3. The price of a small cabin is $$ \$ 100,000 $$. The bank requires a $$ 5 \% $$ down payment. The buyer is offered two mortgage options: 20 -year fixed at $$ 4 \% $$ or 30 -year fixed at $$ 4 \% $$. Calculate the amount of interest paid for each option. How much does the buyer save in interest with the 20-year option?

Question

UpStudy AI · Accepted Answer

**Step 1. Calculate the Mortgage Amount**

The cabin’s price is $$\$100,\!000$$ and the down payment is $$5\%$$. Thus, the financed mortgage amount is

$$
\text{Loan Amount} = \$100,\!000 \times (1 - 0.05) = \$100,\!000 \times 0.95 = \$95,\!000.
$$

---

**Step 2. Set Up the Mortgage Payment Formula**

The monthly payment $$M$$ for a fixed‐rate mortgage is given by

$$
M = P \cdot \frac{r(1+r)^n}{(1+r)^n - 1},
$$

where  
$$P$$ is the principal (here, $$\$95,\!000$$),  
$$r$$ is the monthly interest rate, and  
$$n$$ is the total number of monthly payments.

The annual interest rate is $$4\%$$, so the monthly rate is

$$
r = \frac{0.04}{12} \approx 0.0033333.
$$

---

**Step 3. Option 1 – 20-Year Fixed Rate Mortgage**

For a 20-year mortgage, the number of payments is

$$
n = 20 \times 12 = 240.
$$

Substitute into the formula:

$$
M_{20} = 95000 \cdot \frac{0.0033333 \, (1+0.0033333)^{240}}{(1+0.0033333)^{240} - 1}.
$$

Calculate $$(1+0.0033333)^{240}$$:

Using the approximation

$$
(1+0.0033333)^{240} \approx e^{240 \times 0.0033333} \approx e^{0.8} \approx 2.222,
$$

the numerator becomes

$$
0.0033333 \times 2.222 \approx 0.0074073,
$$

and the denominator is

$$
2.222 - 1 \approx 1.222.
$$

Thus, the fraction is

$$
\frac{0.0074073}{1.222} \approx 0.006058.
$$

Then the monthly payment is

$$
M_{20} \approx 95000 \times 0.006058 \approx \$575.51.
$$

The total amount paid over 20 years is

$$
\text{Total}_{20} = M_{20} \times 240 \approx \$575.51 \times 240 \approx \$138,\!122.
$$

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

$$
\text{Interest}_{20} \approx \$138,\!122 - \$95,\!000 \approx \$43,\!122.
$$

---

**Step 4. Option 2 – 30-Year Fixed Rate Mortgage**

For a 30-year mortgage, the number of payments is

$$
n = 30 \times 12 = 360.
$$

Substitute into the formula:

$$
M_{30} = 95000 \cdot \frac{0.0033333 \, (1+0.0033333)^{360}}{(1+0.0033333)^{360} - 1}.
$$

Calculate $$(1+0.0033333)^{360}$$:

Using the approximation

$$
(1+0.0033333)^{360} \approx e^{360 \times 0.0033333} \approx e^{1.2} \approx 3.310,
$$

the numerator becomes

$$
0.0033333 \times 3.310 \approx 0.0110333,
$$

and the denominator is

$$
3.310 - 1 \approx 2.310.
$$

Thus, the fraction is

$$
\frac{0.0110333}{2.310} \approx 0.004773.
$$

Then the monthly payment is

$$
M_{30} \approx 95000 \times 0.004773 \approx \$453.44.
$$

The total amount paid over 30 years is

$$
\text{Total}_{30} = M_{30} \times 360 \approx \$453.44 \times 360 \approx \$163,\!238.
$$

The total interest paid is

$$
\text{Interest}_{30} \approx \$163,\!238 - \$95,\!000 \approx \$68,\!238.
$$

---

**Step 5. Calculate the Interest Savings**

The savings in interest by choosing the 20-year over the 30-year option is

$$
\text{Savings} \approx \text{Interest}_{30} - \text{Interest}_{20} \approx \$68,\!238 - \$43,\!122 \approx \$25,\!116.
$$

---

**Final Answers:**

- **20-year fixed at $$4\%$$:**  
  Total interest paid $$\approx \$43,\!122.$$

- **30-year fixed at $$4\%$$:**  
  Total interest paid $$\approx \$68,\!238.$$

- **Interest Savings with the 20-year option:**
  
  $$
\approx \$25,\!116.
$$
- **20-Year Mortgage:** Total interest paid ≈ \$43,122. - **30-Year Mortgage:** Total interest paid ≈ \$68,238. - **Interest Savings with 20-Year Option:** ≈ \$25,116

The price of a small cabin is . The bank requires a down
payment. The buyer is offered two mortgage options: 20 -year fixed at or
30 -year fixed at . Calculate the amount of interest paid for each option.
How much does the buyer save in interest with the 20-year option?

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The price of a small cabin is . The bank requires a down payment. The buyer is offered two mortgage options: 20 -year fixed at or 30 -year fixed at . Calculate the amount of interest paid for each option. How much does the buyer save in interest with the 20-year option?