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+r/n)nt−1] over r/n(1+r/n)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

Given:
- Monthly payment P = $350
- Term t = 4 years ⇒ number of payments N = n t = 12·4 = 48
- Formula (as given): A = P · [ (1 + r/n)^{nt} − 1 ] / [ (r/n) (1 + r/n)^{nt} ]
- We will use n = 12 (monthly compounding / monthly payments).

1) New car: APR r = 2.79% = 0.0279
- i = r/n = 0.0279/12 = 0.002325
- 1 + i = 1.002325

Compute (1 + i)^{48}:
- ln(1 + i) ≈ i − i^2/2 + i^3/3 − … ≈ 0.002325 − (0.002325^2)/2 + (0.002325^3)/3
= 0.002325 − 0.000002703125 + 0.000000004204668 ≈ 0.00232230107 (rounded)
- 48·ln(1+i) ≈ 48·0.00232230107 = 0.11146805
- (1 + i)^{48} = e^{0.11146805} ≈ 1.11791782 (rounded to 8 decimal places)

Now evaluate the factor:
- numerator = (1 + i)^{48} − 1 = 1.11791782 − 1 = 0.11791782
- denominator = i · (1 + i)^{48} = 0.002325 · 1.11791782 = 0.00259915895 (rounded)

So
- factor = numerator / denominator = 0.11791782 / 0.00259915895 ≈ 45.367684 (rounded)
- A_new = P · factor = 350 · 45.367684 ≈ $15,878.69

(rounded to the nearest cent)

2) Used car: APR r = 3.29% = 0.0329
- i = r/n = 0.0329/12 = 0.0027416666667
- 1 + i = 1.0027416666667

Compute (1 + i)^{48}:
- ln(1 + i) ≈ i − i^2/2 + i^3/3 ≈ 0.002741666667 − (0.002741666667^2)/2 + … ≈ 0.00273790813 (rounded)
- 48·ln(1+i) ≈ 48·0.00273790813 = 0.131419589
- (1 + i)^{48} = e^{0.131419589} ≈ 1.14044584 (rounded)

Now evaluate the factor:
- numerator = 1.14044584 − 1 = 0.14044584
- denominator = i · (1 + i)^{48} = 0.0027416666667 · 1.14044584 ≈ 0.00312672151 (rounded)

So
- factor = 0.14044584 / 0.00312672151 ≈ 44.917925 (rounded)
- A_used = 350 · 44.917925 ≈ $15,721.27

(rounded to the nearest cent)

Final answers:
- Maximum loan amount for a new car (2.79% APR, 48 months, $350/mo): about $15,878.69
- Maximum loan amount for a used car (3.29% APR, 48 months, $350/mo): about $15,721.27

I showed the intermediate computations and indicated where values were rounded.