Question

For Exercises \(1-10\), calculate the simple interest and final balance. \(\$ 4000\) at \(3 \%\) for 1 year

Short answer

Answer: The final balance after 1 year is $4120.

Step-by-step solution

Identify given values

In this problem, we are given the principal amount (P), interest rate (R), and time (T). Let's list these values explicitly: - Principal (P) = $4000 - Interest Rate (R) = \(3\%\) - Time (T) = 1 year

Convert the interest rate to a decimal

To use the simple interest formula, we need to convert the interest rate from a percentage to a decimal. To do this, we can divide the given interest rate by 100. Interest Rate (R) = \(\frac{3\%}{100}\) = \(0.03\)

Calculate the simple interest

Now that we have the principal, interest rate as a decimal, and the time, we can plug these values into the simple interest formula and calculate the interest: Simple Interest (I) = P × R × T I = \(4000 \times 0.03 \times 1\) I = \(120\) The simple interest is \(\$120\).

Calculate the final balance

To find the final balance, we need to add the simple interest to the principal amount: Final Balance = Principal + Simple Interest Final Balance = \(4000 + 120\) Final Balance = \(4120\) The final balance after 1 year is \(\$4120\).

Key concepts

Simple Interest Formula

Understanding the simple interest formula is crucial for calculating how much extra money you'll earn or owe on a loan or investment over time. Simple interest is determined with the formula:

Simple Interest (I) = Principal (P) \times Interest Rate (R) \times Time (T) In this formula, the principal is the original amount of money loaned or invested. The interest rate typically provided as a percentage must be converted to a decimal to work in the formula. The time is the period over which the interest is calculated. To clarify with an example, if you invest \(4,000 at a 3% interest rate for 1 year, using the formula, the simple interest would be: I = P \times R \times T = \)4000 \times 0.03 \times 1 = \(120. The outcome, which in this case is \)120, reveals the amount of interest earned over the specified period. It's important to note that simple interest is distinct from compound interest, where interest is calculated on the initial principal and also on the accumulated interest of previous periods.

Converting Percentages to Decimals

To employ the simple interest formula, understanding how to convert percentages to decimals is essential. This is because interest rates are typically provided as percentages, but they must be in decimal form to plug into the formula correctly. Here's a simple two-step process to convert a percentage to a decimal:

Division by 100:

Simply take the percentage and divide it by 100. For instance, to convert 3% to a decimal, divide 3 by 100, which gives 0.03.

Removing the Percent Sign:

Alternatively, you can move the decimal point two places to the left. This method equally drops the percent sign, effectively converting the rate. For example, 3% becomes 0.03 once the decimal point is moved. Using these methods ensures accurate calculations in financial formulas. Remember that neglecting to convert an interest rate from a percent to a decimal in financial formulas will result in a calculation that is off by a factor of 100, leading to significant errors.

Calculating Final Balance

Once the simple interest is computed, the next step is finding out the final balance of an investment or loan. It's a straightforward process where the original principal amount is added to the calculated interest. Here's the formula for determining the final balance:

Final Balance = Principal (P) + Simple Interest (I) Applying our example, with a principal amount of \(4000 and simple interest of \)120, the final balance after 1 year would be: Final Balance = \(4000 + \)120 = $4120. The final balance reflects the total value after the interest has been applied to the original amount for the specified time period. It's a simple but powerful calculation that can help in budgeting, financial planning, and understanding the cost or yield of money over time.