A recent college graduate buys a new car by borrowing $20,000 at 8.4%, compounded monthly, for 5 years. She decides to pay $435 instead of the monthly payment required by the loan.

(a) What is the monthly payment required by the loan? (Round your answer to the nearest cent.)
$ _____

How much extra did she pay per month? (Round your answer to the nearest cent.)
$ _____

(b) How many $435 payments will she make to pay off the loan? (Round your answer up to the next whole number.)
_____ payments

(c) How much will she save by paying $435 per month rather than the required payment? (Round your answer to the nearest cent.)
$ _____

To solve the problem, we need to address each part step-by-step.

### Part (a): Monthly Payment Required by the Loan

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

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

Where:
- $M$ is the monthly payment.
- $P$ is the principal amount of the loan ($20,000).
- $r$ is the monthly interest rate.
- $n$ is the total number of payments.

Given:
- $P = 20,000$
- Annual interest rate = 8.4% per year, so the monthly interest rate $r$ is $\frac{8.4\%}{12} = 0.007$.
- The loan term is 5 years, so the total number of payments $n$ is $5 \times 12 = 60$.

Let's calculate the monthly payment.

\[
M = 20000 \times \frac{0.007 \times (1 + 0.007)^{60}}{(1 + 0.007)^{60} - 1}
\]

Let's calculate this now.The monthly payment required by the loan is **$409.37** (rounded to the nearest cent).

Next, let's calculate how much extra she paid per month by paying $435 instead of the required $409.37.

\[
\text{Extra payment per month} = 435 - 409.37
\]She paid **$25.63** extra per month (rounded to the nearest cent).

### Part (b): Number of $435 Payments Needed to Pay Off the Loan

To find the number of payments, we can use the formula for the balance of a loan after a certain number of payments:

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

However, since she's paying more than the required monthly payment, we need to calculate how many $435 payments it will take to pay off the loan. We'll do this iteratively by updating the balance after each payment until it reaches zero or becomes negative.

Let's calculate the number of payments required.She will need to make **56 payments** of $435 to pay off the loan.

### Part (c): Total Savings by Paying $435 Per Month

To determine the savings, we calculate the total amount paid under the required payment plan and compare it to the total amount paid with the $435 monthly payment.

1. **Total amount paid with the required payment:**
   \[
   \text{Total payment with required monthly payment} = 60 \times 409.37
   \]

2. **Total amount paid with the $435 payment:**
   \[
   \text{Total payment with $435 per month} = 56 \times 435
   \]

3. **Savings:**
   \[
   \text{Savings} = \text{Total payment with required monthly payment} - \text{Total payment with $435 per month}
   \]It seems like I can’t do more advanced data analysis right now. Please try again later. 

If you have any other questions or need help with something else, feel free to ask!

Learn how to calculate the required monthly payment for a car loan of $20,000 at 8.4% interest compounded monthly over 5 years. Understand how much extra is paid per month, the total number of payments, and the savings when paying more than the required monthly amount. Suitable for students in Grades 11-12 or college freshmen.

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