A loan of $ 17,450 was repaid at the end of 10 months. What size repayment check​ (principal and​ interest) was​ written, if an 8.6 % annual rate of interest was​ charged? Round the final answer to two decimals.

To find the size of the repayment check (including both the principal and interest), we can use the formula for simple interest:

\[
A = P(1 + rt)
\]

Where:
- $ A $ is the total repayment amount (principal + interest),
- $ P $ is the principal (the original loan amount),
- $ r $ is the annual interest rate (as a decimal),
- $ t $ is the time in years.

### Step 1: Assign the given values
- $ P = 17,450 $ (loan amount),
- $ r = 8.6\% = \frac{8.6}{100} = 0.086 $ (annual interest rate),
- $ t = \frac{10}{12} $ years (since the loan is repaid after 10 months).

### Step 2: Calculate the total repayment amount

\[
A = 17,450 \times \left(1 + 0.086 \times \frac{10}{12}\right)
\]

Let me calculate that for you.

\[
A = 17,450 \times \left(1 + 0.086 \times 0.8333\right)
\]

\[
A = 17,450 \times \left(1 + 0.07167\right)
\]

\[
A = 17,450 \times 1.07167 = 18,710.07
\]

### Final Answer:
The repayment check amount, including both principal and interest, is **$18,710.07**.

Would you like a breakdown of the steps or any further explanation? 

Here are 5 related questions to deepen understanding:
1. How does the time of repayment affect the interest amount in simple interest loans?
2. What is the difference between simple interest and compound interest?
3. How would the repayment amount change if the loan term was 12 months instead of 10 months?
4. How is the annual interest rate converted into a monthly interest rate for shorter repayment periods?
5. What happens to the repayment if interest rates increase during the loan period?

**Tip:** Always verify if the interest rate is applied as simple or compound before making calculations, as they significantly affect the repayment amount.

Learn how to calculate the repayment check for a loan of $17,450 at an 8.6% annual interest rate, repaid over 10 months. This problem uses the simple interest formula A = P(1 + rt) to determine the total amount due, including both principal and interest. Suitable for students in Grades 9-11.

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