Questions: Susan opened a savings account and deposited 100.00. The account earns 2% interest, compounded annually. If she wants to use the money to buy a new bicycle in 3 years, how much will she be able to spend on the bike? Use the formula A=P(1+r/n)^(nt), where A is the balance (final amount), P is the principal (starting amount), r is the interest rate expressed as a decimal, n is the number of times per year that the interest is compounded, and t is the time in years. Round your answer to the nearest cent.

Solution Steps

To solve this problem, we will use the compound interest formula provided: \( A = P \left(1 + \frac{r}{n}\right)^{nt} \). Here, \( P \) is the initial deposit of $100, \( r \) is the annual interest rate of 2% (expressed as 0.02), \( n \) is the number of times the interest is compounded per year (which is 1 for annually), and \( t \) is the time in years (3 years in this case). We will calculate the final amount \( A \) and round it to the nearest cent.

We are given the following values:

• Principal amount, \( P = 100.00 \)
• Annual interest rate, \( r = 0.02 \)
• Number of times interest is compounded per year, \( n = 1 \)
• Time in years, \( t = 3 \)

The formula for compound interest is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Substituting the given values into the formula, we have: \[ A = 100.00 \left(1 + \frac{0.02}{1}\right)^{1 \times 3} \]

Calculate the expression inside the parentheses: \[ 1 + \frac{0.02}{1} = 1.02 \] Raise this result to the power of \( nt \): \[ 1.02^3 = 1.061208 \] Multiply by the principal amount: \[ A = 100.00 \times 1.061208 = 106.1208 \]

Round the final amount to the nearest cent: \[ A \approx 106.12 \]

Final Answer