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