$$ 1

Question

$$ 1< $$ The price of a home is $$ \$ 230,000 $$. 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. $$ \$ 1,703 $$ (Round to the nearest dollar as needed.) Find the monthly payment for the 30-year option. $$ \$ \square $$ (Round to the nearest dollar as needed.)

UpStudy AI · Accepted Answer

**Given:**

- Home price: $$ \$230{,}000 $$  
- Down payment: $$ 15\% $$  
- Loan amount (principal):  
$$
P = 0.85 \times 230{,}000 = \$195{,}500
$$

- Annual interest rate: $$ 6.5\% $$  
- Monthly interest rate:  
$$
\frac{r}{n} = \frac{0.065}{12} \approx 0.00541667
$$
- Payment formula:  
$$
\text{PMT} = \frac{P\left(\frac{r}{n}\right)}{1 - \left(1+\frac{r}{n}\right)^{-n t}}
$$

---

**Step 1. 15‐Year Option**

- For $$ t = 15 $$ years, number of payments:  
$$
n t = 15 \times 12 = 180
$$

- It is given that the monthly payment is:  
$$
\text{PMT}_{15} \approx \$1{,}703
$$

- **Total amount paid over 15 years:**  
$$
\text{Total}_{15} = 1{,}703 \times 180 = \$306{,}540
$$

- **Total interest paid for the 15‐year option:**  
$$
\text{Interest}_{15} = \text{Total}_{15} - P = 306{,}540 - 195{,}500 = \$111{,}040
$$

---

**Step 2. 30‐Year Option**

- For $$ t = 30 $$ years, number of payments:  
$$
n t = 30 \times 12 = 360
$$

- Use the PMT formula:  
$$
\text{PMT}_{30} = \frac{195{,}500 \times (0.065/12)}{1 - \left(1+0.065/12\right)^{-360}}
$$

1. **Calculate the numerator:**  
$$
195{,}500 \times \frac{0.065}{12} \approx 195{,}500 \times 0.00541667 \approx 1{,}058.96
$$

2. **Calculate the denominator:**  
$$
1 - \left(1+0.00541667\right)^{-360}
$$
   - First, compute $$ 1+0.00541667 \approx 1.00541667 $$.  
   - Next, compute the exponent:
     $$
     360 \times \ln(1.00541667) \approx 360 \times 0.0054 \approx 1.944
     $$
   - Therefore,
     $$
     \left(1.00541667\right)^{-360} \approx e^{-1.944} \approx 0.143
     $$
   - Thus, the denominator is:
     $$
     1 - 0.143 \approx 0.857
     $$

3. **Compute the monthly payment:**
$$
\text{PMT}_{30} \approx \frac{1{,}058.96}{0.857} \approx 1{,}235
$$

(Rounding to the nearest dollar.)

- **Total amount paid over 30 years:**  
$$
\text{Total}_{30} = 1{,}235 \times 360 = \$444{,}600
$$

- **Total interest paid for the 30‐year option:**  
$$
\text{Interest}_{30} = 444{,}600 - 195{,}500 = \$249{,}100
$$

---

**Step 3. Savings in Interest**

- **Interest saved by choosing the 15‐year option instead of the 30‐year option:**
$$
\text{Savings} = \text{Interest}_{30} - \text{Interest}_{15} = 249{,}100 - 111{,}040 = \$138{,}060
$$

---

**Final Answers:**

- Monthly payment for the 15‐year option: $$ \$1{,}703 $$ (given)  
- Monthly payment for the 30‐year option: $$ \$1{,}235 $$  
- Interest paid over 15 years: $$ \$111{,}040 $$  
- Interest paid over 30 years: $$ \$249{,}100 $$  
- Interest saved with the 15‐year option: $$ \$138{,}060 $$
- **15-Year Option:** - Monthly Payment: \$1,703 - Total Interest Paid: \$111,040 - **30-Year Option:** - Monthly Payment: \$1,235 - Total Interest Paid: \$249,100 - **Savings with 15-Year Option:** - \$138,060

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