Murphy made a down payment of $2,000 on a car costing $8,000. Murphy will pay the remaining balance in 3 years at an interest rate of 12% per year compounded monthly. (a) What is the monthly payment for the car note? (b) How much interest did Murphy pay?
Murphy made a down payment of $2,000 on a car costing $8,000. Murphy will pay the remaining balance in 3 years at an interest rate of 12% per year compounded monthly. (a) What is the monthly payment for the car note? (b) How much interest did Murphy pay?
Essentials Of Investments
11th Edition
ISBN:9781260013924
Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Chapter1: Investments: Background And Issues
Section: Chapter Questions
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![## Car Loan Calculation Example
Murphy made a down payment of $2,000 on a car costing $8,000. Murphy will pay the remaining balance in 3 years at an interest rate of 12% per year, compounded monthly.
### Questions:
1. **What is the monthly payment for the car note?**
2. **How much interest did Murphy pay?**
### Solution:
**Remaining Balance After Down Payment:**
The initial cost of the car is $8,000, and Murphy made a down payment of $2,000. Therefore, the remaining balance to be financed is:
\[ \text{Remaining Balance} = \$8,000 - \$2,000 = \$6,000 \]
**Monthly Interest Rate:**
The annual interest rate is 12%, compounded monthly. Therefore, the monthly interest rate is:
\[ \text{Monthly Interest Rate} = \frac{12\%}{12} = 1\% = 0.01 \]
**Number of Monthly Payments:**
The loan duration is 3 years. Therefore, the total number of monthly payments is:
\[ \text{Number of Monthly Payments} = 3 \, \text{years} \times 12 \, \text{months/year} = 36 \, \text{months} \]
**Monthly Payment Calculation:**
The formula for calculating the monthly payment \(M\) for a loan is given by the formula:
\[ M = P \frac{r(1+r)^n}{(1+r)^n-1} \]
Where:
- \(M\) is the monthly payment.
- \(P\) is the loan principal (remaining balance).
- \(r\) is the monthly interest rate.
- \(n\) is the number of monthly payments.
Plugging in the values, we get:
\[ M = 6000 \frac{0.01(1+0.01)^{36}}{(1+0.01)^{36} - 1} \]
Using a calculator, let's compute this value.
**Total Interest Paid:**
The total amount paid over the loan duration can be found by multiplying the monthly payment by the number of payments. The interest paid is the total amount paid minus the loan principal.
\[ \text{Total Amount Paid} = M \times n \]
\[ \text{Total Interest Paid} = (\text{Total Amount Paid}) - P \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F72a6c93d-fdfd-4269-a1ad-44ac8a950dd4%2F5310195d-5d8f-42c8-98d4-014c8743fd30%2Fccbatl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Car Loan Calculation Example
Murphy made a down payment of $2,000 on a car costing $8,000. Murphy will pay the remaining balance in 3 years at an interest rate of 12% per year, compounded monthly.
### Questions:
1. **What is the monthly payment for the car note?**
2. **How much interest did Murphy pay?**
### Solution:
**Remaining Balance After Down Payment:**
The initial cost of the car is $8,000, and Murphy made a down payment of $2,000. Therefore, the remaining balance to be financed is:
\[ \text{Remaining Balance} = \$8,000 - \$2,000 = \$6,000 \]
**Monthly Interest Rate:**
The annual interest rate is 12%, compounded monthly. Therefore, the monthly interest rate is:
\[ \text{Monthly Interest Rate} = \frac{12\%}{12} = 1\% = 0.01 \]
**Number of Monthly Payments:**
The loan duration is 3 years. Therefore, the total number of monthly payments is:
\[ \text{Number of Monthly Payments} = 3 \, \text{years} \times 12 \, \text{months/year} = 36 \, \text{months} \]
**Monthly Payment Calculation:**
The formula for calculating the monthly payment \(M\) for a loan is given by the formula:
\[ M = P \frac{r(1+r)^n}{(1+r)^n-1} \]
Where:
- \(M\) is the monthly payment.
- \(P\) is the loan principal (remaining balance).
- \(r\) is the monthly interest rate.
- \(n\) is the number of monthly payments.
Plugging in the values, we get:
\[ M = 6000 \frac{0.01(1+0.01)^{36}}{(1+0.01)^{36} - 1} \]
Using a calculator, let's compute this value.
**Total Interest Paid:**
The total amount paid over the loan duration can be found by multiplying the monthly payment by the number of payments. The interest paid is the total amount paid minus the loan principal.
\[ \text{Total Amount Paid} = M \times n \]
\[ \text{Total Interest Paid} = (\text{Total Amount Paid}) - P \
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