To help open up a wine bar, Chau borrowed money from an online lending company.

He took out a personal, amortized loan for $46,000, at an interest rate of 6.45%, with monthly payments for a term of 6 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 Chau's monthly payment.

X

(b) If Chau pays the monthly payment each month for the full term, find his total amount to repay the loan.

(c)

If Chau pays the monthly payment each month for the full term, find the total amount of interest he will pay.

To solve this problem, we'll use the amortized loan formula to calculate Chau's monthly payment. Afterward, we'll compute the total repayment and the total interest paid.

### Amortized Loan Formula:

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

Where:
- $ M $ = Monthly payment
- $ P $ = Principal loan amount ($46,000)
- $ r $ = Monthly interest rate (annual rate divided by 12)
- $ n $ = Total number of payments (loan term in months)

Given:
- $ P = 46,000 $
- Annual interest rate = 6.45%, so the monthly interest rate $ r = \frac{6.45\%}{12} = 0.0645/12 = 0.005375 $
- Loan term = 6 years, so $ n = 6 \times 12 = 72 $ months

### (a) Find Chau's monthly payment.

Substitute the values into the amortized loan formula to calculate $ M $:

\[
M = \frac{46000 \times 0.005375 \times (1 + 0.005375)^{72}}{(1 + 0.005375)^{72} - 1}
\]

Let me calculate this for you.(a) Chau's monthly payment is approximately **$772.16**.

### (b) Find the total amount Chau will repay the loan.

The total repayment can be found by multiplying the monthly payment by the number of months:

\[
\text{Total repayment} = M \times n = 772.16 \times 72
\]

Let me calculate the total amount Chau will repay.(b) If Chau pays the monthly payment each month for the full term, the total amount he will repay is approximately **$55,595.68**.

### (c) Find the total amount of interest Chau will pay.

The total interest is the difference between the total repayment and the principal amount:

\[
\text{Total interest} = \text{Total repayment} - P = 55,595.68 - 46,000
\]

Let me calculate the total interest.(c) The total amount of interest Chau will pay over the term of the loan is approximately **$9,595.68**.

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Would you like further details or have any questions?

Here are 5 related questions for deeper understanding:

1. What would happen to the monthly payment if the interest rate increased to 8%?
2. How would a shorter loan term (e.g., 4 years) affect the total interest paid?
3. How can you calculate the principal remaining after 3 years of payments?
4. What are the advantages of making additional payments toward the principal?
5. How does an amortized loan differ from a simple interest loan?

**Tip:** To reduce the overall interest on a loan, making extra payments toward the principal can help shorten the loan term and decrease the interest paid.

To help open up a wine bar, Chau borrowed money from an online lending company. He took out a personal, amortized loan for $46,000, at an interest rate of 6.45%, with monthly payments for a term of 6 years. (a) Find Chau's monthly payment. (b) If Chau pays the monthly payment each month for the full term, find his total amount to repay the loan. (c) If Chau 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 a $46,000 amortized loan with a 6.45% interest rate over 6 years. This finance problem uses key formulas to determine the monthly payment and total costs involved, suitable for students in Grades 11-12 or college-level finance courses.

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