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Part 1

A loan of 

$ 17 comma 676$17,676 

was repaid at the end of 

99 

months. What size repayment check​ (principal and​ interest) was​ written, if 

aa 

6.7 %6.7% 

annual rate of interest was​ charged?

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Part 1

The amount of the repayment check was 

​$enter your response here. 

​(Round to two decimal​ places.)

To solve this problem, we need to calculate the monthly payment for a loan using the **formula for the payment of a loan** under compound interest. The formula to calculate the monthly payment $ M $ is:

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

Where:
- $ M $ = monthly payment
- $ P $ = loan amount (principal) = $17,676
- $ r $ = monthly interest rate (annual interest rate divided by 12) = $ \frac{6.7\%}{12} = 0.067/12 $
- $ n $ = number of months = 99 months

### Step 1: Calculate the monthly interest rate
\[
r = \frac{0.067}{12} = 0.0055833
\]

### Step 2: Plug values into the formula

\[
M = 17,676 \times \frac{0.0055833(1 + 0.0055833)^{99}}{(1 + 0.0055833)^{99} - 1}
\]

### Step 3: Simplify the equation and solve

\[
M = 17,676 \times \frac{0.0055833 \times (1.0055833)^{99}}{(1.0055833)^{99} - 1}
\]

We can calculate this step-by-step using a calculator:

1. $ (1.0055833)^{99} = 1.664117 $
2. $ 0.0055833 \times 1.664117 = 0.009293 $
3. $ 1.664117 - 1 = 0.664117 $
4. Now, $ M = 17,676 \times \frac{0.009293}{0.664117} $
5. $ \frac{0.009293}{0.664117} = 0.013991 $
6. Finally, $ M = 17,676 \times 0.013991 = 247.47 $

### Step 4: Conclusion

The size of the monthly repayment check is **$247.47**.

Would you like to see more details or have any questions? 

Here are five related questions:
1. What if the loan term was shorter, like 60 months?
2. How would the repayment change with a 5% interest rate?
3. How do we handle loans with interest rates compounded more frequently (e.g., daily)?
4. What is the total interest paid over the loan term?
5. How does this formula change if you are making payments more frequently than monthly?

**Tip:** Always double-check your interest rate calculations—small rounding errors can impact the final result significantly.

A loan of $17,676 was repaid at the end of 99 months. What size repayment check (principal and interest) was written, if a 6.7% annual rate of interest was charged?

Learn how to calculate the size of a repayment check for a $17,676 loan over 99 months with a 6.7% annual interest rate. This step-by-step solution covers loan amortization, compound interest, and monthly payment formulas, suitable for Grades 10-12.

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