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In this problem, Dennis Lamenti is trying to buy a car for **$15,657.86** and has two loan options, one from a car dealer and one from a bank. Both loans are 4 years in duration, and we are asked to calculate the monthly payments and total interest for both options.

### Loan Details:

1. **Dealer Loan**: 
   - Loan type: 4-year add-on interest loan at **7.75% (7 3/4%)**
   - Requires a down payment of **$1,000**.

2. **Bank Loan**: 
   - Loan type: 4-year simple interest loan at **7.75% (7 3/4%)**
   - Requires a down payment of **$1,000**.

We will calculate the monthly payment and total interest for both loans using the given information.

---

### **Step-by-Step Approach:**

#### 1. **Determine the Loan Amount:**
   - The car's price is **$15,657.86**, but Dennis will make a down payment of **$1,000**. 
   - The loan amount for both loans is:
     \[
     \text{Loan Amount} = 15,657.86 - 1,000 = 14,657.86
     \]

#### 2. **Dealer Loan (Add-On Interest Loan):**
   The add-on interest loan calculates interest on the full amount of the loan upfront for the entire loan term.

   - Loan amount = **$14,657.86**
   - Interest rate = **7.75% per year**
   - Loan term = **4 years**
   
   Interest = Principal × Rate × Time:
   \[
   \text{Interest} = 14,657.86 \times 0.0775 \times 4
   \]

   The total amount (Principal + Interest) will be divided equally into monthly payments over 4 years (48 months).

#### 3. **Bank Loan (Simple Interest Amortized Loan):**
   In a simple interest amortized loan, payments are made monthly, and interest is applied based on the remaining loan balance. We can use the formula for calculating the monthly payment for a simple interest loan:
   \[
   M = P \times \frac{r(1+r)^n}{(1+r)^n-1}
   \]
   Where:
   - $M$ is the monthly payment,
   - $P$ is the loan amount,
   - $r$ is the monthly interest rate (annual interest rate divided by 12),
   - $n$ is the total number of payments (loan term in months).

Now, I'll proceed to calculate the monthly payments and total interest for each loan.### Calculation Results:

1. **Dealer Loan (Add-On Interest Loan)**:
   - **Total Interest**: \$4,543.94
   - **Monthly Payment**: \$400.04

2. **Bank Loan (Simple Interest Amortized Loan)**:
   - **Total Interest**: \$2,436.07
   - **Monthly Payment**: \$356.12

### Conclusion for Part (c):
Dennis should choose the **bank loan** because the interest is lower. 

Would you like more details or have any questions?

Here are some related questions to explore:

1. How does the loan term affect total interest in amortized loans?
2. What is the difference between simple and compound interest?
3. How does a higher down payment affect monthly payments?
4. What other factors should Dennis consider when choosing between the two loans?
5. What happens if Dennis pays off the loan early?

**Tip:** Choosing loans with lower interest rates can save significant amounts over time, especially for longer loan terms.

Dennis Lamenti wants to buy a new car that costs $15,657.86. He has two possible loans in mind. One loan is through the car dealer; it is a four-year add-on interest loan at 7.75% and requires a down payment of $1,000. The second is through his bank; it is a four-year simple interest amortized loan at 7.75% and requires a down payment of $1,000. Find the monthly payment and total interest for each loan.

Learn how to calculate the monthly payments and total interest for a $15,657.86 car loan with two different options: a 7.75% add-on interest loan and a 7.75% simple interest loan. This problem covers key concepts of loan amortization and interest calculation, suitable for high school students.

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