Question

For Exercises \(1-10\), calculate the simple interest and final balance. \(\$ 1200\) at \(6.5 \%\) for 3 years

Short answer

Answer: The final balance after 3 years is $1434.

Step-by-step solution

Understand the simple interest formula

The formula to calculate the simple interest (I) is: I = P * R * T where P is the principal amount ($1200 in this case), R is the rate of interest (6.5% in this case), and T is the time period (3 years in this case).

Convert the interest rate into a decimal

Convert the interest rate from percentage to decimal by dividing it by 100: R = \(\frac{\displaystyle 6.5}{\displaystyle 100}\) = 0.065

Find the simple interest

Now, substitute the values into the simple interest formula: I = P * R * T I = \(1200 \times 0.065 \times 3\) I = \(234\) The simple interest after 3 years is $234.

Calculate the final balance

To find the final balance (B), add the interest to the initial balance: B = P + I B = \(1200 + 234\) B = \(1434\) The final balance after 3 years is $1434.

Key concepts

Simple Interest Formula

When it comes to understanding how money grows over time in a savings account or loan, the simple interest formula is a fundamental concept. It's used to calculate the extra amount you'll earn or owe, separate from the original sum, known as the principal. The formula is represented as:
\( I = P \times R \times T \)
where \( I \) stands for the interest earned or paid, \( P \) represents the principal amount (the initial sum of money), \( R \) is the interest rate in decimal form, and \( T \) signifies the time the money is invested or borrowed for, in years.

Interest Rate Conversion

Interest rates are typically presented as percentages, which is helpful for quick comparisons; however, for calculations, they need to be converted to decimal form. To do this, simply divide the interest rate by 100.
For instance, a rate of \( 6.5\text{%} \) converts to a decimal as follows:
\( R = \frac{6.5}{100} = 0.065 \)
This decimal is used in the simple interest formula. The conversion is crucial; failing to convert will lead to incorrect outcomes and possible misunderstandings about the cost or growth of money over time.

Principal Amount

The principal amount is the starting figure before any interest is applied. It represents the initial sum of money that is either deposited into a savings account or loaned out. Understanding the principal is important because it dictates the potential growth or cost. In the exercise, a principal amount of \( \(1200 \) is used.
In different scenarios, the principal amount could change based on deposits and withdrawals or additional borrowing. However, for the sake of simple interest calculations, it is typically a fixed amount over the set time period.

Calculating Final Balance

The final balance is calculated by adding the interest earned to the principal amount. This gives you the total value at the end of the investment or loan period. With the simple interest formula having been used to find the interest, you can determine the final balance as follows:
\( B = P + I \)
By substituting our known values from the exercise:
\( B = \)1200 + \(234 \)
\( B = \)1434 \)
This balance reflects the total amount after the 3-year period. When planning financial goals or assessing loans, calculating the final balance gives a clear picture of future value.