To determine the future value of an investment of $3,920 at an annual interest rate of 9.6%, compounded monthly for 7 years, we need to use the compound interest formula. Here's a step-by-step solution:
1. Identify the Variables:
- Principal (P): \$3,920
- Annual Interest Rate (r): 9.6% or 0.096 (in decimal form)
- Number of Times Interest is Compounded per Year (n): 12 (monthly)
- Number of Years (t): 7
2. Write the Compound Interest Formula:
The formula for compound interest is:
[tex]\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\][/tex]
Where:
- \( A \) is the future value of the investment/loan, including interest
- \( P \) is the principal investment amount (the initial deposit or loan amount)
- \( r \) 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
3. Substitute the Values into the Formula:
[tex]\[
A = 3920 \left(1 + \frac{0.096}{12}\right)^{12 \times 7}
\][/tex]
4. Calculate the Monthly Interest Rate:
[tex]\[
\frac{0.096}{12} = 0.008
\][/tex]
5. Calculate the Exponent:
[tex]\[
12 \times 7 = 84
\][/tex]
6. Apply the Formula:
[tex]\[
A = 3920 \left(1 + 0.008\right)^{84}
\][/tex]
7. Calculate the Base of the Exponent:
[tex]\[
1 + 0.008 = 1.008
\][/tex]
8. Raise the Base to the Power of the Exponent:
[tex]\[
1.008^{84} \approx 1.953989
\][/tex]
9. Calculate the Future Value:
[tex]\[
A = 3920 \times 1.953989 \approx 7655.4507
\][/tex]
10. Round to the Nearest Cent:
[tex]\[
A \approx 7655.45
\][/tex]
Therefore, the future value of the investment is approximately $7,655.45.
