Question

For Exercises \(1-10\), calculate the simple interest and final balance. \(\$ 18,000\) at \(14.5 \%\) for 4 years

Short answer

Question: Calculate the simple interest and final balance of an initial investment of $18,000 at an interest rate of 14.5% for 4 years. Answer: The simple interest is $10,440, and the final balance after 4 years is $28,440.

Step-by-step solution

Write down the given information

We are given the principal amount (P), interest rate (R), and time (T) as follows: - P = \(\$ 18,000\) - R = \(14.5\%\) - T = 4 years

Convert the interest rate to decimal form

To use the simple interest formula, we need to convert the interest rate from percentage to decimal form. To do this, we divide the interest rate by 100. $$ R = \frac{14.5}{100} = 0.145 $$

Calculate the simple interest (I)

Now we have all the information needed to calculate the simple interest using the formula: $$ I = P \times R \times T = \$18,000 \times 0.145 \times 4 $$ Compute the result: $$ I = \$18,000 \times 0.145 \times 4 = \$10,440 $$ So the simple interest is \(\$10,440\).

Calculate the final balance

In this step, we will add the simple interest (I) to the principal amount (P) to find the final balance: $$ final\:balance = P + I = \$18,000 + \$10,440 $$ Compute the result: $$ final\:balance = \$18,000 + \$10,440 = \$28,440 $$ The final balance after 4 years is \(\$28,440\).

Key concepts

Interest Rate Conversion

When working with simple interest, one of the crucial steps is converting the interest rate from a percentage to a decimal form. This step is necessary for calculations in most financial formulas, including those used to find simple interest.
To perform this conversion, you simply divide the given percentage value by 100.

• For example, if the interest rate is 14.5%, you divide 14.5 by 100 to get 0.145 as the decimal equivalent.
• This allows you to easily use the interest rate in the formula for simple interest.

By converting the interest rate to a decimal, it standardizes the unit, which avoids any miscalculation and prepares the numbers for further arithmetic operations.

Principal Amount

The principal amount is the initial sum of money that is either invested or loaned. It's a foundational piece in calculating simple interest.
In our example, the principal amount is $18,000. Think of it as the starting point of your financial transactions.

• The principal remains constant throughout the transaction period unless additional contributions or withdrawals are made.
• It serves as the base for computing interest, meaning that any interest earned or owed is calculated based on this amount.

Understanding the role of the principal is crucial because it's the base to which our calculated interest will be added to determine the final balance.

Final Balance Calculation

The final balance in the context of simple interest is the total amount you have at the end of the investment or loan period. Calculating it involves adding the computed interest back to the principal amount.

• First, calculate the simple interest using the formula: \[ I = P \times R \times T \]
  where \(P\) is the principal, \(R\) is the interest rate (as a decimal), and \(T\) is the time in years.
• Once you have the interest, as in our example where the simple interest is \(10,440, you sum it with the principal: \[ \text{final balance} = P + I \]
• In our scenario, the calculation is \)18,000 + \(10,440, resulting in a final balance of \)28,440.

This final balance indicates the total money available after the interest period, making it a crucial figure for financial planning.