Mortgage Calculation Ea2573

1. **State the problem:** We have a house price of $375000$, a down payment of $30\%$, a $40$-year mortgage at an annual interest rate of $6.5\%$, and closing costs of $8200$. We need to find: (a) The down payment. (b) The amount of the mortgage. (c) The monthly payment. (d) The total interest paid over the life of the loan. --- 2. **Find the down payment:** The down payment is $30\%$ of the price. $$\text{Down payment} = 0.30 \times 375000 = 112500$$ --- 3. **Find the amount of the mortgage:** The mortgage amount is the price minus the down payment plus closing costs. $$\text{Mortgage amount} = 375000 - 112500 + 8200 = 270700$$ --- 4. **Find the monthly payment:** Use the mortgage payment formula for fixed-rate loans: $$M = P \times \frac{r(1+r)^n}{(1+r)^n - 1}$$ where: - $M$ is the monthly payment - $P = 270700$ is the loan principal - $r = \frac{6.5\%}{12} = \frac{0.065}{12} \approx 0.0054167$ is the monthly interest rate - $n = 40 \times 12 = 480$ is the total number of payments Calculate: $$M = 270700 \times \frac{0.0054167(1+0.0054167)^{480}}{(1+0.0054167)^{480} - 1}$$ Calculate $(1+0.0054167)^{480}$: $$1.0054167^{480} \approx 10.349$$ Then: $$M = 270700 \times \frac{0.0054167 \times 10.349}{10.349 - 1} = 270700 \times \frac{0.05607}{9.349} = 270700 \times 0.006000 = 1624.20$$ So the monthly payment is approximately $1624.20$. --- 5. **Find the total interest paid:** Total payments over $40$ years: $$\text{Total paid} = M \times n = 1624.20 \times 480 = 779616$$ Subtract the principal: $$\text{Interest paid} = 779616 - 270700 = 508916$$ --- **Final answers:** - (a) Down payment = $112500$ - (b) Mortgage amount = $270700$ - (c) Monthly payment = $1624.20$ - (d) Total interest paid = $508916$