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Suppose Alonso needs to purchase a car and is considering the following two options:

- Option 1: A new car that costs [tex]\$21,700[/tex] and can be financed at [tex]1.31\%[/tex] for seven years.
- Option 2: A 1-year-old used car (same model) that costs [tex]\$7,200[/tex] and can be financed at [tex]1.79\%[/tex] for seven years.

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]S\ \square[/tex].

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 [tex]PMT[/tex] 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
To determine the difference between the monthly payments for the two loan options, we need to use the formula for calculating the monthly payment for a fixed-installment loan.

The loan payment formula is given by:
[tex]\[ PMT = \frac{P \left(\frac{r}{n}\right)}{1 - \left(1 + \frac{r}{n}\right)^{-nt}} \][/tex]
where:
- [tex]\(PMT\)[/tex] is the periodic payment (monthly payment in this case).
- [tex]\(P\)[/tex] is the principal amount (loan amount).
- [tex]\(r\)[/tex] is the annual interest rate (as a decimal).
- [tex]\(n\)[/tex] is the number of payments per year.
- [tex]\(t\)[/tex] is the total number of years.

Let's go through this step by step for both car options.

### Option 1: New Car
- Principal ([tex]\(P_{\text{new}}\)[/tex]): \[tex]$21,700 - Annual Interest Rate (\(r_{\text{new}}\)): 1.31% or 0.0131 as a decimal - Time (\(t\)): 7 years - Number of payments per year (\(n\)): 12 Substitute these values into the formula: \[ PMT_{\text{new}} = \frac{21700 \left(\frac{0.0131}{12}\right)}{1 - \left(1 + \frac{0.0131}{12}\right)^{-12 \times 7}} \] ### Option 2: Used Car - Principal (\(P_{\text{used}}\)): \$[/tex]7,200
- Annual Interest Rate ([tex]\(r_{\text{used}}\)[/tex]): 1.79% or 0.0179 as a decimal
- Time ([tex]\(t\)[/tex]): 7 years
- Number of payments per year ([tex]\(n\)[/tex]): 12

Substitute these values into the formula:
[tex]\[ PMT_{\text{used}} = \frac{7200 \left(\frac{0.0179}{12}\right)}{1 - \left(1 + \frac{0.0179}{12}\right)^{-12 \times 7}} \][/tex]

### Calculating the Monthly Payments
After substituting the values and performing the calculations (as mentioned in the numerical results):

- The monthly payment for the new car is: [tex]\(\$270.50\)[/tex] (rounded to the nearest cent).
- The monthly payment for the used car is: [tex]\(\$91.26\)[/tex] (rounded to the nearest cent).

### Difference in Monthly Payments
The difference in monthly payments between the new car and the used car is:
[tex]\[ S = 270.50 - 91.26 = 179.24 \][/tex]

### Final Answer
The difference in the monthly payments is:

[tex]\[ \boxed{179.24} \][/tex]

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