1. **State the problem:** Engr. Peter Aventajado borrowed 100000 pesos at 10% annual interest compounded annually. He must pay monthly for 30 years, with payments at the beginning of each month. We need to find the monthly payment amount. 2. **Identify the formula:** This is an annuity due problem with compound interest. The formula for the monthly payment $PMT$ when payments are made at the beginning of each period (annuity due) is: $$PMT = \frac{P \times r}{1 - (1 + r)^{-n}} \times (1 + r)$$ where: - $P$ is the loan amount, - $r$ is the interest rate per payment period, - $n$ is the total number of payments. 3. **Convert annual interest rate to monthly rate:** Since interest is compounded annually but payments are monthly, we first find the effective monthly interest rate. Annual interest rate $i = 0.10$ Monthly interest rate $r = (1 + i)^{\frac{1}{12}} - 1 = (1 + 0.10)^{\frac{1}{12}} - 1$ Calculate: $$r = 1.10^{\frac{1}{12}} - 1 \approx 1.007974 - 1 = 0.007974$$ 4. **Calculate total number of payments:** $$n = 30 \text{ years} \times 12 \text{ months/year} = 360$$ 5. **Plug values into the formula:** $$PMT = \frac{100000 \times 0.007974}{1 - (1 + 0.007974)^{-360}} \times (1 + 0.007974)$$ 6. **Calculate denominator:** $$1 - (1.007974)^{-360} = 1 - \frac{1}{(1.007974)^{360}}$$ Calculate $(1.007974)^{360}$: $$ (1.007974)^{360} = (1.10)^{30} \approx 17.4494$$ So: $$1 - \frac{1}{17.4494} = 1 - 0.0573 = 0.9427$$ 7. **Calculate numerator:** $$100000 \times 0.007974 = 797.4$$ 8. **Calculate fraction:** $$\frac{797.4}{0.9427} \approx 846.0$$ 9. **Multiply by $(1 + r)$ for annuity due:** $$846.0 \times 1.007974 \approx 852.8$$ 10. **Final answer:** The monthly payment is approximately **852.80 pesos**. **Summary:** Engr. Peter must pay about 852.80 pesos at the beginning of each month for 30 years to fully repay the loan with 10% annual interest compounded annually.