Question

For Exercises \(11-14\), solve. Roger notes that his savings account earned \(\$ 24.56\) in 9 months. If the simple interest rate for his account is \(4.8 \%,\) what was his principal?

Short answer

Answer: The initial deposit in Roger's savings account was approximately $2045.

Step-by-step solution

Understand the Simple Interest Formula

The simple interest formula is given by: I = P * R * T, where I is the interest earned, P is the principal amount, R is the interest rate per period, and T is the number of periods. In this case, we have I = 24.56, R = 4.8% per year, and T = 9 months. Since we are working with an annual interest rate but the time is given in months, we need to adjust these values to be consistent.

Convert Time Periods to Years

To convert the time period from months to years, we simply divide the number of months by the total number of months in a year: T = 9 months / 12 months/year = 3/4 years

Convert the Interest Rate to Decimal Form

We also need to convert the given interest rate from percentage to decimal for calculation purposes. To do this, simply divide the rate by 100: R = 4.8% / 100 = 0.048

Substitute Values into the Simple Interest Formula

Now, we have I = 24.56, R = 0.048 and T = 3/4 years. Our goal is to find the principal, P. Substitute the given values into the simple interest formula: 24.56 = P * 0.048 * (3/4)

Solve for the Principal, P

Now, we need to solve for P in the equation 24.56 = P * 0.048 * (3/4): To do this, divide both sides of the equation by 0.048 * (3/4): P = 24.56 / (0.048 * (3/4)) Calculating this, we get: P ≈ 2045 The principal in Roger's savings account was approximately $2045.

Key concepts

Interest Rate

The interest rate is a percentage that tells you how much money the account earns over a certain period. It is essential to understand that the interest rate provided is often expressed annually. For simple interest, the rate implies a straightforward calculation, where only the principal earns interest each period.

To use the interest rate in calculations, you must convert it to a decimal format. This means dividing it by 100. For example, in Roger's case, with an interest rate of 4.8%, it becomes 0.048.

By using this decimal form, you ease up the calculation while using the simple interest formula:

• It ensures the rate fits into the formula properly without errors.
• It helps maintain uniformity, especially when different periods are involved.

Knowing how to manipulate the interest rate correctly is crucial for solving problems like these efficiently.

Principal Amount

The principal amount is the initial money that you deposit in a savings account or invest elsewhere. It is the base sum before any interest is added.

Understanding how to find the principal is key in calculating simple interest, particularly when you know the interest earned over a specific period.

• In problems like Roger's, finding the principal means resolving the equation for P after you plug in other variables like interest and rate.
• The formula simplifies into dividing the interest by the product of the decimal interest rate and the time period expressed in years.

This approach offers clarity in isolating the principal in the equation, ensuring accurate financial assessments or predictions.

Time Conversion

Time conversion is an important step when adjusting the time period in a problem to match the annual interest rate. Often, the period given is not in years, necessitating conversion.

For instance, when given a time of nine months, as in Roger's example, convert this to years to align with the annual interest rate:

• Divide the months (9) by the number of months in a year (12), resulting in the fraction \( \frac{3}{4} \).
• This fraction represents the portion of a year.

Correct time conversion ensures that all elements of the interest formula work together seamlessly, preventing errors and allowing easy calculation of the interest or principal.