Asked by joe
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.
All Answers 1
Answered by
GPT-5 mini
AI
Use the loan-present-value form of the given formula (with P = monthly payment and A = loan amount):
A = P * [1 − (1 + r/n)^(−nt)] / (r/n)
Given: P = $350, n = 12 (monthly), t = 4 years (nt = 48).
1) New car: r = 2.79% = 0.0279
- monthly rate i = r/n = 0.0279 / 12 = 0.002325
- (1 + i)^(nt) = (1.002325)^(48) ≈ 1.11791844 (rounded)
- (1 + i)^(−nt) = 1 / 1.11791844 ≈ 0.89451266
- numerator: 1 − (1 + i)^(−nt) ≈ 1 − 0.89451266 = 0.10548734
- divide by i: 0.10548734 / 0.002325 ≈ 45.37103
- A = 350 * 45.37103 ≈ $15,879.86
(Rounded intermediate values to 8 significant digits; final answer rounded to cents.)
2) Used car: r = 3.29% = 0.0329
- monthly rate i = r/n = 0.0329 / 12 ≈ 0.0027416666667
- (1 + i)^(nt) = (1.0027416666667)^(48) ≈ 1.1404456 (rounded)
- (1 + i)^(−nt) = 1 / 1.1404456 ≈ 0.87685003
- numerator: 1 − (1 + i)^(−nt) ≈ 1 − 0.87685003 = 0.12314997
- divide by i: 0.12314997 / 0.0027416666667 ≈ 44.91792
- A = 350 * 44.91792 ≈ $15,721.27
(Final answer rounded to cents; intermediate values rounded as shown.)
Summary:
- Maximum loan for a new car (2.79% APR, 72-month advertisement but she wants 4 years): ≈ $15,879.86
- Maximum loan for a used car (3.29% APR): ≈ $15,721.27
Quick check (total paid over 48 months = 350 * 48 = $16,800):
- Interest paid (new) ≈ 16,800 − 15,879.86 = $920.14
- Interest paid (used) ≈ 16,800 − 15,721.27 = $1,078.73
If you want, I can recompute with more digits of precision or show the same steps using the exact algebraic form you gave.
A = P * [1 − (1 + r/n)^(−nt)] / (r/n)
Given: P = $350, n = 12 (monthly), t = 4 years (nt = 48).
1) New car: r = 2.79% = 0.0279
- monthly rate i = r/n = 0.0279 / 12 = 0.002325
- (1 + i)^(nt) = (1.002325)^(48) ≈ 1.11791844 (rounded)
- (1 + i)^(−nt) = 1 / 1.11791844 ≈ 0.89451266
- numerator: 1 − (1 + i)^(−nt) ≈ 1 − 0.89451266 = 0.10548734
- divide by i: 0.10548734 / 0.002325 ≈ 45.37103
- A = 350 * 45.37103 ≈ $15,879.86
(Rounded intermediate values to 8 significant digits; final answer rounded to cents.)
2) Used car: r = 3.29% = 0.0329
- monthly rate i = r/n = 0.0329 / 12 ≈ 0.0027416666667
- (1 + i)^(nt) = (1.0027416666667)^(48) ≈ 1.1404456 (rounded)
- (1 + i)^(−nt) = 1 / 1.1404456 ≈ 0.87685003
- numerator: 1 − (1 + i)^(−nt) ≈ 1 − 0.87685003 = 0.12314997
- divide by i: 0.12314997 / 0.0027416666667 ≈ 44.91792
- A = 350 * 44.91792 ≈ $15,721.27
(Final answer rounded to cents; intermediate values rounded as shown.)
Summary:
- Maximum loan for a new car (2.79% APR, 72-month advertisement but she wants 4 years): ≈ $15,879.86
- Maximum loan for a used car (3.29% APR): ≈ $15,721.27
Quick check (total paid over 48 months = 350 * 48 = $16,800):
- Interest paid (new) ≈ 16,800 − 15,879.86 = $920.14
- Interest paid (used) ≈ 16,800 − 15,721.27 = $1,078.73
If you want, I can recompute with more digits of precision or show the same steps using the exact algebraic form you gave.
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.