Question

For Exercises \(11-14\), solve. Sharice notes that her savings account earned \(\$ 21.28\) in 1 year. If the account earns \(5.5 \%\) simple interest, what was her principal?

Short answer

Answer: \$386.91

Step-by-step solution

Identify the given values

We are given the following information: - Simple Interest (Interest earned in a year): $21.28 - Rate (Interest rate per year as a decimal): \(5.5\% = 0.055\) - Time (Number of years): 1 year

Insert the given values into the formula

The formula to calculate the Principal amount is: Principal = (Simple Interest) ÷ (Rate × Time) Substitute the given values into the formula: Principal = (\(21.28) ÷ (0.055 × 1)\)

Calculate the Principal

Now, we just need to perform the calculation to find the principal: Principal = \(21.28 ÷ 0.055\) Principal = \(386.91\)

Present the result

The principal amount in Sharice's savings account was \(\$386.91\).

Key concepts

Principal Calculation

Understanding the principal amount in financial contexts is crucial, as it represents the initial sum of money before any interest is applied. In Sharice's scenario, she needs to backtrack from the interest earned to determine her original deposit or the principal.

To solve for principal when dealing with simple interest, a specific formula is used. This is rearranged from the standard simple interest formula. With simple interest, unlike compound interest, the interest earned is directly proportional to the original principal, the rate of interest, and the time for which the money is invested or borrowed.

For the problem at hand, we employ the rearranged formula based on the simple interest calculation which confirms to us that Principal = Simple Interest ÷ (Rate × Time). The simplicity of this formula makes it a valuable tool for anyone looking to reverse-engineer interest scenarios to find out the original amount involved.

Simple Interest Formula

The simple interest formula is a straightforward way to calculate the interest earned or paid on a sum of money over a certain period. The formula is expressed as SI = P × R × T, where SI stands for simple interest, P represents the principal amount, R is the interest rate per period, and T is the time the money is invested or borrowed for.

In the case of Sharice, we already know the interest she earned and are solving for P, the principal. It is critical to recognize that the formula is versatile and can be rearranged to solve for any of the variables as required. The basic principle is to multiply the original amount of money (principal) by the interest rate and then by the time to find the total interest earned if the principal was the unknown factor.

Interest Rate Conversion

Converting the interest rate from a percentage to a decimal is an essential step in the process of calculating simple interest. The reason for this conversion lies in how mathematical operations are performed. To convert a percentage to a decimal, one simply divides the percentage by 100.

For example, Sharice's account earns 5.5%, which as a decimal becomes 0.055 (5.5 ÷ 100). This decimal form is then used in the simple interest formula. It's important to always perform this conversion before plugging the rate into any formula, as using the percentage directly would result in an incorrect calculation. Also, remember to convert it back to a percentage form if needed when presenting the final result, to make it more understandable in the context of interest rates, which are typically discussed as percentages.