se PMT $$ =\frac{\mathrm{P}\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} $$ to determine the regular payment amount, rounded to the nearest dollar. The price of a ome is $$ \$ 183,000 $$. The bank requires a $$ 20 \% $$ down payment and three points at the time of closing. The cost of the ome is financed with a 30 -year fixed-rate mortgage at $$ 10 \% $$. Complete parts (a) through (e) below. a. Find the required down payment. $$ \$ 36,600 $$ b. Find the amount of the mortgage. $$ \$ 146,400 $$ c. How much must be paid for the three points at closing? $$ \$ 4,392 $$ (Round to the nearest dollar as needed.) d. Find the monthly payment (excluding escrowed taxes and insurance). $$ \$ 1,285 $$ (Round to the nearest dollar as needed.) e. Find the total cost of interest over 30 years. $$ \$ \square $$ (Round to the nearest dollar as needed.)

Question

UpStudy AI · Accepted Answer

**Step 1. Down Payment**

The down payment is $$20\%$$ of the home price. So,
$$
\text{Down Payment} = 0.20 \times \$183,\!000 = \$36,\!600.
$$

**Step 2. Mortgage Amount**

The mortgage amount is the remaining $$80\%$$ of the home price:
$$
\text{Mortgage Amount} = \$183,\!000 - \$36,\!600 = \$146,\!400.
$$

**Step 3. Points at Closing**

Three points cost $$3\%$$ of the mortgage amount. Hence,
$$
\text{Cost of Points} = 0.03 \times \$146,\!400 = \$4,\!392.
$$

**Step 4. Monthly Payment Calculation**

We use the formula:
$$
\text{PMT} = \frac{P\left(\frac{r}{n}\right)}{1-\left(1+\frac{r}{n}\right)^{-n t}},
$$
where  
$$P = \$146,\!400$$ (mortgage amount),  
$$r = 0.10$$ (annual interest rate),  
$$n = 12$$ (monthly compounding),  
$$t = 30$$ (years).

Substitute the values:
$$
\text{PMT} = \frac{146,\!400\left(\frac{0.10}{12}\right)}{1-\left(1+\frac{0.10}{12}\right)^{-12 \times 30}}.
$$

Compute the monthly rate:
$$
\frac{r}{n} = \frac{0.10}{12} \approx 0.0083333.
$$

The numerator becomes:
$$
146,\!400 \times 0.0083333 \approx 1,\!220.
$$

For the denominator compute:
$$
1-\left(1+0.0083333\right)^{-360}.
$$
First, note:
$$
1+0.0083333 \approx 1.0083333.
$$
Then,
$$
1.0083333^{360} \approx 19.82,
$$
so
$$
\left(1.0083333\right)^{-360} \approx \frac{1}{19.82} \approx 0.05045.
$$
Thus, the denominator is:
$$
1-0.05045 = 0.94955.
$$

Now,
$$
\text{PMT} \approx \frac{1,\!220}{0.94955} \approx 1,\!285.
$$

So, the monthly payment is approximately:
$$
\$1,\!285.
$$

**Step 5. Total Cost of Interest Over 30 Years**

The total amount paid over 30 years is:
$$
\text{Total Paid} = \text{Monthly Payment} \times \text{Number of Months} = 1,\!285 \times 360 = \$462,\!600.
$$
Subtract the mortgage amount to find the total interest cost:
$$
\text{Total Interest} = \$462,\!600 - \$146,\!400 = \$316,\!200.
$$

**Final Answers:**

a. Required down payment: $$\$36,\!600$$  
b. Mortgage amount: $$\$146,\!400$$  
c. Points at closing: $$\$4,\!392$$  
d. Monthly payment: $$\$1,\!285$$  
e. Total cost of interest over 30 years: $$\$316,\!200$$
**Summary of Calculations:** a. **Down Payment:** \$36,600 b. **Mortgage Amount:** \$146,400 c. **Points at Closing:** \$4,392 d. **Monthly Payment:** \$1,285 e. **Total Interest Over 30 Years:** \$316,200

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Arithmetic Questions