Find the monthly payment for a mortgage of $$ \$ 85,500 $$ at $$ 6^{\frac{3}{4}} \% $$ for 20 years. Select one: a. $$ \$ 6,500.00 $$ b. $$ \$ 650.66 $$ c. $$ \$ 876.54 $$ d. $$ \$ 393.60 $$

Question

UpStudy AI · Accepted Answer

We use the monthly payment formula for a fixed-rate mortgage:

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

where

- $$P = 85500$$ is the principal,
- $$i$$ is the monthly interest rate,
- $$n$$ is the total number of payments.

**Step 1. Calculate the monthly interest rate**

The annual interest rate is given as $$6^{\frac{3}{4}}\%$$, which is equivalent to $$6.75\%$$ in decimal form $$0.0675$$. The monthly rate $$i$$ is:

$$
i = \frac{0.0675}{12} \approx 0.005625
$$

**Step 2. Calculate total number of payments**

For a term of 20 years, with 12 payments per year:

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

**Step 3. Evaluate $$(1+i)^n$$**

$$
(1+i)^n = (1.005625)^{240}
$$

Using logarithms or a calculator, we find:

$$
(1.005625)^{240} \approx 3.845
$$

**Step 4. Substitute values into the payment formula**

Substitute $$P = 85500$$, $$i = 0.005625$$, and $$(1+i)^n \approx 3.845$$ into the formula:

$$
M = 85500 \cdot \frac{0.005625 \times 3.845}{3.845 - 1}
$$

Calculate the numerator:

$$
0.005625 \times 3.845 \approx 0.02164
$$

Multiply with the principal:

$$
85500 \times 0.02164 \approx 1849.62
$$

Calculate the denominator:

$$
3.845 - 1 = 2.845
$$

Thus, the monthly payment $$M$$ is:

$$
M \approx \frac{1849.62}{2.845} \approx 650.66
$$

The correct monthly payment is:

$$
\boxed{\$650.66}
$$

So, the answer is option b: $$ \$650.66 $$.
The monthly payment is approximately \$650.66.

Find the monthly payment for a mortgage of \( \$ 85,500 \) at \( 6^{\frac{3}{4}} \% \) for 20 years. Select one: a. \( \$ 6,500.00 \) b. \( \$ 650.66 \) c. \( \$ 876.54 \) d. \( \$ 393.60 \)

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