## Solving Simple Interest Applications

Contents

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 tutorial. 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

Nathaly deposited \(\text{\$12,500}\) in her bank account where it will earn \(\text{4%}\) 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.

\(\begin{array}{ccc}\hfill I& =& ?\hfill \\ \hfill P& =& \text{\$12,500}\hfill \\ \hfill r& =& \text{4%}\hfill \\ \hfill t& =& \text{5 years}\hfill \end{array}\)

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 \$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

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?

### Solution

We are asked to find the rate of interest, \(r.\)

Organize the given information.

\(\begin{array}{ccc}\hfill I& =& 660\hfill \\ \hfill P& =& \text{\$3,000}\hfill \\ \hfill r& =& ?\hfill \\ \hfill t& =& \text{4 years}\hfill \end{array}\)

Write the formula. | \(I=Prt\) |

Substitute the given information. | \(660=\left(3,000\right)r\left(4\right)\) |

Multiply. | \(660=\left(12,000\right)r\) |

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

Simplify. | \(0.055=r\) |

Change to percent form. | \(\text{5.5%}=r\) |

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

\(I=Prt\) | |

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

\(660=660✓\) | |

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

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?

### Solution

We are asked to find the principal, \(P.\)

Organize the given information.

\(\begin{array}{ccc}\hfill I& =& 6,596.25\hfill \\ \hfill P& =& ?\hfill \\ \hfill r& =& \text{7.5%}\hfill \\ \hfill t& =& \text{5 years}\hfill \end{array}\)

Write the formula. | \(I=Prt\) |

Substitute the given information. | \(6,596.25=P\left(0.075\right)\left(5\right)\) |

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? | |

\(I=Prt\) | |

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

\(6,596.25=6,596.25✓\) | |

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

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?

### Solution

We are asked to find the interest, \(I.\)

Organize the given information.

\(\begin{array}{ccc}\hfill I& =& ?\hfill \\ \hfill P& =& \text{\$900}\hfill \\ \hfill r& =& \text{2.1%}\hfill \\ \hfill t& =& \text{10 months}\hfill \end{array}\)

Write the formula. | \(I=Prt\) |

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

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. |