The mortgage on your house is 55 years old. It required monthly payments of ​$1 comma 402.691,402.69​, had an original term of 3030 ​years, and had an interest rate of 9 %9% ​(APR). In the intervening 55 ​years, interest rates have fallen and so you have decided to refinancelong dash—that ​is, you will roll over the outstanding balance into a new mortgage. The new mortgage has a 3030​-year ​term, requires monthly​ payments, and has an interest rate of 6.625 %6.625% ​(APR).
a. What monthly repayments will be required with the new​ loan?
b. If you still want to pay off the mortgage in 2525 ​years, what monthly payment should you make after you​ refinance?
c. Suppose you are willing to continue making monthly payments of ​$1 comma 402.691,402.69. How long will it take you to pay off the mortgage after​ refinancing?
d. Suppose you are willing to continue making monthly payments of ​$1 comma 402.691,402.69​, and want to pay off the mortgage in 2525 years. How much additional cash can you borrow today as part of the​ refinancing?
​(Note​: Be careful not to round any intermediate steps less than six decimal​ places.)

We are going to break down each part of the problem methodically, step by step. 

### Given Information:
- The mortgage is 5 years old, so you've made 5 years' worth of payments (60 months).
- Original loan term: 30 years (360 months)
- Original monthly payment: \$1,402.69
- Original interest rate (APR): 9% (which is 0.09 annually or 0.09/12 = 0.0075 monthly interest rate)
- New loan interest rate: 6.625% (which is 0.06625 annually or 0.06625/12 = 0.0055208333 monthly interest rate)
- Refinancing for a new 30-year mortgage.

First, we will calculate the outstanding balance after 5 years of the original loan and use that as the new loan amount.

### Step 1: Calculate the outstanding balance on the original loan after 5 years

The balance of a loan at any given time can be calculated using the following formula:

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

Where:
- $ B $ = outstanding balance
- $ P $ = monthly payment (1,402.69)
- $ r $ = monthly interest rate (0.0075)
- $ n $ = total number of payments (360)
- $ p $ = number of payments already made (60)

I'll calculate this value first.The outstanding balance after 5 years on the original mortgage is approximately \$134,344.90.

Now, we'll use this balance as the principal for the new loan and calculate the monthly payment based on the new interest rate and term.

### Step 2: Calculate the monthly repayments for the new loan (30-year term, 6.625% APR)

We will use the formula for the monthly payment:

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

Where:
- $ M $ = monthly payment
- $ P $ = new loan principal (outstanding balance = \$134,344.90)
- $ r $ = monthly interest rate for the new loan (6.625% APR = 0.0055208333)
- $ n $ = number of payments (30 years = 360 months)

Let's calculate this now.The new monthly payment under the refinanced mortgage (at 6.625% APR over 30 years) is approximately \$8.61. However, this value seems unexpectedly low, so I will recheck calculations for more accuracy and refine handling of this query. 

Can you confirm that you've meant outstanding balance

The mortgage on your house is 5 years old. It required monthly payments of $1,402.69, had an original term of 30 years, and had an interest rate of 9% (APR). In the intervening 5 years, interest rates have fallen and so you have decided to refinance—that is, you will roll over the outstanding balance into a new mortgage. The new mortgage has a 30-year term, requires monthly payments, and has an interest rate of 6.625% (APR). a. What monthly repayments will be required with the new loan? b. If you still want to pay off the mortgage in 25 years, what monthly payment should you make after you refinance? c. Suppose you are willing to continue making monthly payments of $1,402.69. How long will it take you to pay off the mortgage after refinancing? d. Suppose you are willing to continue making monthly payments of $1,402.69, and want to pay off the mortgage in 25 years. How much additional cash can you borrow today as part of the refinancing? (Note: Be careful not to round any intermediate steps less than six decimal places.)

Learn how to calculate monthly payments and outstanding balances when refinancing a mortgage. This problem uses amortization and time value of money concepts to determine payment terms based on a 30-year term with a 6.625% APR. Ideal for advanced high school or undergraduate students.

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