To calculate the amount owed after 3 years when [tex]$2000 is loaned at an annual interest rate of 19.5%, compounded monthly, we can use the compound interest formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Here:
- \( P \) is the principal amount (initial loan or investment), which is $[/tex]2000.
- [tex]\( r \)[/tex] is the annual interest rate in decimal form, so 19.5% becomes 0.195.
- [tex]\( n \)[/tex] is the number of times interest is compounded per year. Since the interest is compounded monthly, [tex]\( n = 12 \)[/tex].
- [tex]\( t \)[/tex] is the time the money is invested or borrowed for in years, which is 3 years.
Let's substitute these values into the formula:
[tex]\[ A = 2000 \left(1 + \frac{0.195}{12}\right)^{12 \cdot 3} \][/tex]
First, calculate the monthly interest rate:
[tex]\[ \frac{0.195}{12} = 0.01625 \][/tex]
Next, add 1 to the monthly interest rate:
[tex]\[ 1 + 0.01625 = 1.01625 \][/tex]
Then, raise this amount to the power of the total number of compounding periods over 3 years. Since there are 12 compounding periods per year and the investment is for 3 years, the total number of periods is:
[tex]\[ 12 \cdot 3 = 36 \][/tex]
So, we raise 1.01625 to the power of 36:
[tex]\[ 1.01625^{36} \approx 1.78657 \][/tex]
Finally, multiply this result by the principal amount to find the final amount:
[tex]\[ A = 2000 \cdot 1.78657 = 3573.14 \][/tex]
After 3 years, the amount owed, rounded to the nearest cent, is $3573.14.
