Question

For Exercises \(1-10\), calculate the simple interest and final balance. \(\$ 2400\) at \(8 \%\) for six months \(\left(\frac{1}{2} \text { year }\right)\)

Short answer

Based on step by step solution, the simple interest on a principal amount of $2400 at an interest rate of 8% for a period of six months is $96. The final balance after six months, which includes the principal amount and the simple interest, is equal to $2496.

Step-by-step solution

Convert the interest rate to a decimal

To convert the interest rate of 8% to a decimal, we need to divide by 100: \(R = \frac{8}{100} = 0.08\)

Calculate the time in years

Since the investment period is six months, we need to divide by 12 to find the time in years: \(T = \frac{6}{12} = 0.5\)

Find Simple Interest

Using the principal (P), interest rate (R), and time (T) calculated, we can find the Simple Interest (SI): \(SI = P \times R \times T = 2400 \times 0.08 \times 0.5 = 96\) Therefore, the simple interest is $96.

Calculate the final balance

Now that we know the simple interest (\(96), we can add it to the principal amount (\)2400): \(Final\ Balance = Principal\ Amount + Simple\ Interest = 2400 + 96 = 2496\) Thus, the final balance after six months is $2496.

Key concepts

Interest Rate Conversion

Before diving into calculations of simple interest, we must first convert the interest rate, usually given as a percentage, into a decimal. An interest rate expresses the cost of borrowing or the gain from investing as a percentage per year. However, to use it in our formulas, we need to work with it in decimal form. This conversion simplifies further computations.

Here's how you do it:

• Take the given percentage rate. For example, if you have a rate of 8%, you start with 8.
• Divide this by 100 to convert it into a decimal form. So, 8 divided by 100 becomes 0.08.

This conversion is crucial as it allows us to use the interest rate in mathematical formulas effortlessly. Without this step, you might easily end up with incorrect calculations.

Time Conversion

Time conversion is the process of expressing a period of money investment or loan in a standard unit used in mathematical formulas. For simple interest calculations, the standard unit is usually the year. If the given time is not in years, conversion is needed to ensure accurate computation.

Here's the process for time conversion:

• Identify the given time period. In our exercise, it was six months.
• Then, convert it to years. Since there are 12 months in a year, you divide by 12. So, six months becomes 0.5 years.

This step ensures that the time factor aligns with the yearly rate of interest, maintaining consistency across the formula and preventing errors.

Final Balance Calculation

After determining the simple interest using the principal amount, converted interest rate, and time in years, the final step is calculating the final balance. This is essentially finding out how much money you have at the end of the investment or loan term.

Here's what you should do:

• Add the calculated simple interest to the principal amount. The simple interest is what you earn or pay over time.
• In the given example, the principal amount was $2400, and the simple interest calculated was $96.
• Therefore, the final balance would be the principal $2400 plus the simple interest $96, which totals to $2496.

This final amount gives you a clear picture of what you will end up with after the interest has been applied over the time period considered.