The price of a home is $$ \$ 200,000 $$. The bank requires a $$ 15 \% $$ down payment. The buyer is offered two mortgage options: 15-year fixed at $$ 6.5 \% $$ or 30 -year fixed at $$ 6.5 \% $$. Calculate the amount of interest paid for each option. How much does the buyer save in interest with the 15 -year option? Use the following formula to determine the regular payment amount. PMT $$ =\frac{\mathrm{P}\left(\frac{\mathrm{r}}{\mathrm{n}}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} $$ Find the monthly payment for the 15-year option. $$ \$ \square $$ (Round to the nearest dollar as needed.)

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We begin by noting that the purchase price is $$\$200,\!000$$ but the bank requires a $$15\%$$ down payment. The down payment is

$$
0.15 \times 200,\!000=\$30,\!000,
$$

so the mortgage principal is

$$
P=200,\!000-30,\!000=\$170,\!000.
$$

The general formula for the regular (monthly) payment is

$$
\text{PMT}=\frac{P\left(\frac{r}{n}\right)}{1-\left(1+\frac{r}{n}\right)^{-nt}},
$$

where  
$$P=$$ loan principal,  
$$r=$$ annual interest rate (as a decimal),  
$$n=$$ number of payments per year, and  
$$t=$$ number of years.

Because payments are monthly, we take $$n=12$$ and $$r=0.065$$.

---

### 1. 15-Year Mortgage Option

For the 15-year option, $$t=15$$.

#### a. Monthly Payment Calculation

First compute the periodic interest rate:

$$
\frac{r}{n}=\frac{0.065}{12}\approx0.00541667.
$$

Then, the number of payments is

$$
n t=12\times15=180.
$$

Plug these into the payment formula:

$$
\text{PMT}_{15}=\frac{170,\!000\,(0.00541667)}{1-\left(1+0.00541667\right)^{-180}}.
$$

Let’s compute the numerator:

$$
170,\!000\times0.00541667\approx920.83.
$$

Next, compute the denominator. First, note that

$$
1+0.00541667\approx1.00541667.
$$

Raise this to the $$-180$$ power:

$$
\left(1.00541667\right)^{-180}\approx\frac{1}{\left(1.00541667\right)^{180}}.
$$

Using a calculator or logarithm estimates, one finds

$$
\left(1.00541667\right)^{180}\approx2.644,
$$

so

$$
\left(1.00541667\right)^{-180}\approx\frac{1}{2.644}\approx0.378.
$$

Thus the denominator becomes

$$
1-0.378=0.622.
$$

Now, the monthly payment is

$$
\text{PMT}_{15}\approx\frac{920.83}{0.622}\approx1481.
$$

Hence, the monthly payment for the 15-year option is approximately $$\$1481$$.

#### b. Total Interest Paid

The total amount paid over 15 years is

$$
\text{Total Paid}_{15}=1481\times180\approx\$266,\!580.
$$

The total interest paid is the total amount paid minus the principal:

$$
\text{Interest}_{15}\approx266,\!580-170,\!000\approx\$96,\!580.
$$

---

### 2. 30-Year Mortgage Option

For the 30-year option, $$t=30$$.

#### a. Monthly Payment Calculation

The periodic interest rate remains

$$
\frac{r}{n}=0.00541667,
$$

and the number of payments is

$$
n t=12\times30=360.
$$

Plug into the formula:

$$
\text{PMT}_{30}=\frac{170,\!000\,(0.00541667)}{1-\left(1.00541667\right)^{-360}}.
$$

The numerator remains approximately $$920.83$$.

Now, calculate the denominator. Raise $$1.00541667$$ to the $$-360$$ power. First,

$$
\left(1.00541667\right)^{360}\approx6.994,
$$

so

$$
\left(1.00541667\right)^{-360}\approx\frac{1}{6.994}\approx0.143.
$$

Thus the denominator is

$$
1-0.143=0.857.
$$

Now, the monthly payment is

$$
\text{PMT}_{30}\approx\frac{920.83}{0.857}\approx1076.
$$

#### b. Total Interest Paid

The total amount paid over 30 years is

$$
\text{Total Paid}_{30}=1076\times360\approx\$387,\!360.
$$

Subtracting the principal gives the total interest:

$$
\text{Interest}_{30}\approx387,\!360-170,\!000\approx\$217,\!360.
$$

---

### 3. Interest Savings

The buyer saves in interest by choosing the 15-year option. The savings are

$$
\text{Savings}= \text{Interest}_{30}-\text{Interest}_{15}\approx217,\!360-96,\!580\approx\$120,\!780.
$$

---

### Final Answers

- Monthly payment for the 15-year option: $$\boxed{\$1481}$$
- Total interest paid over 15 years: $$\approx \$96,\!580$$
- Total interest paid over 30 years: $$\approx \$217,\!360$$
- Interest saved by choosing the 15-year option: $$\approx \$120,\!780$$
- Monthly payment for the 15-year option: \$1481 - Total interest paid over 15 years: \$96,580 - Total interest paid over 30 years: \$217,360 - Interest saved by choosing the 15-year option: \$120,780

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