Question

Jacqueline took out a 200,000 mortgage, at an interest rate of 3.5%
, for 30 years. What will her monthly payments be? When solving, round numbers to the nearest hundred thousandth. Round your final answer to the nearest cent.

Answer 1

To calculate Jacqueline's monthly mortgage payment, we can use the formula for a fixed-rate mortgage payment:

\[
M = P \frac{r(1 + r)^n}{(1 + r)^n - 1}
\]

where:
- \( M \) = monthly payment
- \( P \) = principal loan amount (the amount borrowed)
- \( r \) = monthly interest rate (annual interest rate divided by 12)
- \( n \) = number of payments (loan term in months)

Given:
- \( P = 200,000 \)
- Annual interest rate = 3.5% → \( r = \frac{3.5}{100} \div 12 = 0.00291667 \)
- Term of the loan = 30 years → \( n = 30 \times 12 = 360 \)

Now substituting these values into the formula:

1. Calculate \( r(1 + r)^n \):

\[
(1 + r)^n = (1 + 0.00291667)^{360}
\]

\[
(1 + 0.00291667)^{360} \approx 2.849229
\]

Hence, \( r(1 + r)^n \) can be calculated as:

\[
r(1 + r)^n = 0.00291667 \times 2.849229 \approx 0.008314
\]

2. Calculate the denominator:

\[
(1 + r)^n - 1 = 2.849229 - 1 \approx 1.849229
\]

3. Now calculate \( M \):

\[
M \approx 200,000 \times \frac{0.008314}{1.849229}
\]

\[
M \approx 200,000 \times 0.00449 \approx 898.153
\]

Finally, rounding to the nearest cent, Jacqueline's monthly payment will be:

\[
\text{Monthly Payment} \approx \$898.15
\]

So, Jacqueline's monthly mortgage payment is **$898.15**.