Question

For Exercises \(15-30,\) find the final balance and interest. \(\$ 5000\) at \(8 \%\) compounded annually for 2 years

Short answer

Answer: The final balance is $5832 and the interest earned is $832.

Step-by-step solution

Identify the given values

In this problem, we are given: - Principal amount (P) = $5000 - Annual interest rate (r) = \(8\%\) = $0.08 - Number of times interest is compounded per year (n) = 1 (compounded annually) - Number of years (t) = 2

Use the compound interest formula to find the final balance (A)

The compound interest formula is \(A = P(1 + r/n)^{nt}\). Substitute the given values: \(A = 5000(1 + 0.08/1)^{1 \times 2}\)

Simplify and solve for A

Simplify the expression inside the parentheses and the exponent: \(A = 5000(1 + 0.08)^2\) \(A = 5000(1.08)^2\) Now, evaluate the expression: \(A = 5000 \times 1.1664\) \(A = 5832\) So, the final balance after 2 years will be $5832.

Calculate the interest

To find the interest, subtract the principal amount from the final balance: Interest = Final balance - Principal amount Interest = \(5832 - 5000\) Interest = \(832\) The interest earned is $832. So, the final balance is \(5832, and the interest earned is \)832.

Key concepts

Compound Interest Formula

When it comes to growing your savings, the compound interest formula is a critical mathematical tool that can help you understand how your money can grow over time. The formula for compound interest is expressed as \[ A = P(1 + \frac{r}{n})^{nt} \].
In this equation,

• \( A \) represents the future value of the investment/loan, including interest,
• \( P \) stands for the principal amount,
• \( r \) is the annual interest rate (in decimal form),
• \( n \) is the number of times that interest is compounded per year, and
• \( t \) symbolizes the time the money is invested or borrowed for, in years.

To calculate the final balance using this formula, you follow a series of steps to break down the formula and insert the given values from the exercise.

Annual Interest Rate

Understanding the annual interest rate is key to mastering financial calculations involving compound interest. The rate is expressed as a percentage and typically denotes the cost of borrowing money or the profit gained from an investment for one year.
In our example, the annual interest rate is given as 8%, which must be converted into a decimal form to be used in the formula. This is done by dividing the percentage by 100, leading to an annual rate of \( r = 0.08 \).
It's important to note the rate should reflect the compounding period, meaning if the interest is compounded more than once per year, the rate must be adjusted accordingly in the compound interest formula. However, for an annual compounding, the rate remains as stated per year.

Principal Amount

The principal amount, represented by \( P \) in the compound interest formula, is essentially the initial sum of money on which the interest is calculated. It's the starting point of any investment or loan and the base figure from which all the compound interest is grown.
In the given exercise, the principal amount is $5000. This is the amount that is initially deposited and will be subject to compounding over the course of the investment. It's vital to correctly identify this amount, as it affects the overall calculation of the final balance after interest. The distinction from any accumulated interest is key, as it helps us find out just the amount earned from interest by subtracting the principal from the final balance after the designated period.