When Shen bought his house, he got his mortgage through an online lending company.
The mortgage was a personal, amortized loan for , at an interest rate of , with monthly payments for a term of  years.

For each part, do not round any intermediate computations and round your final answers to the nearest cent.
If necessary, refer to the list of financial formulas.
(a)	Find Shen's monthly payment.

(b)	If Shen pays the monthly payment each month for the full term, find his total amount to repay the loan.
(c)	If Shen pays the monthly payment each month for the full term, find the total amount of interest he will pay.

To solve this problem, we will use the formulas related to amortized loans. Let’s go step by step, assuming the missing values are placeholders for relevant information, which we'll denote as follows:

- **Loan amount (P)**: Let this be the principal amount of the loan.
- **Interest rate (r)**: The annual interest rate (in decimal form).
- **Loan term (n)**: The number of years Shen has to repay the loan.
- **Number of payments per year (m)**: Typically, for a monthly payment schedule, \( m = 12 \).

We'll need the following formulas:

### (a) Monthly Payment Formula:
The monthly payment (M) for an amortized loan is calculated by the formula:

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

Where:
- \( P \) is the loan amount.
- \( r \) is the annual interest rate (expressed as a decimal).
- \( m \) is the number of payments per year (12 for monthly payments).
- \( n \) is the number of years of the loan term.

### (b) Total Amount to Repay:
If Shen makes monthly payments for the full term, the total amount to repay is the monthly payment multiplied by the number of payments:

\[
\text{Total Repayment} = M \cdot n \cdot m
\]

### (c) Total Interest Paid:
The total interest paid is the difference between the total repayment and the loan amount:

\[
\text{Total Interest} = \text{Total Repayment} - P
\]

Once you provide the missing values (the loan amount, interest rate, and loan term), I can calculate the exact figures for Shen’s monthly payment, total repayment, and total interest. 

Let me know if you have those values!

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Here are some related questions that expand on the concepts discussed:

1. How does the loan term (number of years) affect the monthly payment?
2. What impact does increasing the interest rate have on the total interest paid over the life of a loan?
3. How would Shen’s total interest change if he makes additional payments towards the principal?
4. What happens to the monthly payment if the number of payments per year changes from monthly to biweekly?
5. How would refinancing affect Shen’s monthly payment and total interest?

**Tip**: Making extra payments towards the principal in an amortized loan can significantly reduce the total interest paid over time!

When Shen bought his house, he got his mortgage through an online lending company. The mortgage was a personal, amortized loan for , at an interest rate of , with monthly payments for a term of years. (a) Find Shen's monthly payment. (b) If Shen pays the monthly payment each month for the full term, find his total amount to repay the loan. (c) If Shen pays the monthly payment each month for the full term, find the total amount of interest he will pay.

Learn how to calculate the monthly payment, total repayment, and interest for Shen's amortized mortgage loan using the formula for loan amortization. This problem covers essential financial math concepts like loan amortization, interest rates, and algebra, suitable for advanced high school or college-level students.

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