Philip buys a new car at the price of $32500, where he makes 20% down payment and finances the remaining balance at the interest rate of 6% compounded monthly. The loan will be amortized (paid back) with 48 monthly payments.
How much does Philip have to pay back each month?

To determine how much Philip has to pay back each month, we can follow these steps:

### Step 1: Calculate the Down Payment
Philip makes a 20% down payment on the car's purchase price.

\[
\text{Down Payment} = 20\% \times 32,500 = 0.20 \times 32,500 = 6,500
\]

### Step 2: Calculate the Loan Amount
The loan amount will be the purchase price minus the down payment.

\[
\text{Loan Amount} = 32,500 - 6,500 = 26,000
\]

### Step 3: Identify the Loan Terms
- Loan amount: $26,000
- Interest rate: 6% compounded monthly (0.06 annual interest rate)
- Number of payments: 48 months

We will use the following formula for calculating the monthly payment for an amortized loan:

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

Where:
- $M$ is the monthly payment
- $P$ is the loan amount ($26,000)
- $r$ is the monthly interest rate (annual rate divided by 12)
  \[
  r = \frac{6\%}{12} = \frac{0.06}{12} = 0.005
  \]
- $n$ is the total number of monthly payments (48 months)

### Step 4: Plug in the Values
Now we substitute the values into the formula:

\[
M = \frac{26,000 \times 0.005 \times (1 + 0.005)^{48}}{(1 + 0.005)^{48} - 1}
\]

Let me calculate this for you.Philip will have to pay approximately **$610.61** per month for 48 months to repay the loan.

Would you like more details or have any questions?

Here are 5 related questions to expand this concept:
1. How much total interest will Philip pay over the 48 months?
2. What would be the monthly payment if the loan term were extended to 60 months?
3. How would the monthly payment change if the interest rate was 4% instead of 6%?
4. What is the total amount Philip will pay for the car including both the down payment and monthly payments?
5. How much principal is remaining after 24 months of payments?

**Tip:** When taking out a loan, understanding the difference between the interest rate and the loan term helps you manage your monthly payments and overall cost effectively.

Philip buys a new car at the price of $32500, where he makes 20% down payment and finances the remaining balance at the interest rate of 6% compounded monthly. The loan will be amortized (paid back) with 48 monthly payments. How much does Philip have to pay back each month?

Calculate the monthly payment for a car loan with a principal of $26,000, a 6% interest rate compounded monthly, and a term of 48 months. Learn about amortization, compound interest, and financial mathematics in this detailed solution, suitable for Grades 11-12.

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