# 6.4 Solve Simple Interest Applications

The topics covered in this section are:

## 6.4.1 Use the Simple Interest Formula

Do you know that banks pay you to let them keep your money? The money you put in the bank is called the principal, $P$, and the bank pays you interest, $I$. The interest is computed as a certain percent of the principal; called the rate of interest, $r$. The rate of interest is usually expressed as a percent per year, and is calculated by using the decimal equivalent of the percent. The variable for time, $t$, represents the number of years the money is left in the account.

### SIMPLE INTEREST

If an amount of money, $P$, the principal, is invested for a period of $t$ years at an annual interest rate $r$, the amount of interest, $I$, earned is

$I=Prt$

where

$I= \mathrm{interest}$

$P= \mathrm{principal}$

$r= \mathrm{rate}$

$t= \mathrm{time}$

Interest earned according to this formula is called simple interest.

The formula we use to calculate simple interest is $I=Prt$. To use the simple interest formula we substitute in the values for variables that are given, and then solve for the unknown variable. It may be helpful to organize the information by listing all four variables and filling in the given information.

#### Example 1

Find the simple interest earned after $3$ years on $\$500$at an interest rate of$6 \%$. Solution Organize the given information in a list.$I=?P= \$500$

$r=6 \%$

$t=3$ years

We will use the simple interest formula to find the interest.

In the next example, we will use the simple interest formula to find the principal.

#### Example 2

Find the principal invested if $\$178$interest was earned in$2$years at an interest rate of$4 \%$. Solution Organize the given information in a list.$I= \$178$

$P= ?$

$r=4 \%$

$t=2$ years

We will use the simple interest formula to find the principal.

Now we will solve for the rate of interest.

#### Example 3

Find the rate if a principal of $\$ 8,200$earned$ \$3,772$ interest in $4$ years.

Solution

Organize the given information.

$I= \$ 3,772P= \$8,200$

$r=?$

$t=4$ years

We will use the simple interest formula to find the rate.

## 6.4.2 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 start by solving a simple interest application to find the interest.

#### Example 4

Nathaly deposited $\$ 12,500$in her bank account where it will earn$4 \%$interest. How much interest will Nathaly earn in$5$years? Solution We are asked to find the Interest,$I$. Organize the given information in a list.$I= ?P= \$12,500$

$r=4 \%$

$t=5$ years

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 5

Loren lent his brother $\$3,000$to help him buy a car. In$4$years his brother paid him back the$ \$3,000$ plus $\$ 660$in interest. What was the rate of interest? Solution We are asked to find the rate of interest,$r$. Organize the given information.$I= 660P= \$3,000$

$r=?$

$t=4$ years

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.

Eduardo noticed that his new car loan papers stated that with an interest rate of $7.5 \%$, he would pay $\$6,596.25$in interest over$5$years. How much did he borrow to pay for his car? Solution We are asked to find the principal,$P$. Organize the given information.$I= 6,596.25P= ?r=7.5 \%t=5$years 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 7 Caroline got$ \$900$ as graduation gifts and invested it in a $10$-month certificate of deposit that earned $2.1 \%$ interest. How much interest did this investment earn?

Solution

We are asked to find the interest, $I$.

Organize the given information.

$I= ?$

$P= \$ 900r=2.1 \%t=10\$ months