**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.

Write the formula. | $I=Prt$ |

Substitute the given information. Remember to write the percent in decimal form. | $I=(500)(0.06)(3)$ |

Simplify. | $I=90$ |

Check your answer. Is $ \$90$ a reasonable interest earned on $ \$ 500$ in $3$ years? | |

In $3$ years the money earned is $18 \%$. If we rounded to $20 \%$, the interest would have been $500(0.20)$ or $ \$100$. Yes, $ \$90$ is reasonable. | |

Write a complete sentence that answers the question. | The simple interest is $ \$ 90$ |

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.

Write the formula. | $I=Prt$ |

Substitute the given information. Remember to write the percent in decimal form. | $178=P(0.04)(2)$ |

Divide | $\frac{178}{0.08} = \frac{0.08P}{0.08}$ |

Simplify. | $2,225=P$ |

Check your answer. Is it reasonable that $ \$ 2,225$ would earn $ \$178$ in $2$ years? | |

$I+Prt$ | |

$178 \stackrel{?}{=} 2,225 (0.04)(2)$ | |

$178=178$✓ | |

Write a complete sentence that answers the question. | The principal is $ \$2,225$. |

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.

$I= \$ 3,772$

$P= \$ 8,200$

$r=?$

$t=4$ years

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

Write the formula. | $I=Prt$ |

Substitute the given information. | $3,772=8,200r(4)$ |

Multiply. | $3,772=32,800r$ |

Divide. | $ \frac{3,772}{32,800} = \frac{32,800r}{32,800}$ |

Simplify. | $0.115=r$ |

Write as a percent. | $11.5 \% =r$ |

Check your answer. Is $11.5 \%$ a reasonable rate if $ \$3,772$ was earned in $4$ years? | |

$I=Prt$ | |

$3,772 \stackrel{?}{=} 8,200(0.115)(4)$ | |

$3,772=3,722$✓ | |

Write a complete sentence that answers the question. | The rate was $11.5 \%$. |

**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

Write the formula. | $I=Prt$ |

Substitute the given information. | $I=(12,500)(0.04)(5)$ |

Simplify. | $I=2,500$ |

Check your answer. Is $ \$2,500$ a reasonable interest on $ \$12,500$ over $5$ years? | |

At $4 \%$ interest per year, in $5$ years the interest would be $20 \%$ of the principal. Is $20 \%$ of $ \$12,500$ equal to $ \$2,500?$ Yes. | |

Write a complete sentence that answers the question. | The interest is $ \$2,500$. |

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.

$I= 660$

$P= \$ 3,000$

$r=?$

$t=4$ years

Write the formula. | $I=Prt$ |

Substitute the given information. | $660=(3,300)r(4)$ |

Multiply. | $660=(12,000)r$ |

Divide. | $\frac{660}{12,000} = \frac{(12,000)r}{12,000}$ |

Simplify. | $0.055=r$ |

Change to percent form. | $5.5 \% =r$ |

Check your answer. Is $5.5 \%$ a reasonable interest rate to pay your brother? | |

$I=Prt$ | |

$660 \stackrel{?}{=} (3,000)(0.055)(4)$ | |

$660=660$✓ | |

Write a complete sentence that answers the question. | The rate of interest was $5.5 \%$. |

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 6**

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.

$I= 6,596.25$

$P= ?$

$r=7.5 \%$

$t=5$ years

Write the formula. | $I=Prt$ |

Substitute the given information. | $6,596.25=P(0.075)(5)$ |

Multiply. | $6,596.25=0.375P$ |

Divide. | $\frac{6,596.25}{0.375} = \frac{0.375P}{0.375}$ |

Simplify. | $17,590=P$ |

Check your answer. Is $ \$ 17,590$ a reasonable amount to borrow to buy a car? | |

$I=Prt$ | |

$6,596.25 \stackrel{?}{=} (17,590)(0.075)(5)$ | |

$6,596.25=6,596.25$ ✓ | |

Write a complete sentence that answers the question. | The amount borrowed was $ \$17,590$ |

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.

$I= ?$

$P= \$ 900$

$r=2.1 \%$

$t=10$ months

Write the formula. | $I=Prt$ |

Substitute the given information, converting $10$ months to $\frac{10}{12}$ of a year. | $I= \$900(0.021)( \frac{10}{12} )$ |

Multiply. | $I=15.75$$ |

Check your answer. Is $ \$15.75$ a reasonable amount of interest? | |

If Caroline had invested the $ \$900$ for a full year at $2 \%$ interest, the amount of interest would have been $ \$18$. Yes, $ \$15.75$ is reasonable. | |

Write a complete sentence that answers the question. | The interest earned was $ \$15.75$. |

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*Marecek, L., Anthony-Smith, M., & Mathis, A. H. (2020). Use the Language of Algebra. In Prealgebra 2e. OpenStax. https://openstax.org/books/prealgebra-2e/pages/6-4-solve-simple-interest-applications*.*License: CC BY 4.0. Access for free at https://openstax.org/books/prealgebra-2e/pages/1-introduction*