## Using the Simple Interest Formula

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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.

### Definition: 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

where

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

Find the simple interest earned after \(3\) years on \(\text{\$500}\) at an interest rate of \(\text{6%.}\)

### Solution

Organize the given information in a list.

\(\begin{array}{ccc}\hfill I& =& ?\hfill \\ \hfill P& =& \text{\$500}\hfill \\ \hfill r& =& \text{6%}\hfill \\ \hfill t& =& \text{3 years}\hfill \end{array}\)

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. | \(I=\left(500\right)\left(0.06\right)\left(3\right)\) |

Simplify. | \(I=90\) |

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

In 3 years the money earned 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

Find the principal invested if \(\text{\$178}\) interest was earned in \(2\) years at an interest rate of \(\text{4%.}\)

### Solution

Organize the given information in a list.

\(\begin{array}{ccc}\hfill I& =& \text{\$178}\hfill \\ \hfill P& =& ?\hfill \\ \hfill r& =& \text{4%}\hfill \\ \hfill t& =& \text{2 years}\hfill \end{array}\)

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

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

Substitute the given information. | \(178=P\left(0.04\right)\left(2\right)\) |

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? | |

\(I=Prt\) | |

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

\(178=178✓\) | |

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

Now we will solve for the rate of interest.

## Example

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

### Solution

Organize the given information.

\(\begin{array}{ccc}\hfill I& =& \text{\$3,772}\hfill \\ \hfill P& =& \text{\$8,200}\hfill \\ \hfill r& =& ?\hfill \\ \hfill t& =& \text{4 years}\hfill \end{array}\)

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

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

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

Multiply. | \(3,772=32,800r\) |

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

Simplify. | \(0.115=r\) |

Write as a percent. | \(\text{11.5%}=r\) |

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

\(I=Prt\) | |

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

\(3,772=3,772✓\) | |

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

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