This problem is a complex financial problem that requires several skills, perhaps some from previous sections.

Clark and Lana take a 30-year home mortgage of $123,000 at 7.9%, compounded monthly. They make their regular monthly payments for 5 years, then decide to pay $1200 per month.

(a) Find their regular monthly payment. (Round your answer to the nearest cent.)

(b) Find the unpaid balance when they begin paying the $1200. (Round your answer to the nearest cent.)

(c) How many payments of $1200 will it take to pay off the loan? Give the answer correct to two decimal places.

 _____ monthly payments

(d) Use your answer to part (c) to find how much interest they save by paying the loan this way. (Round your answer to the nearest cent.)

This problem involves a mortgage calculation with monthly compounded interest and changing payments. Let's break it down step by step.

### Given information:
- Loan amount: $ P = 123,000 $
- Annual interest rate: $ r = 7.9\% $ (compounded monthly)
- Loan term: $ 30 $ years
- First 5 years: Regular monthly payments
- After 5 years: $1,200 per month

We will approach each part one by one:

### Part (a) - Finding the regular monthly payment:

To find the monthly payment $ M $ on a mortgage, we use the formula for an amortizing loan:

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

Where:
- $ M $ = Monthly payment
- $ P $ = Principal (loan amount)
- $ r $ = Monthly interest rate = $ \frac{7.9\%}{12} = \frac{0.079}{12} $
- $ n $ = Total number of payments = $ 30 \times 12 = 360 $

Let's calculate the monthly payment.

### Part (b) - Finding the unpaid balance after 5 years:

The unpaid balance after 5 years can be calculated using the formula for the remaining balance on an amortized loan:

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

Where:
- $ B $ = Remaining balance
- $ m $ = Number of payments made after 5 years = $ 5 \times 12 = 60 $

We'll calculate this next.

### Part (c) - Finding how many payments of $1,200 will be needed:

Once the remaining balance is calculated in part (b), we can use the new payment amount ($ M = 1200 $) to find the number of payments needed. The formula is derived from the loan payment formula rearranged to solve for the number of payments:

\[
m = \frac{\log\left(\frac{M}{M - rB}\right)}{\log(1 + r)}
\]

Where:
- $ M $ = New monthly payment = 1200
- $ B $ = Remaining balance from part (b)
- $ r $ = Monthly interest rate

### Part (d) - Finding how much interest is saved:

To calculate the total interest savings, we will compare the total interest paid under the regular payment plan (30 years of regular payments) versus the total interest paid under the accelerated payment plan (paying off the loan faster with $1,200 payments after 5 years).

First, we will calculate the total interest paid under both scenarios, and subtract the amounts to find the savings.

Now, let's calculate these values step by step.Here are the solutions to each part of the problem:

### Part (a) - Regular Monthly Payment:
The regular monthly payment is:
\[
M = 893.97
\]
(Rounded to the nearest cent)

### Part (b) - Unpaid Balance after 5 years:
The unpaid balance after 5 years of regular payments is:
\[
B = 116,827.84
\]
(Rounded to the nearest cent)

### Part (c) - Number of Payments with $1,200:
The number of payments of $1,200 it will take to pay off the remaining balance is approximately:
\[
156.09 \text{ months}
\]
(Or about 13 years and 0.09 months)

### Part (d) - Interest Saved:
By switching to the $1,200 monthly payment plan, the interest saved is:
\[
\text{Interest saved} = 128,346.62
\]
(Rounded to the nearest cent)

---

Let me know if you'd like more details on any part of this process. Also, here are five related questions to expand on this:

1. How would the monthly payment change if the interest rate was 6% instead of 7.9%?
2. What is the total interest paid over the life of the loan at $893.97 per month?
3. How does increasing the payment to $1,500 after 5 years affect the total interest savings?
4. What would happen if they made a lump-sum payment instead of increasing monthly payments?
5. How would the loan terms change if it was a 20-year loan instead of a 30-year loan?

### Tip:
When making additional payments on a mortgage, more of your payment goes toward the principal, reducing the interest you'll pay over time!

Clark and Lana take a 30-year home mortgage of $123,000 at 7.9%, compounded monthly. They make their regular monthly payments for 5 years, then decide to pay $1200 per month.

(a) Find their regular monthly payment. (Round your answer to the nearest cent.)
(b) Find the unpaid balance when they begin paying the $1200. (Round your answer to the nearest cent.)
(c) How many payments of $1200 will it take to pay off the loan? Give the answer correct to two decimal places.
(d) Use your answer to part (c) to find how much interest they save by paying the loan this way. (Round your answer to the nearest cent.)

Learn how to calculate mortgage payments and remaining balances with a 30-year loan at 7.9%, compounded monthly. Explore detailed step-by-step solutions for finding monthly payments, unpaid balance after 5 years, and interest saved by paying $1200 per month. Suitable for finance students and advanced math learners.

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