​E-Loan, an online lending​ service, recently offered 60​-month auto loans at 4.2 % compounded monthly to applicants with good credit ratings. If you have a good credit rating and can afford monthly payments of $ 458​, how much can you borrow from​ E-Loan? What is the total interest you will pay for this​ loan?


To determine how much you can borrow from E-Loan and the total interest you'll pay, we can use the **loan amortization formula**:

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

Where:
- $ A $ = monthly payment (\$458)
- $ P $ = loan principal (amount you can borrow)
- $ r $ = monthly interest rate (annual rate divided by 12)
- $ n $ = total number of payments (loan term in months)

### Step 1: Convert the interest rate to a monthly rate
The annual interest rate is 4.2%, so the monthly interest rate is:

\[
r = \dfrac{4.2\%}{12} = \dfrac{0.042}{12} = 0.0035
\]

### Step 2: Total number of payments
Since this is a 60-month loan, the total number of payments is $ n = 60 $.

### Step 3: Rearrange the loan formula to solve for $ P $
We know the monthly payment $ A $, so we solve for $ P $:

\[
P = \dfrac{A \cdot \left( (1 + r)^n - 1 \right)}{r \cdot (1 + r)^n}
\]

### Step 4: Calculate the loan principal

Let's compute $ P $ using the given values.

The amount you can borrow from E-Loan is approximately **$24,747.51**.

### Step 5: Calculate the total interest paid
The total interest is the difference between the total amount paid over the loan period and the amount borrowed:

\[
\text{Total Interest} = A \times n - P
\]

Let's calculate the total interest.The total interest you will pay for this loan is approximately **$2,732.49**.

### Summary:
- **Loan amount**: $24,747.51
- **Total interest paid**: $2,732.49

Would you like further details or have any other questions? Here are five related questions for deeper understanding:

1. What happens if you make extra payments on the loan?
2. How would the loan change if the interest rate was different?
3. What is the impact of extending the loan term on the total interest?
4. How is the monthly interest rate calculated from the annual percentage rate (APR)?
5. How does compounding frequency affect loan interest?

**Tip**: Making extra payments on a loan can significantly reduce the total interest paid over time!

E-Loan, an online lending service, recently offered 60-month auto loans at 4.2 % compounded monthly to applicants with good credit ratings. If you have a good credit rating and can afford monthly payments of $458, how much can you borrow from E-Loan? What is the total interest you will pay for this loan?

Learn how to calculate the maximum amount you can borrow for a 60-month auto loan at 4.2% compounded monthly. This example includes monthly payments of $458, covering loan amortization and total interest calculation. Suitable for advanced high school or college-level financial math learners.

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