Marta takes a loan of \( \$ 12,450 \) at an interest rate of \( 4.56 \% \), compounded quarterly. Assuming no payments are made, find the amount owed after 5 years. Round to the nearest dollar. Type your numerical answer below.
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To find the amount owed after 5 years, we can use the formula for compound interest: \[ A = P(1 + \frac{r}{n})^{nt} \] Where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial loan amount). - \( r \) is the annual interest rate (decimal). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the number of years the money is invested or borrowed. For this loan: - \( P = 12450 \) - \( r = 0.0456 \) - \( n = 4 \) (since the interest is compounded quarterly) - \( t = 5 \) Plugging these values into the formula gives: \[ A = 12450 \left(1 + \frac{0.0456}{4}\right)^{4 \cdot 5} \] Calculating the values step-by-step: 1. Calculate \( \frac{0.0456}{4} = 0.0114 \) 2. So \( A = 12450 \left(1 + 0.0114\right)^{20} \) 3. That simplifies to \( A = 12450 (1.0114)^{20} \) 4. Calculating \( (1.0114)^{20} \approx 1.2522 \) 5. Finally, \( A \approx 12450 \times 1.2522 \approx 15561.34 \) Rounding to the nearest dollar, the amount owed after 5 years is \( \$15,561 \).
