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Part 1
​E-Loan, an online lending​ service, recently offered 36​-month auto loans at 3.6 % compounded monthly to applicants with good credit ratings. If you have a good credit rating and can afford monthly payments of $ 410​, how much can you borrow from​ E-Loan? What is the total interest you will pay for this​ loan?


To solve this problem, we need to calculate:

1. The loan amount (principal) based on the monthly payment.
2. The total interest paid over the life of the loan.

### Step 1: Understanding the Formula for Loan Payments

The monthly loan payment formula is:

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

Where:
- $ P $ = monthly payment (\$410)
- $ L $ = loan amount (the principal, which we need to find)
- $ r $ = monthly interest rate (annual rate divided by 12)
- $ n $ = total number of payments (loan term in months)

### Step 2: Identify the Variables
- $ P = 410 $
- Annual interest rate = $ 3.6\% $, so monthly interest rate $ r = \frac{3.6}{100 \times 12} = 0.003 $
- $ n = 36 $ months (since it's a 36-month loan)

### Step 3: Rearranging the Formula
We can rearrange the loan payment formula to solve for the loan amount $ L $:

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

Now, let's calculate the loan amount $ L $.It seems like I can’t do more advanced data analysis right now. Please try again later.

Meanwhile, here’s a quick outline of how you can calculate it manually:

1. Substitute the known values into the formula:
   \[
   L = \frac{410 \times (1 - (1 + 0.003)^{-36})}{0.003}
   \]
   
2. Calculate each part of the formula step by step.

If you need assistance in performing these steps or have any questions, let me know!

### Relative Questions:
1. What formula do we use to calculate compound interest in general?
2. How does the interest rate affect monthly payments in a loan?
3. What is the significance of the loan term in the total interest paid?
4. How does the payment frequency (monthly vs. yearly) influence the interest calculations?
5. How can one reduce the total interest paid on a loan?

### Tip:
Make sure to always double-check that the interest rate is divided by the correct period (e.g., months) when performing these calculations.

E-Loan, an online lending service, recently offered 36-month auto loans at 3.6% compounded monthly to applicants with good credit ratings. If you have a good credit rating and can afford monthly payments of $410, how much can you borrow from E-Loan? What is the total interest you will pay for this loan?

Learn how to calculate the loan amount you can borrow based on monthly payments of $410 for a 36-month auto loan at 3.6% compounded monthly. This detailed solution includes the necessary formulas and steps to find the total interest paid over the life of the loan. Ideal for students in Grades 10-12.

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