The loan statement for Margot’s family’s car arrived in the mail. Margot noticed that the amount due was $175.67 if paid by October 25. However, the loan originator noted that if payment arrived after this date the amount due would be $186.21. What simple interest rate does the loan originator charge on the loan if it is late?

In this concept, you will learn to use the simple interest equation to find an interest rate.

### Using the Simple Interest Equation to Find Interest Rates

**Interest** is a charge for money that is borrowed. When you borrow money, you pay the lender interest for the privilege of borrowing the money. On the flipside, when you deposit money into a saving account at a bank, the bank pays you interest since the bank is borrowing money from you. So interest can be something either that you have to pay or that is paid to you.

The amount of money that is invested or borrowed is called the **principal**. If you borrow $500, this is the principal. It is the initial amount before any interest is added on.

The **rate of interest** is the percent charged or earned. You also have to consider the time in years that the money is borrowed or deposited when computing interest.

You can calculate interest by using this equation:

\begin{align*}\begin{array}{rcl} \text{Interest} &=& \text{Principal} \times \text{rate} \times \text{time} \\ I &=& Prt \end{array}\end{align*}

If you know the amount of interest, the principal, and the time in years, you can find the interest rate.

Let’s look at an example.

If you deposit $2,000 at USA Savings Bank, at the end of two years you will have received $240 in simple interest. What is the interest rate at USA Savings Bank?

\begin{align*}\begin{array}{rcl} I &=& Prt \\ $240 & = & $2, 000 \times r \times 2 \qquad \ \leftarrow \qquad \text{Substitute values}. \\ $240 & = & $4, 000 \times r \qquad \qquad \leftarrow \qquad \text{Simplify}. \\ 0.06 & = & r \qquad \qquad \qquad \qquad \leftarrow \qquad \text{Solve for} \ r. \end{array}\end{align*}

Change 0.06 to the percent 6%.

The answer is the interest rate is 6%.

Let’s take a look at another example.

If you borrow $3,600 from USA Savings Bank for 18 months, at the end of the 18 months you will repay $4,059 to the bank. What is the interest rate for this loan?

First, note that the amount to be repaid includes the principal plus the interest.

Next, subtract the principal from the amount to be repaid to find the amount of the interest.

\begin{align*}$4, 059 - $3, 600 = $459\end{align*}

Then, since there are 12 months in a year, 18 months is \begin{align*}18 \div 12\end{align*} or \begin{align*}1 \frac{1}{2}\end{align*}.

\begin{align*}\begin{array}{rcl} I & = & Prt \\ $459 & = & $3,600 \times r \times 1 \frac{1}{2}\qquad \leftarrow \quad \text{Substitute values}. \\ $459 & = & $5,400 \times r \qquad \qquad \ \ \ \leftarrow \quad \text{Simplify}. \\ 0.085 & = & r \qquad \qquad \qquad \qquad \quad \leftarrow \quad \text{Solve for} \ r. \end{array}\end{align*}

Finally, change 0.085 to the percent 8.5%

The answer is the interest rate is 8.5%.

### Examples

#### Example 1

Earlier, you were given a problem about Margot’s family’s car loan statement.

The loan originator charges interest if the loan is not paid by a certain date. If paid by October 25, the amount owed is $175.67. If paid after this date the amount owed is $186.21. What simple interest rate does the loan originator charge on the loan if it is late?

First, change 1 month to years: \begin{align*}\frac{1}{12} = 0.0833\end{align*}

Next, find the interest that will be owed by subtracting: \begin{align*}$186.21 - $175.67 = $10.54\end{align*}

Then, plug the values into the equation.

\begin{align*}\begin{array}{rcl} I &=& Prt \\ $10.54 &=& $175.67 \times r \times 0.0833 \qquad \qquad \leftarrow \quad \text{Substitute values}. \\ $10.54 &=& $14.63 \times r \qquad \qquad \qquad \qquad \ \ \leftarrow \quad \text{Simplify}. \\ 0.7204 &=& r \qquad \qquad \qquad \qquad \qquad \qquad \ \ \leftarrow \quad \text{Solve for} \ r. \end{array}\end{align*}

Finally, change 0.7204 to the percent 72.04%.

The answer is the interest rate is 72.04%.

#### Example 2

Wanda borrowed $5,000.00 from the bank. At the end of three years, she had paid $450.00 in interest. What is Wanda’s interest rate?

To figure this out, you can use the formula for finding simple interest.

\begin{align*}I = Prt\end{align*}

First, fill in what you know.

\begin{align*}450 = 5,000 \times r \times 3\end{align*}

Next, you are trying to figure out the rate, so that is your variable. \begin{align*}450 = 15,000 r\end{align*}

\begin{align*}\frac{450}{15,000} = 0.03 \ \text{or} \ 3\%\end{align*}

The answer is Wanda’s interest rate is 3%.

#### Example 3

Jesse borrowed $500.00. At the end of the year he paid back $50.00 in interest. What was the interest rate?

\begin{align*}\begin{array}{rcl} I & = & Prt \\ $ 50 & = & $ 500 \times r \times 1 \qquad \leftarrow \quad \text{Substitute values}. \\ $ 50 & = & $ 500 \times r \qquad \quad \ \ \ \leftarrow \quad \text{Simplify}. \\ $ 0.10 & = & r \qquad \qquad \qquad \quad \leftarrow \quad \text{Solve for} \ r. \end{array}\end{align*}

Change 0.10 to the percent 10%.

The answer is the interest rate is 10%.

#### Example 4

Karen earned $200.00 in two years of simple interest on her initial investment of $400.00. What was the interest rate?

\begin{align*}\begin{array}{rcl} I & = & Prt \\ $200 & = & $400 \times r \times 2 \qquad \qquad \leftarrow \quad \text{Substitute values}. \\ $200 & = & $800 \times r \qquad \qquad \quad \ \ \ \leftarrow \quad \text{Simplify}. \\ 0.25 & = & r \qquad \qquad \qquad \qquad \quad \leftarrow \quad \text{Solve for} \ r. \end{array}\end{align*}

Change 0.25 to the percent 25%.

The answer is the interest rate is 25%.

#### Example 5

Principal: $5,600; Time: 9 months; Simple interest: $357

First, change 9 months to years: \begin{align*}\frac{9}{12} = \frac{3}{4} = 0.75\end{align*}

\begin{align*}\begin{array}{rcl} I & = & Prt \\ $357 & = & $5,600 \times r \times 0.75 \qquad \quad \leftarrow \quad \text{Substitute values}. \\ $357 & = & $4, 200 \times r \qquad \qquad \qquad \leftarrow \quad \text{Simplify}. \\ 0.085 & = & r \qquad \qquad \qquad \qquad \qquad \ \leftarrow \quad \text{Solve for} \ r. \end{array}\end{align*}

Change 0.085 to the percent 8.5%.

The answer is the interest rate is 8.5%.

### Review

Find the simple interest on each amount.

- $500.00 at 4% for 2 years
- $200.00 at 5% for 3 years
- $5,000.00 at 2% for 2 years
- $600.00 at 10% for 1 year
- $1,200.00 at 4% for 2 years
- $1,500.00 at 3% for 1 year
- $2,300.00 at 2% for 2 years
- $500.00 at 4% for 2 years
- $2,500.00 at 5% for 5 years
- $1,500.00 at 11% for 2 years
- $3,500 at 3% for 5 years
- $3,500 at 4% for 15 years
- $13,000 at 4.5% for 6 years
- $23,000 at 3.5% for 10 years
- $50,000 at 2.5% for 20 years

### Review (Answers)

To see the Review answers, open this PDF file and look for section 6.20.