## Applications With Simple Interest

### Learning Outcomes

• Solve simple interest applications

Applications with simple interest usually involve either investing money or borrowing money. To solve these applications, we continue to use the same strategy for applications that we have used earlier in this chapter. The only difference is that in place of translating to get an equation, we can use the simple interest formula. We will provide examples of how to find interest earned, calculate the rate of interest, and how to find the principal given a rate and the interest earned.

### Calculating Interest Earned

We will start by solving a simple interest application to find the interest.

### example

Nathalie deposited $\text{\12,500}$ in her bank account where it will earn $\text{4%}$ interest. How much interest will Nathalie earn in $5$ years?

Solution
We are asked to find the Interest, $I$.
Organize the given information in a list.

Interest: $I=$ unknown

Principal: $P=12,500$

Rate: $r=4\%=0.04$

Time: $t=5$ years

 Write the formula. $I=Prt$ Substitute the given information. $I=\left(12,500\right)\left(0.04\right)\left(5\right)$ Simplify. $I=2,500$ Check your answer. Is $\text{\2,500}$ a reasonable interest on $\text{\12,500}$ over $5$ years? At $4\text{%}$ interest per year, in $5$ years the interest would be $20\text{%}$ of the principal. Is $20\text{%}$ of $\text{\12,500}$ equal to $\text{\2,500}$? Yes. Write a complete sentence that answers the question. The interest is $\text{\2,500}$.

### Calculating Rate

There may be times when you know the amount of interest earned on a given principal over a certain length of time, but you don’t know the rate. For instance, this might happen when family members lend or borrow money among themselves instead of dealing with a bank. In the next example, we’ll show how to solve for the rate.

### example

Loren lent his brother $\text{\3,000}$ to help him buy a car. In $\text{4 years}$ his brother paid him back the $\text{\3,000}$ plus $\text{\660}$ in interest. What was the rate of interest?

### try it

In the next video we use the simple interest formula to find the rate of interest given an amount of money borrowed and the amount if interest paid.

### Calculating Principal

There may be times when you take a loan for a large purchase and the amount of the principal is not clear. This might happen, for instance, in making a car purchase when the dealer adds the cost of a warranty to the price of the car. In the next example, we will solve a simple interest application for the principal.

### example

Eduardo noticed that his new car loan papers stated that with an interest rate of $\text{7.5%}$, he would pay $\text{\6,596.25}$ in interest over $5$ years. How much did he borrow to pay for his car?

### TRY IT

In the simple interest formula, the rate of interest is given as an annual rate, the rate for one year. So the units of time must be in years. If the time is given in months, we convert it to years.

### example

Caroline got $\text{\900}$ as graduation gifts and invested it in a $\text{10-month}$ certificate of deposit that earned $\text{2.1%}$ interest. How much interest did this investment earn?

### Example

A friend asks to borrow $240$, offering to repay you $250$ in 1 month. What annual interest rate is this equivalent to?

The example video that follows shows how to determine the annual simple interest rate.

Another application of interest rate is calculating Treasury Notes.

### Example

Treasury Notes (T-notes) are bonds issued by the federal government to cover its expenses. Suppose you obtain a $1,000$ T-note with a  $4\%$ annual rate, with a maturity in  $2$ years. How much interest will you earn?

in the following video we show an example of how to calculate the amount of interest earned on a treasury note.

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