Suppose a $64,000 mortgage is to be amortized at 5.2% interest. Find the total amount that would be paid for a 15-year term.

To find the total amount paid for a 15-year mortgage with a principal of $64,000 and an annual interest rate of 5.2%, we need to determine the monthly payment and then calculate the total payment over the term of the loan.

We will use the formula for the monthly mortgage payment:

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

where:
- $M$ is the monthly payment,
- $P$ is the principal loan amount ($64,000),
- $r$ is the monthly interest rate (annual interest rate divided by 12),
- $n$ is the total number of payments (loan term in years multiplied by 12).

Given:
- $P = 64,000$
- Annual interest rate = 5.2%, so the monthly interest rate $r = \frac{5.2\%}{12} = \frac{0.052}{12} \approx 0.0043333$
- Loan term $n = 15 \text{ years} \times 12 \text{ months/year} = 180 \text{ months}$

Let's plug these values into the formula to find the monthly payment $M$:

\[
M = 64,000 \frac{0.0043333(1+0.0043333)^{180}}{(1+0.0043333)^{180} - 1}
\]

First, calculate $(1 + r)^n$:

\[
(1 + 0.0043333)^{180} \approx 2.123678
\]

Now, calculate the numerator:

\[
0.0043333 \times 2.123678 \approx 0.0092008
\]

Next, calculate the denominator:

\[
2.123678 - 1 \approx 1.123678
\]

Now, find the monthly payment $M$:

\[
M = 64,000 \frac{0.0092008}{1.123678} \approx 64,000 \times 0.0081902 \approx 524.37
\]

The monthly payment $M$ is approximately $524.37.

To find the total amount paid over the term of the loan, multiply the monthly payment by the number of payments:

\[
\text{Total amount paid} = M \times n = 524.37 \times 180 \approx 94,386.60
\]

Therefore, the total amount that would be paid for the 15-year mortgage is approximately $94,386.60.

If you have any questions or need further details, please let me know.

Here are 8 related questions:

1. How does the total amount paid change if the interest rate increases to 6%?
2. What would the monthly payment be for a 30-year mortgage with the same principal and interest rate?
3. How can you calculate the remaining balance on the mortgage after 5 years?
4. What is the impact of making additional principal payments on the total amount paid?
5. How does changing the loan term affect the monthly payment and total interest paid?
6. How do you calculate the interest portion of a specific mortgage payment?
7. What is the formula for calculating the amortization schedule for a mortgage?
8. How does the APR (Annual Percentage Rate) differ from the nominal interest rate in mortgage calculations?

**Tip:** When considering a mortgage, always look at the total cost over the term of the loan, including interest, to fully understand the financial commitment.

Learn how to calculate the total amount paid for a 15-year mortgage of $64,000 at 5.2% interest. Includes detailed steps to find the monthly payment and total payment over the loan term. Suitable for advanced high school students studying finance or mathematics.

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