To find how much will still be owed after 5 years of payments, we will follow these steps:

1. Calculate the monthly interest rate: The annual interest rate is 5.1%, so the monthly interest rate is: $\text{Monthly interest rate} = \frac{5.1\%}{12} = \frac{0.051}{12} \approx 0.00425$

2. Determine the number of payments: Since the loan term is 30 years, the total number of payments is: $N = 30 \times 12 = 360 \text{ payments}$ After 5 years, the number of payments made is: $N_{\text{made}} = 5 \times 12 = 60 \text{ payments}$

3. Calculate the remaining balance using the formula for the remaining balance of a mortgage: The formula for the remaining balance $B$ after $N_{\text{made}}$ payments is: $B = P \left( \frac{(1 + r)^N - (1 + r)^{N_{\text{made}}}}{(1 + r)^N - 1} \right)$ where:

   • $P$ is the loan amount ($258,000),
   • $r$ is the monthly interest rate (0.00425),
   • $N$ is the total number of payments (360),
   • $N_{\text{made}}$ is the number of payments made (60).

Now let’s plug in the numbers:

$B = 258,000 \times \left( \frac{(1 + 0.00425)^{360} - (1 + 0.00425)^{60}}{(1 + 0.00425)^{360} - 1} \right)$

This will give the remaining balance after 60 payments. I'll calculate that for you.

The remaining balance on the mortgage after making payments for 5 years will be approximately $237,252.

Would you like more details or have any further questions?

Here are 5 related questions you might find interesting:

1. How does the remaining balance change with a lower interest rate?
2. What happens to the monthly payment if the loan term is shortened?
3. How much total interest will you pay over the life of the loan?
4. How does making extra payments affect the loan balance after 5 years?
5. What if the interest rate was compounded differently (quarterly, annually)?

Tip: Making additional payments toward the principal can significantly reduce the total interest paid over the life of the mortgage.