To calculate the future value of Melanie's investment, we can use the compound interest formula:

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

Where:

• $A$ is the amount of money accumulated after interest (the future value).
• $P$ is the principal amount (initial investment).
• $r$ is the annual interest rate (decimal form).
• $n$ is the number of times interest is compounded per year.
• $t$ is the time the money is invested or borrowed for, in years.

Given:

• $P = 7600$
• $r = 4.0\% = 0.04$
• $n = 12$ (compounded monthly)
• $t = 2$ years

Now, let's substitute the values into the formula:

$A = 7600 \left(1 + \frac{0.04}{12}\right)^{12 \times 2}$

Let's calculate this step by step:

1. First, compute $\frac{0.04}{12}$: $\frac{0.04}{12} = 0.003333\ldots$

2. Add 1 to this value: $1 + 0.003333\ldots = 1.003333\ldots$

3. Multiply $12 \times 2 = 24$ (the number of compounding periods).

4. Raise $1.003333\ldots$ to the 24th power: $1.003333\ldots^{24} \approx 1.083282$

5. Multiply this by the principal $P = 7600$: $A = 7600 \times 1.083282 \approx 8229.73$

So, the value of Melanie's investment after 2 years will be $8229.73.

Would you like further details on the steps or need clarification? Here are a few questions you might find useful:

1. How do you convert the percentage interest rate into a decimal?
2. What would the value be if the interest was compounded annually instead of monthly?
3. How does increasing the compounding frequency affect the final investment value?
4. What happens if the interest rate changes over the course of the investment?
5. How do you calculate compound interest for investments with different time periods?

Tip: Always ensure the interest rate is converted to decimal form before plugging it into the compound interest formula!