Darla purchased a new car during a special sales promotion by the manufacturer. She secured a loan from the manufacturer in the amount of $20,000 at a rate of 4.8%/year compounded monthly. Her bank is now charging 6.2%/year compounded monthly for new car loans. Assuming that each loan would be amortized by 36 equal monthly installments, determine the amount of interest she would have paid at the end of 3 years for each loan. How much less will she have paid in interest payments over the life of the loan by borrowing from the manufacturer instead of her bank? (Round your answers to the nearest cent.)

To solve this problem, we need to calculate the interest paid on each loan over the 36 months and then compare the two amounts. Here’s how we do it step-by-step:

### Step 1: Determine the monthly payment for each loan

The formula to calculate the monthly payment $ M $ for a loan is:

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

Where:
- $ P $ is the principal amount of the loan.
- $ r $ is the monthly interest rate (annual interest rate divided by 12).
- $ n $ is the total number of payments (loan term in months).

### For the Manufacturer's Loan:
- Principal $ P = 20,000 $
- Annual interest rate $ = 4.8\% $
- Monthly interest rate $ r = \frac{4.8\%}{12} = 0.004 $ (or 0.048/12)
- Number of payments $ n = 36 $

### For the Bank's Loan:
- Principal $ P = 20,000 $
- Annual interest rate $ = 6.2\% $
- Monthly interest rate $ r = \frac{6.2\%}{12} = 0.005167 $ (or 0.062/12)
- Number of payments $ n = 36 $

### Step 2: Calculate the total payment over the life of each loan

The total payment for the loan is calculated by multiplying the monthly payment by the number of payments.

\[
\text{Total Payment} = M \times n
\]

### Step 3: Calculate the total interest paid

The total interest paid is the difference between the total payment and the principal.

\[
\text{Total Interest} = \text{Total Payment} - P
\]

### Let's perform these calculations:

1. **For the Manufacturer's Loan:**
   - Monthly payment $ M_m $:

   \[
   M_m = \frac{20000 \times 0.004 \times (1 + 0.004)^{36}}{(1 + 0.004)^{36} - 1}
   \]

2. **For the Bank's Loan:**
   - Monthly payment $ M_b $:

   \[
   M_b = \frac{20000 \times 0.005167 \times (1 + 0.005167)^{36}}{(1 + 0.005167)^{36} - 1}
   \]

After calculating the monthly payments, we can then determine the total interest paid and compare them.

Let me calculate these values for you.Here are the results:

1. **Total Interest Paid with the Manufacturer's Loan:**
   - $ \$1,514.45 $

2. **Total Interest Paid with the Bank's Loan:**
   - $ \$1,969.10 $

3. **Interest Savings by Borrowing from the Manufacturer:**
   - $ \$454.65 $

By securing the loan from the manufacturer instead of the bank, Darla will save **$454.65** in interest payments over the life of the loan.

Would you like any further details or have any questions?

Here are some related questions:

1. How would the interest payments change if the loan term was 48 months instead of 36?
2. What would be the effect of a 0.5% increase in the bank's interest rate?
3. How would a down payment affect the total interest paid on either loan?
4. What is the impact of changing from monthly compounding to quarterly compounding?
5. How do bi-weekly payments instead of monthly payments impact the total interest?

**Tip:** Making extra payments towards your loan principal can significantly reduce the total interest you pay over the life of the loan.

Calculate and compare the interest paid on a $20,000 car loan at 4.8% and 6.2% interest rates over 36 months. This detailed solution covers loan amortization, compound interest, and interest rate comparison, suitable for students in Grades 10-12.

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