1. **State the problem:** A group of private investors purchased a condominium complex for 5,000,000. They made a down payment of 15% and financed the rest with a loan amortized over 13 years at 5.1% annual interest compounded quarterly. We need to find the required quarterly payment. 2. **Calculate the loan amount:** Down payment = 15% of 5,000,000 = 0.15 \times 5,000,000 = 750,000 Loan amount = 5,000,000 - 750,000 = 4,250,000 3. **Identify the formula for amortized loan payment:** The formula for the periodic payment $R$ is: $$ R = P \times \frac{i}{1 - (1 + i)^{-n}} $$ where: - $P$ = loan principal = 4,250,000 - $i$ = periodic interest rate - $n$ = total number of payments 4. **Calculate periodic interest rate and number of payments:** Annual interest rate = 5.1% = 0.051 Compounded quarterly means 4 periods per year. $$ i = \frac{0.051}{4} = 0.01275$$ Number of payments: $$ n = 13 \times 4 = 52$$ 5. **Calculate the denominator:** $$ 1 - (1 + i)^{-n} = 1 - (1 + 0.01275)^{-52} = 1 - (1.01275)^{-52} $$ Calculate $(1.01275)^{-52}$: $$ (1.01275)^{52} \approx e^{52 \times \ln(1.01275)} \approx e^{52 \times 0.01267} = e^{0.6588} \approx 1.932 $$ So, $$ (1.01275)^{-52} = \frac{1}{1.932} \approx 0.5177 $$ Therefore, $$ 1 - 0.5177 = 0.4823 $$ 6. **Calculate the payment $R$:** $$ R = 4,250,000 \times \frac{0.01275}{0.4823} = 4,250,000 \times 0.02644 = 112,370 $$ 7. **Final answer:** The required quarterly payment is approximately **112,370.00**.