Problem 129 - Elementary Algebra

Question

In 10 years, a bank account that paid 5.25\% earned \(\$ 18,375\) interest. What was the principal of the account?

Step-by-Step Solution

Answer

The principal of the account was \ \$35,000 \ or \(\$35,000\).

Step 1: Identify the Given Information

Interest earned in 10 years is \( I = \$18,375 \). The annual interest rate is \( r = 5.25\% = 0.0525 \). The time period is \( t = 10 \) years.

Step 2: Recall the Simple Interest Formula

The formula to calculate simple interest is \[ I = P \times r \times t \], where \( I \) is the interest earned, \( P \) is the principal amount, \( r \) is the annual interest rate, and \( t \) is the time period in years.

Step 3: Rearrange the Formula to Solve for Principal

Rearrange the simple interest formula to solve for the principal amount \( P \): \[ P = \frac{I}{r \times t} \]

Step 4: Plug in the Values

Substitute the given values into the rearranged formula: \[ P = \frac{18375}{0.0525 \times 10} \]

Step 5: Calculate the Principal

Perform the calculations: \[ P = \frac{18375}{0.525} = 35000 \]. Therefore, the principal of the account was \ \$35,000 \.

Key Concepts

Simple Interest Formula

Simple interest is a straightforward way of calculating the interest earned or paid on a principal amount over a specific period. The formula for calculating simple interest is: \( I = P \times r \times t \)Here:

• \( I \) is the interest earned or paid.
• \( P \) is the principal amount (initial investment or loan).
• \( r \) is the annual interest rate (as a decimal).
• \( t \) is the time period in years.

Using this formula helps you determine how much interest you'll earn on an investment or have to pay on a loan over time.

Interest Rate

The interest rate is the percentage of the principal that is paid as interest for a specific time period. In simple interest calculations, it is usually expressed as an annual rate and denoted by \( r \). To use the interest rate in the simple interest formula, you need to convert it from a percentage to a decimal. For example, if the interest rate is 5.25%, convert it by dividing by 100: 5.25% = 0.0525.

The interest rate can differ based on various factors, including economic conditions and the lender's policies. Always make sure to use the correct decimal form of the interest rate when performing your calculations.

Principal Amount

The principal amount, denoted by \( P \), is the initial sum of money that is either invested or borrowed before any interest is applied. In the given exercise, the objective is to find out the principal amount based on the total interest earned over time. To determine the principal, you can rearrange the simple interest formula to: \( P = \frac{I}{r \times t} \).

For example, if you know the interest earned is \( \$18,375 \), the annual interest rate is 5.25% (or 0.0525 in decimal form), and the time period is 10 years, you would substitute these values into the formula to find the principal: \( P = \frac{18375}{0.0525 \times 10} = 35000 \). This means the original principal amount was \( \$35,000 \).

Time Period

The time period \( t \) is the duration over which the interest is calculated. It is typically measured in years for simple interest calculations. In our example, the time period is 10 years. The time period directly influences the amount of interest earned or paid. The longer the time period, the more interest accumulates. This relationship is reflected in the simple interest formula:

\( I = P \times r \times t \).It's important to ensure the time period aligns with the annual interest rate. If the time period is given in months or days, you must convert it to years to use it correctly in the formula. For instance, if the time period were 6 months, you would convert it to years by dividing by 12: 6 months = 0.5 years.