To solve the problem shown in the image, let's go through each step as you've asked.

Problem Analysis

We need to calculate the monthly payment ($PMT$) for a loan using the formula:

$PMT = \frac{P \cdot \left( \frac{r}{n} \right)}{1 - \left( 1 + \frac{r}{n} \right)^{-nt}}$

where:

• $P$ is the principal (loan amount),
• $r$ is the annual interest rate,
• $n$ is the number of compounding periods per year (monthly, so $n = 12$),
• $t$ is the loan term in years.

Given:

• Loan Amount $P = 11,000$,
• Annual Interest Rate $r = 0.07$,
• Compounding Periods $n = 12$ (monthly),
• Loan Term $t = 3$ years.

Step 1: Substitute the Values

1. Calculate $\frac{r}{n}$: $\frac{0.07}{12} = 0.0058333\ldots$

2. Calculate $-nt$: $-12 \times 3 = -36$

Step 2: Substitute into the PMT Formula

$PMT = \frac{11,000 \times 0.0058333}{1 - \left( 1 + 0.0058333 \right)^{-36}}$

Step 3: Solve the Expression Inside the Denominator

1. Calculate $1 + \frac{r}{n}$: $1 + 0.0058333 = 1.0058333$

2. Calculate $\left( 1.0058333 \right)^{-36}$: $(1.0058333)^{-36} \approx 0.811366$

3. Calculate $1 - 0.811366$: $1 - 0.811366 = 0.188634$

Step 4: Complete the Division

1. Calculate $11,000 \times 0.0058333$: $11,000 \times 0.0058333 \approx 64.1663$

2. Calculate $\frac{64.1663}{0.188634}$: $\frac{64.1663}{0.188634} \approx 340.15$

So, the monthly payment $PMT$ is approximately $340.15.

Step 5: Calculate Total Interest

1. Total Paid Over Loan Term: $340.15 \times 12 \times 3 = 12,245.40$

2. Total Interest: $12,245.40 - 11,000 = 1,245.40$

Thus, the monthly payment is $340.15 and the total interest for the loan is $1,245.40.

Would you like additional details on any step? Here are some related questions:

1. What if the loan term was extended to 5 years instead of 3?
2. How would the monthly payment change if the interest rate was reduced to 5%?
3. What is the impact of making bi-weekly payments instead of monthly payments?
4. Can you explain why compound interest grows faster than simple interest?
5. What are some effective ways to reduce total interest paid on a loan?

Tip: Always try to make extra payments on a loan's principal if possible; it reduces total interest.