Problem 12
Question
Car loans. Katie purchases a new Jeep Wrangler Sport for \(\$ 23,000 .\) She makes a \(\$ 5000\) down payment and finances the remainder through an amortized loan at an annual interest rate of \(5.7 \%,\) compounded monthly for 7 yr. a) Find Katie's monthly car payment. b) Assume that Katie makes every payment for the life of the loan. Find her total payments. c) How much interest does Katie pay over the life of the loan?
Step-by-Step Solution
Verified Answer
Katie's monthly payment is approximately \(\$ 261.36\). Her total payments are approximately \(\$ 21,953.76\), and she pays about \(\$ 3,953.76\) in interest.
1Step 1: Calculate Loan Amount
First, we need to determine the loan amount Katie is financing. She purchased the car for \(\\( 23,000\) and made a \(\\) 5000\) down payment. The loan amount is: \[ \\( 23,000 - \\) 5000 = \\( 18,000 \] So, Katie is financing \(\\) 18,000\).
2Step 2: Determine Monthly Interest Rate and Number of Payments
The annual interest rate is \(5.7\%\), and it's compounded monthly. Therefore, the monthly interest rate is \(\frac{5.7}{12}\% = 0.475\%\) or \(\frac{5.7}{100 \times 12} = 0.00475\) in decimal form. Since the loan term is 7 years, the number of months (payments) is: \[ 7 \times 12 = 84 \text{ months} \]
3Step 3: Use Amortization Formula for Monthly Payment
The monthly payment \(M\) is calculated using the amortization formula: \[ M = P \frac{r(1+r)^n}{(1+r)^n - 1} \] where \(P = \\(18,000\) is the principal, \(r = 0.00475\) is the monthly interest rate, and \(n = 84\) is the total number of payments. Substituting these values in, we get: \[ M = 18000 \frac{0.00475(1 + 0.00475)^{84}}{(1 + 0.00475)^{84} - 1} \] Calculating this, the monthly payment \(M\) is approximately \(\\) 261.36\).
4Step 4: Calculate Total Payments
To find the total payment amount over the life of the loan, multiply the monthly payment by the total number of payments: \[ \text{Total Payments} = 261.36 \times 84 \approx \\( 21,953.76 \] Thus, the total amount Katie pays over the loan period is approximately \(\\) 21,953.76\).
5Step 5: Calculate Total Interest Paid
The total interest paid is the difference between the total payments and the principal amount of the loan: \[ \text{Total Interest} = \text{Total Payments} - \text{Principal} = 21,953.76 - 18,000 = \\( 3,953.76 \] Hence, Katie pays approximately \(\\) 3,953.76\) in interest over the life of the loan.
Key Concepts
Monthly payment calculationTotal interest calculationLoan principal
Monthly payment calculation
Calculating monthly payments for an amortized loan involves understanding the loan terms and applying a specific formula. Here, Katie started by identifying that her loan amount is \(18,000, since the car costs \)23,000 and she made a down payment of \(5,000. This initial step gives her the principal, or the starting loan balance.
With the annual interest rate of 5.7%, compounded monthly, Katie needed to convert this to a monthly rate. This was done by dividing 5.7% by 12, which results in about 0.475% per month, and in decimal form, it is 0.00475.
The number of payments over the 7-year term is 84, since 7 years times 12 months per year equals 84 months, or payment periods. With this information, Katie used the amortization formula:
\[ M = P \frac{r(1+r)^n}{(1+r)^n - 1} \]
Substituting in her values: \(P = 18,000\), \(r = 0.00475\), and \(n = 84\), she calculated her monthly payment \(M\), which turned out to be approximately \)261.36. This is her monthly commitment for the lifetime of her loan.
With the annual interest rate of 5.7%, compounded monthly, Katie needed to convert this to a monthly rate. This was done by dividing 5.7% by 12, which results in about 0.475% per month, and in decimal form, it is 0.00475.
The number of payments over the 7-year term is 84, since 7 years times 12 months per year equals 84 months, or payment periods. With this information, Katie used the amortization formula:
\[ M = P \frac{r(1+r)^n}{(1+r)^n - 1} \]
Substituting in her values: \(P = 18,000\), \(r = 0.00475\), and \(n = 84\), she calculated her monthly payment \(M\), which turned out to be approximately \)261.36. This is her monthly commitment for the lifetime of her loan.
Total interest calculation
Calculating the total interest paid over the life of an amortized loan is an important step in understanding how much the loan will cost beyond the principal. After finding the monthly payment, the next step is to calculate the total amount paid over the entire loan period. Katie calculated her total payments by multiplying her monthly payment, \(261.36, by the total number of payments, 84 months.
This gave her a total payment amount of approximately \)21,953.76.
To determine the total interest paid, Katie then needed to find the difference between this total payment and the principal amount of her loan, which was \(18,000.
The formula for total interest is:
\[ \text{Total Interest} = \text{Total Payments} - \text{Principal} \]
Using her calculated values, Katie's total interest is approximately \)3,953.76. This amount represents the extra cost she pays to the lender for borrowing the money over the 7-year period.
This gave her a total payment amount of approximately \)21,953.76.
To determine the total interest paid, Katie then needed to find the difference between this total payment and the principal amount of her loan, which was \(18,000.
The formula for total interest is:
\[ \text{Total Interest} = \text{Total Payments} - \text{Principal} \]
Using her calculated values, Katie's total interest is approximately \)3,953.76. This amount represents the extra cost she pays to the lender for borrowing the money over the 7-year period.
Loan principal
The concept of the loan principal is central to understanding amortized loans. The principal is the amount of money that is initially borrowed, not accounting for any interest or fees. For Katie, who purchased a car, her principal was the leftover cost of the car after making a down payment.
Initially, the price of the Jeep Wrangler Sport was $23,000. Katie made a down payment of $5,000, which is a common practice to reduce the amount that needs to be financed. By subtracting this down payment from the car's purchase price, the amount left to be financed becomes $18,000. This $18,000 is her principal.
Understanding the principal amount is crucial because it is the basis upon which the interest is charged. The principal, combined with the interest rate and the loan term (number of payments), determines the monthly payments and ultimately how much interest is paid over the life of the loan. Proper assessment of the principal allows borrowers to plan better and ensure they can manage the payments effectively.
Initially, the price of the Jeep Wrangler Sport was $23,000. Katie made a down payment of $5,000, which is a common practice to reduce the amount that needs to be financed. By subtracting this down payment from the car's purchase price, the amount left to be financed becomes $18,000. This $18,000 is her principal.
Understanding the principal amount is crucial because it is the basis upon which the interest is charged. The principal, combined with the interest rate and the loan term (number of payments), determines the monthly payments and ultimately how much interest is paid over the life of the loan. Proper assessment of the principal allows borrowers to plan better and ensure they can manage the payments effectively.
Other exercises in this chapter
Problem 11
Differentiate. $$ f(x)=3^{x^{4}+1} $$
View solution Problem 11
Solve for \(x\). $$ \log _{x} 32=5 $$
View solution Problem 12
Annual net sales. Green Mountain Coffee Roasters produces many varieties of flavored coffees, teas, and K-cups. Since \(2008,\) the net sales \(S\) of the compa
View solution Problem 12
Substance \(A\) decomposes at a rate proportional to the amount of \(A\) present. a) Write an equation that gives the amount \(A\) left of an initial amount \(A
View solution