Questions: On Melissa's 6th birthday, she gets a 3000 CD that earns 3% interest, compounded semiannually. If the CD matures on her 12th birthday, how much money will be available? The amount available will be .

Solution Steps

To solve this problem, we need to calculate the future value of a compound interest investment. The formula for compound interest is:

\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

where:

• \( A \) is the amount of money accumulated after n years, including interest.
• \( P \) is the principal amount (initial investment).
• \( r \) is the annual interest rate (decimal).
• \( n \) is the number of times that interest is compounded per year.
• \( t \) is the time the money is invested for in years.

In this case:

• \( P = 3000 \)
• \( r = 0.03 \)
• \( n = 2 \) (since the interest is compounded semiannually)
• \( t = 6 \) (from her 6th to her 12th birthday)

We are given the following values for the compound interest calculation:

• Principal amount \( P = 3000 \)
• Annual interest rate \( r = 0.03 \)
• Compounding frequency per year \( n = 2 \) (semiannually)
• Time in years \( t = 6 \)

The future value \( A \) can be calculated using the formula:

\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

Substituting the known values into the formula:

\[ A = 3000 \left(1 + \frac{0.03}{2}\right)^{2 \times 6} \]

First, calculate \( \frac{r}{n} \):

\[ \frac{0.03}{2} = 0.015 \]

Now, calculate \( nt \):

\[ nt = 2 \times 6 = 12 \]

Now substitute these values back into the formula:

\[ A = 3000 \left(1 + 0.015\right)^{12} \]

Calculating \( 1 + 0.015 \):

\[ 1 + 0.015 = 1.015 \]

Now raise \( 1.015 \) to the power of \( 12 \):

\[ A = 3000 \times (1.015)^{12} \]

Calculating \( (1.015)^{12} \):

\[ (1.015)^{12} \approx 1.1867 \]

Finally, calculate \( A \):

\[ A \approx 3000 \times 1.1867 \approx 3586.85 \]

Final Answer