Question

For Exercises \(1-10\), calculate the simple interest and final balance. \(\$ 600\) at \(4.9 \%\) for 30 days

Short answer

Answer: The final balance of the investment after 30 days is approximately $602.41.

Step-by-step solution

Convert the interest rate to decimal form

To convert the interest rate from percentage to decimal, divide the rate by 100. So, the decimal form of 4.9% is: R = \(\frac{4.9}{100}\) = 0.049

Convert the time to years

Since there are 365 days in a year and the investment period is 30 days, we can convert the time to years by dividing the number of days by 365. So, T = \(\frac{30}{365}\) ≈ 0.0822 years (rounded to four decimal places)

Calculate the simple interest

Now, plug the principal, rate, and time values into the simple interest formula: I = P × R × T I = \(600 × 0.049 × 0.0822 ≈ \$2.41\) So, the simple interest for the investment is about $2.41.

Calculate the final balance

Now that we have the simple interest, we can calculate the final balance by adding the interest to the principal amount: Final balance = Principal amount + Interest Final balance = \(600 + \)2.41 ≈ \$602.41$ Thus, the final balance after 30 days is approximately $602.41.

Key concepts

Interest Rate Conversion

Understanding how to convert an interest rate from a percentage to a decimal is crucial when calculating simple interest. Here's a basic guide:

To convert an interest rate to a decimal, divide the percentage by 100. For example, if you're given an interest rate of 4.9%, you can convert this to decimal form by computing:
\[\begin{equation}R = \frac{4.9}{100} = 0.049\end{equation}\]This decimal form is then used in interest calculations. It's important to make this conversion because the formulas used in finance require the rate to be in decimal form for accurate computations. In practice, this step ensures that you're working with the precise rate for an accurate depiction of the financial scenario.

Mistakes in this conversion can lead to significant errors in interest calculation, which, in financial terms, could mean a difference of hundreds or even thousands of dollars depending on the scale of the transaction.

Time Conversion in Finance

Time conversion is another key aspect in the domain of finance, particularly when dealing with interest calculations for periods that are not standard to banking conventions. The time period, which is often given in days, months, or years, must sometimes be converted to a consistent unit of measurement. In our case, interest is calculated on an annual basis, thus converting days into years is essential for the correct calculation of simple interest.

To convert time from days to years, given there are usually 365 days in a year, divide the number of days by 365, like so:
\[\begin{equation}T = \frac{Number\ of\ Days}{365}\end{equation}\]In the given example of a 30-day period, the conversion would be:\[\begin{equation}T = \frac{30}{365} \approx 0.0822\ years\end{equation}\]This provides a proportion of a year which then can be correctly applied in our simple interest formula. Time conversion errors can result in a miscalculated interest that doesn't match the actual time frame for the investment or loan.

Simple Interest Formula

The simple interest formula is a fundamental concept used to calculate the interest generated on a loan or investment based on the principal amount, the rate, and the time frame of the investment. The formula is given by:
\[\begin{equation}I = P \times R \times T\end{equation}\]where I represents the interest earned, P is the principal amount (initial sum of money), R is the annual interest rate in decimal form, and T is the time the money is invested or borrowed for, in years.

Applying this in our example with a principal amount of \(600 at a rate of 4.9% for 30 days, we first convert the rate and then the time period, resulting in:\[\begin{equation}I = 600 \times 0.049 \times 0.0822 \approx \$2.41\end{equation}\]This simple interest of approximately \)2.41, when added to the principal, gives us the final balance after the investment period:
\[\begin{equation}Final\ balance = Principal\ amount + Interest = 600 + 2.41 \approx \$602.41\end{equation}\]Understanding and applying the simple interest formula is key to determining how much extra money will be earned or paid when it comes to loans and investments. It directly affects financial decisions and the assessment of potential returns on investments.