https://file.aiquickdraw.com/mathbot/file/1730653595176ihgysypl.jpg

To solve for the monthly payment (PMT) on the loan, let's break down the formula provided in the image:

\[
\text{PMT} = \frac{P \left( \frac{r}{n} \right)}{1 - \left( 1 + \frac{r}{n} \right)^{-nt}}
\]

where:
- $ P = 10,000 $ (the principal or loan amount),
- $ r = 0.06 $ (the annual interest rate, expressed as a decimal),
- $ n = 12 $ (the number of payments per year, since payments are monthly),
- $ t = 3 $ (the loan term in years).

### Step 1: Substitute the values into the formula
1. **Calculate** $\frac{r}{n}$:
   \[
   \frac{r}{n} = \frac{0.06}{12} = 0.005
   \]

2. **Plug in the values** to the formula:
   \[
   \text{PMT} = \frac{10000 \times 0.005}{1 - \left(1 + 0.005\right)^{-12 \times 3}}
   \]

3. **Simplify the exponent** in the denominator:
   \[
   \text{PMT} = \frac{10000 \times 0.005}{1 - (1.005)^{-36}}
   \]

4. **Calculate** $ (1.005)^{-36} $:
   - $ (1.005)^{-36} \approx 0.8356 $

5. **Complete the calculation**:
   - Denominator: $ 1 - 0.8356 = 0.1644 $
   - Numerator: $ 10000 \times 0.005 = 50 $

\[
\text{PMT} = \frac{50}{0.1644} \approx 304.41
\]

### Final Answer:
The monthly payment is approximately **$304.41**.

Would you like further details on the calculation process or help with understanding the total interest?

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Here are some related questions that might help expand understanding:

1. How is the total interest on the loan calculated over the full term?
2. What would happen to the monthly payment if the interest rate increased to 8%?
3. How would a shorter loan term, like 2 years, affect the monthly payment?
4. Why do we use the monthly rate in loan calculations instead of the annual rate directly?
5. What would be the monthly payment if payments were made quarterly instead?

**Tip:** Always break down loan payment formulas by handling the exponents and fractional terms carefully to avoid calculation errors.

Suppose that you borrow $10,000 for three years at 6% toward the purchase of a car. Use the formula PMT = $\frac{P(\frac{r}{n})}{1 - (1 + \frac{r}{n})^{-nt}}$ to find the monthly payments and the total interest for the loan.

Learn how to calculate monthly payments on a $10,000 car loan with a 6% interest rate over three years. This problem demonstrates the use of the amortization formula to find the monthly payment amount, making it ideal for high school students studying financial mathematics.

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