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. $$ \text { PMT }=\frac{\mathrm{P}\left(\frac{\mathrm{r}}{\mathrm{n}}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} $$ Find the monthly payment for the 15 -year option. $$ \$ 1,481 $$ (Round to the nearest dollar as needed.) Find the monthly payment for the 30-year option.

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UpStudy AI · Accepted Answer

We start with a home price of \$200,000 and a required down payment of $$15\%$$. This means the loan amount is

$$
P = \$200,\!000 \times (1-0.15)= \$200,\!000 \times 0.85 = \$170,\!000.
$$

The 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
- $$P$$ is the loan principal,
- $$r$$ is the annual interest rate (in decimal form),
- $$n$$ is the number of payments per year (for monthly payments, $$n=12$$),
- $$t$$ is the term (in years).

In both options the annual rate is $$6.5\%$$ or $$r=0.065$$.

### The 15-Year Option

For a 15-year mortgage:
- $$t=15$$,
- $$n=12$$.

The monthly interest rate is

$$
\frac{r}{n}=\frac{0.065}{12}\approx 0.00541667.
$$

The total number of payments is

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

The monthly payment is then

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

The problem gives that this payment is approximately

$$
\text{PMT}_{15}\approx \$1,\!481.
$$

The total amount paid over 15 years is

$$
\$1,\!481\times 180 = \$266,\!580,
$$

so the total interest paid is

$$
266,\!580 - 170,\!000 = \$96,\!580.
$$

### The 30-Year Option

For a 30-year mortgage:
- $$t=30$$,
- $$n=12$$.

Again, the monthly interest rate is

$$
\frac{0.065}{12}\approx 0.00541667,
$$

and the total number of payments is

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

The monthly payment is computed as

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

Let us calculate the components:

1. **Numerator:**

$$
170,\!000\times 0.00541667\approx 920.83.
$$

2. **Denominator:**

We first compute

$$
1+\frac{r}{n}\approx 1.00541667.
$$

Raising this to the 360th power:

$$
\left(1.00541667\right)^{360}\approx \exp\left(360\times\ln(1.00541667)\right).
$$

An approximate calculation gives

$$
\left(1.00541667\right)^{360}\approx 6.99.
$$

Then

$$
\left(1.00541667\right)^{-360}\approx \frac{1}{6.99}\approx 0.143.
$$

Thus, the denominator is

$$
1-0.143\approx 0.857.
$$

3. **Monthly Payment for 30 Years:**

$$
\text{PMT}_{30} \approx \frac{920.83}{0.857}\approx 1,\!074.75.
$$

Rounded to the nearest dollar, the monthly payment is

$$
\text{PMT}_{30}\approx \$1,\!075.
$$

### Interest Comparisons

**15-year option:**
- Total paid: $$1,\!481\times 180=\$266,\!580$$.
- Interest paid: $$266,\!580-170,\!000=\$96,\!580$$.

**30-year option:**
- Total paid: $$1,\!075\times 360=\$387,\!000$$.
- Interest paid: $$387,\!000-170,\!000=\$217,\!000$$.

**Savings with the 15-year option:**

$$
217,\!000 - 96,\!580 = \$120,\!420.
$$

Thus, the buyer saves approximately $$\$120,\!420$$ in interest by choosing the 15-year option.

### Final Answers

- The monthly payment for the 30-year option is $$\$1,\!075$$ (rounded to the nearest dollar).
- The buyer saves about $$\$120,\!420$$ in interest by choosing the 15-year option.
The monthly payment for the 30-year option is \$1,075. By choosing the 15-year option, the buyer saves approximately \$120,420 in interest.

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