1. Let's state the problem: We want to understand how to calculate mortgage payments. 2. The formula for a fixed-rate mortgage monthly payment is: $$M = P \times \frac{r(1+r)^n}{(1+r)^n - 1}$$ where: - $M$ is the monthly payment - $P$ is the loan principal (amount borrowed) - $r$ is the monthly interest rate (annual rate divided by 12) - $n$ is the total number of payments (loan term in months) 3. Important rules: - Convert the annual interest rate to a decimal before dividing by 12. - The number of payments is years times 12. 4. Example: Suppose you borrow 200000 at an annual interest rate of 5% for 30 years. - $P = 200000$ - $r = \frac{0.05}{12} = 0.0041667$ - $n = 30 \times 12 = 360$ 5. Calculate the numerator: $$r(1+r)^n = 0.0041667 \times (1 + 0.0041667)^{360}$$ 6. Calculate the denominator: $$(1+r)^n - 1 = (1 + 0.0041667)^{360} - 1$$ 7. Compute $(1 + 0.0041667)^{360}$: $$ (1.0041667)^{360} \approx 4.4677 $$ 8. Substitute back: $$\text{numerator} = 0.0041667 \times 4.4677 = 0.018615$$ $$\text{denominator} = 4.4677 - 1 = 3.4677$$ 9. Calculate the fraction: $$\frac{0.018615}{3.4677} \approx 0.005367$$ 10. Finally, calculate the monthly payment: $$M = 200000 \times 0.005367 = 1073.40$$ So, the monthly mortgage payment is approximately $1073.40$.