Question

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.

Answer 1

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 $r=2.79$
• For used cars, the annual interest rate $r=3.29$
• 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:

1. Calculate $rn$: $$rn=\frac{0.0279}{12}=0.00232583333$$

2. Now plug the values into the formula to find $A$: $$A=350\left[\frac{{(1+0.00232583333)}^{48}-1}{0.00232583333{(1+0.00232583333)}^{48}}\right]$$

3. Calculate ${(1+0.00232583333)}^{48}$:

   • ${(1+0.00232583333)}^{48}\approx 1.116305383$

4. 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:

1. Calculate $rn$ for the used car: $$rn=\frac{0.0329}{12}=0.00274166667$$

2. Using the same formula: $$A=350\left[\frac{{(1+0.00274166667)}^{48}-1}{0.00274166667{(1+0.00274166667)}^{48}}\right]$$

3. Calculate ${(1+0.00274166667)}^{48}$:

   • ${(1+0.00274166667)}^{48}\approx 1.144557833$

4. 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