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Suppose Matthew has decided to buy a new car for [tex]\$ 65,200[/tex], and the dealership has presented him with two incentive options:

- Option A: [tex]\$ 3,600[/tex] discount off the price of the new car accompanied by a [tex]5.39\%[/tex] loan for seven years.
- Option B: No discount off the price of the car, but the car can be financed at [tex]0\%[/tex] interest for seven years.

Matthew plans to make a [tex]\$ 7,500[/tex] down payment, regardless of the incentive option he chooses.

1. Determine the difference between the monthly payments of the two loan options. Round solutions to the nearest cent, if necessary.

The difference in the monthly payments is [tex]\$\square[/tex].

2. Determine which incentive option is most economical over the life of the loan.

A. Incentive Option A is most economical.
B. Incentive Option B is most economical.
C. The incentive options are equal in cost.

Hint: Related Formula
The loan payment formula for fixed installment loans is given by the expression

[tex]\[
PMT = \frac{P \left( \frac{r}{n} \right)}{ \left[ 1 - \left( 1 + \frac{r}{n} \right)^{-nt} \right] }
\][/tex]

where:
- PMT is the periodic payment required to repay a loan of [tex]P[/tex] dollars,
- paid [tex]n[/tex] times per year over [tex]t[/tex] years,
- at an annual interest rate of [tex]r\%[/tex].



Answer :

GinnyAnswer
Alright, let's break down the problem and find the solution step-by-step.

### Step 1: Calculate Loan Amount for Each Option
- Option A:
- Discount on car: \[tex]$3,600 - Price after discount: \$[/tex]65,200 - \[tex]$3,600 = \$[/tex]61,600
- Down payment: \[tex]$7,500 - Loan amount \(P_A\): \$[/tex]61,600 - \[tex]$7,500 = \$[/tex]54,100

- Option B:
- No discount, so the price remains: \[tex]$65,200 - Down payment: \$[/tex]7,500
- Loan amount [tex]\(P_B\)[/tex]: \[tex]$65,200 - \$[/tex]7,500 = \[tex]$57,700 ### Step 2: Define the Payment Formula The loan payment formula for fixed installment loans is: \[ PMT = \frac{P \left(\frac{r}{n}\right)}{\left[1 - \left(1 + \frac{r}{n}\right)^{-nt}\right]} \] Where: - \(P\) is the loan amount - \(r\) is the annual interest rate - \(n\) is the number of payments per year (monthly payments mean \(n = 12\)) - \(t\) is the loan term in years ### Step 3: Calculate Monthly Payments for Each Option #### Option A: With Interest \( P_A = \$[/tex]54,100 \)
[tex]\( r_A = 5.39\% = 0.0539 \)[/tex]
[tex]\( n = 12 \)[/tex]
[tex]\( t = 7 \)[/tex] years

Substitute these values into the formula:
[tex]\[ PMT_A = \frac{5400 \left(\frac{0.0539}{12}\right)}{\left[1 - \left(1 + \frac{0.0539}{12}\right)^{-12 \times 7}\right]} \][/tex]

First, calculate the monthly interest rate:
[tex]\[ \frac{0.0539}{12} = 0.004491667 \][/tex]

Next, calculate the denominator:
[tex]\[ 1 + 0.004491667 = 1.004491667 \][/tex]
[tex]\[ 1.004491667^{-84} \approx 0.665662374 \][/tex]
[tex]\[ 1 - 0.665662374 = 0.3343376264 \][/tex]

Therefore:
[tex]\[ PMT_A = \frac{5400 \times 0.004491667}{0.3343376264} = \frac{242.943867}{0.3343376264} \approx 726.65 \][/tex]

#### Option B: No Interest
[tex]\( P_B = \$57,700 \)[/tex]
[tex]\( r_B = 0.00 \)[/tex]
[tex]\( n = 12 \)[/tex]
[tex]\( t = 7 \)[/tex]

Substitute these values into the formula:
[tex]\[ PMT_B = \frac{57700 \left(\frac{0}{12}\right)}{\left[1 - \left(1 + \frac{0}{12}\right)^{-12 \times 7}\right]} = \frac{57700 \times 0}{\left[1 - 1\right]} = \frac{0}{0} \approx 57700 / 84 \approx 686.90 \][/tex]

### Step 4: Find the Difference in Monthly Payments
Subtract the monthly payments:
[tex]\[ \text{Difference} = PMT_A - PMT_B = 726.65 - 686.90 = 39.75 \][/tex]

The difference in the monthly payments is [tex]$39.75 ### Step 5: Determine the Most Economical Option #### Total Cost of Option A: \[ \text{Total Cost}_A = (726.65 \times 12 \times 7) + 7500 = \approx 72776.60 + 7500 = 80276.60 \] #### Total Cost of Option B: \[ \text{Total Cost}_B = (686.90 \times 12 \times 7) + 7500 = \approx 69157.6 + 7500 = 76657.60 \] Comparing the total costs: \[ \text{Option A Total Cost} = \$[/tex] 80276.60 \]
[tex]\[ \text{Option B Total Cost} = \$ 76657.60 \][/tex]

Option B is the most economical.

### Final Answers:
- Difference in monthly payments: [tex]\(\$ 39.75\)[/tex]
- Most economical option: [tex]\(\text{Option B}\)[/tex]