Financial institutions often offer lower auto loan rates for new cars than used ones. A local credit union advertises new car loans at 2.79% APR and used car loans at 3.29% both for up to 72 months. Tyresa wants to buy a car but doesn’t want to spend more than $350 a month for a maximum of four years. What is the maximum loan amount she can take out for a new car and a used car using the advertised rates? Use the formula, A=P[(1+rn)nt−1]rn(1+rn)nt where P is the monthly payment, r is the annual interest rate, n is the number times interest is compounded in one year, and t is the number of years. To receive full credit, be sure to show all steps and calculations used, included where you rounded.

Question

GPT-4o mini · Answer

To determine the maximum loan amount Tyresa can take out for a new car and a used car, we will use the loan payment formula:

\[ A = P \left[ \frac{(1 + rn)^{nt} - 1}{rn(1 + rn)^{nt}} \right] \]

Where:

$ A $ is the loan amount (which we are solving for)
$ P $ is the monthly payment
$ r $ is the annual interest rate (as a decimal)
$ n $ is the number of times interest is compounded per year
$ t $ is the number of years

Given the scenario:

Monthly payment $ P = 350 $
For new cars, the annual interest rate $ r = 2.79% = 0.0279 $
For used cars, the annual interest rate $ r = 3.29% = 0.0329 $
Compounding frequency $ n = 12 $
The total term $ t = 4 $ years

First, we need to calculate $ nt $: \[ nt = 12 \times 4 = 48 \]

Now we can update the formula:

For the new car:

Calculate $ rn $: \[ rn = \frac{0.0279}{12} = 0.00232583333 \]
Now plug the values into the formula to find $ A $: \[ A = 350 \left[ \frac{\left(1 + 0.00232583333\right)^{48} - 1}{0.00232583333\left(1 + 0.00232583333\right)^{48}} \right] \]
Calculate $ \left(1 + 0.00232583333\right)^{48} $:
- $ \left(1 + 0.00232583333\right)^{48} \approx 1.116305383 $
Now substituting that back in: \[ A = 350 \left[ \frac{1.116305383 - 1}{0.00232583333 \times 1.116305383} \right] \] \[ = 350 \left[ \frac{0.116305383}{0.00259361773} \right] \approx 350 \times 44.83257865 \approx 15791.41 \]

Maximum loan amount for new car: \[ A_{new} \approx 15791.41 \]

For the used car, we repeat the process:

Calculate $ rn $ for the used car: \[ rn = \frac{0.0329}{12} = 0.00274166667 \]
Using the same formula: \[ A = 350 \left[ \frac{\left(1 + 0.00274166667\right)^{48} - 1}{0.00274166667\left(1 + 0.00274166667\right)^{48}} \right] \]
Calculate $ \left(1 + 0.00274166667\right)^{48} $:
- $ \left(1 + 0.00274166667\right)^{48} \approx 1.144557833 $
Substituting that back in: \[ A = 350 \left[ \frac{1.144557833 - 1}{0.00274166667 \times 1.144557833} \right] \] \[ = 350 \left[ \frac{0.144557833}{0.00313539365} \right] \approx 350 \times 46.07363143 \approx 16125.76 \]

Maximum loan amount for used car: \[ A_{used} \approx 16125.76 \]

Final Answer:

Maximum loan amount for a new car: $15,791.41
Maximum loan amount for a used car: $16,125.76