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make sure the answer is not an approximate

Let's solve this problem step by step using the loan payment formula provided:

The formula for the monthly payment (PMT) is:
\[
\text{PMT} = \frac{P \cdot \frac{r}{n}}{1 - \left(1 + \frac{r}{n}\right)^{-nt}}
\]
where:
- $ P = 15,000 $: the loan principal (amount borrowed).
- $ r = 0.05 $: the annual interest rate (5%).
- $ n = 12 $: the number of payments per year (monthly payments, so 12 per year).
- $ t = 4 $: the loan term in years.

Now, let's calculate each component:

### Step 1: Substitute values
\[
\text{PMT} = \frac{15000 \cdot \frac{0.05}{12}}{1 - \left(1 + \frac{0.05}{12}\right)^{-12 \cdot 4}}
\]

### Step 2: Simplify the interest rate per period
\[
\frac{r}{n} = \frac{0.05}{12} = 0.0041667
\]

### Step 3: Calculate the denominator
Since $ n \cdot t = 12 \cdot 4 = 48 $:
\[
1 - \left(1 + 0.0041667\right)^{-48} = 1 - (1.0041667)^{-48}
\]

Let's calculate $ (1.0041667)^{-48} $:
\[
(1.0041667)^{-48} \approx 0.8171
\]
Thus, the denominator becomes:
\[
1 - 0.8171 = 0.1829
\]

### Step 4: Calculate the numerator
\[
15000 \cdot 0.0041667 = 62.5
\]

### Step 5: Calculate PMT
\[
\text{PMT} = \frac{62.5}{0.1829} \approx 341.91
\]

### Final Answer
The monthly payment is **$341.91**.

Would you like details on calculating total interest or have any other questions?

Here are five related questions:
1. How would the monthly payment change if the interest rate increased to 6%?
2. What would be the total cost of the loan over four years?
3. How much interest is paid in the first year compared to the last year?
4. What happens to the monthly payment if the loan term is extended to five years?
5. How is the monthly payment formula derived from the annuity formula?

**Tip:** When calculating monthly loan payments, always check for rounding instructions at the end to ensure accuracy in your final answer.

Suppose that you borrow $15,000 for four years at 5% toward the purchase of a car. Use the formula provided to find the monthly payments and the total interest for the loan.

Learn how to calculate the monthly payment and total interest for a $15,000 loan over 4 years at a 5% annual interest rate. This example uses the loan payment formula, covering compound interest concepts and step-by-step calculations.

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