What monthly payment is required to amortize a loan of $35,000 over 14 years if interest at the rate of 8%/year is charged on the unpaid balance and interest calculations are made at the end of each month? (Round your answer to the nearest cent.)


To calculate the monthly payment required to amortize the loan, we use the formula for the monthly payment of an amortizing loan:

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

Where:
- $ M $ is the monthly payment.
- $ P $ is the loan amount.
- $ r $ is the monthly interest rate (annual rate divided by 12).
- $ n $ is the total number of payments (loan term in years multiplied by 12).

### Step-by-step calculation:

1. **Loan amount $ P $**: $35,000
2. **Annual interest rate**: 8%, so the monthly interest rate $ r $ is:
   \[
   r = \frac{8\%}{12} = \frac{0.08}{12} = 0.0066667
   \]
3. **Loan term**: 14 years, so the number of monthly payments $ n $ is:
   \[
   n = 14 \times 12 = 168 \text{ months}
   \]

4. Now, substitute these values into the formula:
   \[
   M = \frac{35,000 \times 0.0066667(1 + 0.0066667)^{168}}{(1 + 0.0066667)^{168} - 1}
   \]

### Calculation:

First, calculate $ (1 + r)^{168} $:
\[
(1 + 0.0066667)^{168} \approx 3.030115
\]

Then, substitute into the equation:
\[
M = \frac{35,000 \times 0.0066667 \times 3.030115}{3.030115 - 1} = \frac{35,000 \times 0.0202004}{2.030115} \approx \frac{707.014}{2.030115} \approx 348.30
\]

### Final Answer:
The monthly payment required is approximately **$348.30**.

Would you like a detailed breakdown of how to derive this formula or more information on loan amortization? 

Here are 5 related questions:
1. What is the total amount paid over the lifetime of the loan?
2. How much interest is paid in total over the 14 years?
3. What would the monthly payment be if the interest rate were reduced to 6%?
4. How does the loan balance change after 5 years of payments?
5. Can the loan be paid off early, and how would that affect the total interest?

**Tip:** You can reduce the overall interest paid by making additional payments toward the principal balance each month.

What monthly payment is required to amortize a loan of $35,000 over 14 years if interest at the rate of 8%/year is charged on the unpaid balance and interest calculations are made at the end of each month? (Round your answer to the nearest cent.)

Find out how to calculate the monthly payment needed to amortize a $35,000 loan over 14 years with 8% annual interest. This step-by-step solution covers key concepts in loan amortization, interest rate calculation, and algebra, suitable for Grades 10-12.

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