Suppose that under the Plan of Repayment one should pay off the debt in a number of equal​ end-of-month installments​ (principal and​ interest). This is the customary way to pay off loans on​ automobiles, house​ mortgages, etc. A friend of yours has financed ​$19 comma 000 on the purchase of a new​ automobile, and the annual interest rate is 18​% ​(1.5​% per​ month).
a. Monthly payments over a 24​-month loan period will be how​ much?
b. How much interest and principal will be paid within three month of this​ loan?

To solve this problem, we can use the **loan amortization formula** for fixed monthly payments, which is often used to calculate monthly payments on loans like car loans and mortgages.

### Step 1: Formula for Monthly Payments
The formula for calculating the monthly payment is:

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

Where:
- $ M $ is the monthly payment.
- $ P $ is the loan principal (the amount of the loan), which is $19,000.
- $ r $ is the monthly interest rate (1.5% or 0.015 per month).
- $ n $ is the total number of monthly payments (24 months).

### Part (a) Calculation of Monthly Payment

Using the formula:
\[
P = 19,000, \quad r = 0.015, \quad n = 24
\]

Substituting these values into the formula will give us the monthly payment $ M $.

### Step 2: Calculation of Interest and Principal in the First Three Months
After calculating the monthly payment, we'll compute how much of the payment goes towards interest and how much towards the principal.

For each month, the interest is calculated as:
\[
\text{Interest for month} = \text{Remaining loan balance} \times r
\]

The principal is the remainder of the monthly payment after paying off the interest:
\[
\text{Principal for month} = M - \text{Interest for month}
\]

We will calculate these for the first three months.

Let me calculate that for you.The monthly payment for the loan is approximately **$948.56**.

### Part (b) Interest and Principal Breakdown in the First Three Months

Next, let's calculate how much of the monthly payment goes towards interest and principal during the first three months.

In each month:
1. **Interest** is calculated as $ \text{Interest} = \text{Remaining balance} \times r $.
2. **Principal** is the remainder of the payment after paying the interest, calculated as $ \text{Principal} = M - \text{Interest} $.

Let me compute the interest and principal for each of the first three months.Within the first three months of the loan, here’s how the payments break down:

- **Total Interest Paid**: \$824.99
- **Total Principal Paid**: \$2,020.68

Breakdown by month:
1. **Month 1**:
   - Interest: \$285.00
   - Principal: \$663.56
   - Remaining Balance: \$18,336.44
2. **Month 2**:
   - Interest: \$275.05
   - Principal: \$673.51
   - Remaining Balance: \$17,662.93
3. **Month 3**:
   - Interest: \$264.94
   - Principal: \$683.61
   - Remaining Balance: \$16,979.32

Let me know if you'd like more details or have any questions.

Here are 5 related questions to consider:
1. How would the payments change if the loan period was extended to 36 months?
2. What is the total interest paid over the full 24 months?
3. How would a higher or lower interest rate affect the monthly payment?
4. What is the loan balance after 12 months?
5. How much principal would be paid off in the last 3 months of the loan?

**Tip**: Paying off more of the principal early can reduce the total interest paid over the life of the loan!

Suppose that under the Plan of Repayment one should pay off the debt in a number of equal​ end-of-month installments​ (principal and​ interest). This is the customary way to pay off loans on​ automobiles, house​ mortgages, etc. A friend of yours has financed $19,000 on the purchase of a new​ automobile, and the annual interest rate is 18% (1.5% per month). a. Monthly payments over a 24​-month loan period will be how​ much? b. How much interest and principal will be paid within three months of this​ loan?

Discover how to calculate monthly payments for a $19,000 auto loan with an 18% annual interest rate, using the loan amortization formula. Learn the breakdown of interest and principal paid within the first three months of the loan. Suitable for Grades 11-12 and those learning about personal finance and loans.

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