Philip buys a new car, where he makes $500 down payment and finances the remaining balance at the interest rate of 15% compounded monthly. The loan will be amortized (paid back) with 48 monthly payments of $180 each
What is the cash price of this car?

To find the cash price of the car, we need to calculate the total loan amount that Philip is financing and add his down payment.

### Step 1: Breakdown of the Payment Formula
The formula for the monthly payment of an amortized loan with monthly compounding interest is:

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

Where:
- $M$ is the monthly payment (\$180)
- $P$ is the principal (amount Philip borrowed)
- $r$ is the monthly interest rate (annual interest rate divided by 12)
- $n$ is the total number of payments (48 months)

We are solving for $P$, the principal (loan amount).

### Step 2: Calculate the Monthly Interest Rate
The annual interest rate is 15%, so the monthly interest rate is:

\[
r = \frac{15\%}{12} = \frac{0.15}{12} = 0.0125
\]

### Step 3: Rearranging the Formula to Solve for $P$
Rearrange the formula to solve for $P$:

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

Now, substitute the known values:

\[
M = 180, \quad r = 0.0125, \quad n = 48
\]

### Step 4: Calculate the Loan Amount
Let me now compute the loan amount $P$.The loan amount that Philip is financing is approximately \$6,467.67.

### Step 5: Add the Down Payment
The total cash price of the car includes the financed amount plus the down payment of \$500. Therefore:

\[
\text{Total Cash Price} = 6,467.67 + 500 = 6,967.67
\]

Thus, the cash price of the car is approximately **\$6,967.67**.

Would you like further details or clarification on any step?

Here are 5 related questions to explore:
1. How does compounding frequency affect the total interest paid?
2. What would be the new monthly payment if the interest rate were lower?
3. How does a longer loan term impact the total amount paid?
4. If Philip made a larger down payment, how would the loan amount change?
5. How is the amortization schedule affected by extra payments?

**Tip:** When financing, consider the interest paid over time, as reducing the loan term can save on interest costs.

Philip buys a new car, where he makes $500 down payment and finances the remaining balance at the interest rate of 15% compounded monthly. The loan will be amortized (paid back) with 48 monthly payments of $180 each. What is the cash price of this car?

Find out how to calculate the cash price of a car when financing the balance with a 15% interest rate, compounded monthly, and a $500 down payment. This detailed solution includes key concepts like loan amortization, monthly payments, and interest compounding. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation