To calculate the final balance of an investment when interest is compounded monthly, we use the compound interest formula:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( A \)[/tex] is the future value of the investment/loan, including interest.
- [tex]\( P \)[/tex] is the principal investment amount (the initial deposit or loan amount).
- [tex]\( r \)[/tex] is the annual interest rate (in decimal form).
- [tex]\( n \)[/tex] is the number of times that interest is compounded per year.
- [tex]\( t \)[/tex] is the time the money is invested or borrowed for, in years.
Given:
- [tex]\( P = 6800 \)[/tex] dollars (initial investment),
- [tex]\( r = 0.04 \)[/tex] (annual interest rate as a decimal),
- [tex]\( n = 12 \)[/tex] (interest is compounded monthly),
- [tex]\( t = 5 \)[/tex] years (investment duration),
We substitute these values into the formula:
[tex]\[ A = 6800 \left(1 + \frac{0.04}{12}\right)^{12 \times 5} \][/tex]
First, calculate the monthly interest rate:
[tex]\[ \frac{0.04}{12} = 0.003333\overline{3} \][/tex]
Next, calculate the total number of compounding periods:
[tex]\[ 12 \times 5 = 60 \][/tex]
Now, our expression becomes:
[tex]\[ A = 6800 \left(1 + 0.003333\overline{3}\right)^{60} \][/tex]
The final balance, after performing the calculations, is approximately:
[tex]\[ A \approx 8302.776838806425 \][/tex]
So, the final balance of the account after 5 years of investing [tex]$6,800 at an annual interest rate of 4% compounded monthly will be approximately $[/tex]8,302.78.
