To calculate the amount owed after 2 years with compound interest compounded quarterly, we can use the formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \( A \) is the amount owed after time \( t \),
- \( P \) is the principal amount (initial debt),
- \( r \) is the annual interest rate (decimal),
- \( n \) is the number of times interest is compounded per year,
- \( t \) is the time in years.
Given:
- \( P = 5000 \)
- \( r = 0.20 \) (20%)
- \( n = 4 \) (quarterly)
- \( t = 2 \)
Now, substituting the values into the formula:
\[ A = 5000 \left(1 + \frac{0.20}{4}\right)^{4 \times 2} \]
Calculating:
- Calculate \(\frac{0.20}{4}\):
\[ \frac{0.20}{4} = 0.05 \]
- Substitute back into the formula:
\[ A = 5000 \left(1 + 0.05\right)^{8} \]
- Calculate \(1 + 0.05\):
\[ 1 + 0.05 = 1.05 \]
- Now we find \(A\):
\[ A = 5000 \left(1.05\right)^{8} \]
Calculating \(1.05^8\):
\[ 1.05^8 \approx 1.477455 \]
- Now multiply by the principal:
\[ A \approx 5000 \times 1.477455 \approx 7387.28 \]
Therefore, the total amount owed after 2 years, compounded quarterly, is 7387.28.