find the interest earned.

### Solution

Always take a moment to identify the values given in the problem. Here we are given:

- Time is 2 years: \(t = 2\)
- Initial amount is $500: \(P = 500\)
- The rate is 4%. Write this as a decimal: \(r = 0.04\)

Now apply the formula:

\(\begin{align}I &= Prt \\ &= 500(0.04)(2) \\ &= \bbox[border: 1px solid black; padding: 2px]{40}\end{align}\)

**Answer**: The interest earned is $40.

In this example, the time given was in years, just as in the formula. But what if you are only given a number of months? Let’s use another example to see how this might be different.

### Example

A total of $1,200 is invested at a simple interest rate of 6% for 4 months. How much interest is earned on this investment?

### Solution

Before we can apply the formula, we will need to write the time of 4 months in terms of years. Since there are 12 months in a year:

\(\begin{align}t &= \dfrac{4}{12} \\ &= \dfrac{1}{3}\end{align}\)

With this adjusted to years, we can now apply the formula with \(P = 1200\) and \(r = 0.06\).

\(\begin{align}I &= Prt \\ &= 1200(0.06)\left(\dfrac{1}{3}\right) \\ &= \bbox[border: 1px solid black; padding: 2px]{24}\end{align}\)

**Answer**: The interest earned is $24.

If you hadn’t converted here, you would have found the interest for 4 years, which would be much higher. So, always make sure to check that the time is in years before applying the formula.

Important!The time must be in years to apply the simple interest formula. If you are given months, use a fraction to represent it as years.

Another type of problem you might run into when working with simple interest is finding the total amount owed or the total value of an investment after a given amount of time. This is known as the future value, and can be calculated in a couple of different ways.

## Finding the future value for simple interest

One way to calculate the future value would be to just find the interest and then add it to the principal. The quicker method however, is to use the following formula.

You know to use this formula when you are asked questions like “what is the total amount to be repaid” or “what is the value of the investment” -anything that seems to refer to the overall total after interest is considered.

### Example

A business takes out a simple interest loan of $10,000 at a rate of 7.5%. What is the total amount the business will repay if the loan is for 8 years?

### Solution

The total amount they will repay is the future value, \(A\). We are also given that:

- \(t = 8\)
- \(r = 0.075\)
- \(P = 10\,000\)

Using the simple interest formula for future value:

\(\begin{align}A &= P(1 + rt)\\ &= 10\,000(1 + 0.075(8)) \\ &= \bbox[border: 1px solid black; padding: 2px]{16\,000}\end{align}\)

**Answer**: The business will pay back a total of $16,000.

This may seem high, but remember that in the context of a loan, interest is really just a fee for borrowing the money. The larger the interest rate and the longer the time period, the more expensive the loan.

Also note that you could calculate this by first finding the interest, I = Prt = 10000(0.075(8)) = $6000, and adding it to the principal of $10000. The final answer is the same using either method.

## Continue your study of interest

Now that you have studied the simple interest formula, you can learn the more advanced idea of compound interest. Most savings accounts, credit cards, and loans are based on compound instead of simple interest. You can review this idea here:

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