Questions: Suppose that you decide to borrow 15,000 for a new car. You can select one of the following loans, each requiring regular monthly payments. Installment Loan A: three-year loan at 5.9% Installment Loan B: five-year loan at 5.8% Use PMT = P(r/n) / [1 - (1 + r/n)^-nt] to complete parts (a) through (c) below. a. Find the monthly payments and the total interest for Loan A. The monthly payment for Loan A is 455.65. The total interest for Loan A is 1403.39. b. Find the monthly payments and the total interest for Loan B. The monthly payment for Loan B is 288.60. The total interest for Loan B is 2315.95 c. Compare the monthly payments and the total interest for the two loans. Determine which loan is more economical. Choose the correct answer below. A. The three-year loan at 5.9% is more economical. B. The five-year loan at 5.8% is more economical.

Suppose that you decide to borrow 15,000 for a new car. You can select one of the following loans, each requiring regular monthly payments. Installment Loan A: three-year loan at 5.9% Installment Loan B: five-year loan at 5.8% Use PMT = P(r/n) / [1 - (1 + r/n)^-nt] to complete parts (a) through (c) below. a. Find the monthly payments and the total interest for Loan A. The monthly payment for Loan A is 455.65. The total interest for Loan A is 1403.39. b. Find the monthly payments and the total interest for Loan B. The monthly payment for Loan B is 288.60. The total interest for Loan B is 2315.95 c. Compare the monthly payments and the total interest for the two loans. Determine which loan is more economical. Choose the correct answer below. A. The three-year loan at 5.9% is more economical. B. The five-year loan at 5.8% is more economical.
Transcript text: Suppose that you decide to borrow $\$ 15,000$ for a new car. You can select one of the following loans, each requiring regular monthly payments. Installment Loan A: three-year loan at $5.9 \%$ Installment Loan B: five-year loan at 5.8\% Use PMT $=\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]}$ to complete parts (a) through (c) below. a. Find the monthly payments and the total interest for Loan $A$. The monthly payment for Loan A is $\$ 455.65$. The total interest for Loan A is $\$ 1403.39$. b. Find the monthly payments and the total interest for Loan $B$. The monthly payment for Loan B is $\$ 288.60$. The total interest for Loan B is $\$ 2315.95$ c. Compare the monthly payments and the total interest for the two loans. Determine which loan is more economical. Choose the correct answer below. A. The three-year loan at $5.9 \%$ is more economical. B. The five-year loan at $5.8 \%$ is more economical.
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Solution

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To solve the problem, we will use the PMT formula to calculate the monthly payments for both loans. Then, we will calculate the total interest paid over the life of each loan by subtracting the principal from the total payments made. Finally, we will compare the total interest and monthly payments to determine which loan is more economical.

Paso 1: Calcular el pago mensual para el Préstamo A

Usamos la fórmula PMT para calcular el pago mensual:

\[ \text{PMT} = \frac{P \left(\frac{r}{n}\right)}{1 - \left(1 + \frac{r}{n}\right)^{-nt}} \]

Para el Préstamo A:

  • \( P = 15000 \)
  • \( r = 5.9\% = 0.059 \)
  • \( n = 12 \) (pagos mensuales)
  • \( t = 3 \) años

\[ \text{PMT}_A = \frac{15000 \left(\frac{0.059}{12}\right)}{1 - \left(1 + \frac{0.059}{12}\right)^{-36}} \approx 455.65 \]

Paso 2: Calcular el interés total para el Préstamo A

El interés total es la diferencia entre el pago total y el principal:

\[ \text{Interés Total}_A = (\text{PMT}_A \times 36) - 15000 \approx 1403.39 \]

Paso 3: Calcular el pago mensual para el Préstamo B

Para el Préstamo B:

  • \( P = 15000 \)
  • \( r = 5.8\% = 0.058 \)
  • \( n = 12 \)
  • \( t = 5 \) años

\[ \text{PMT}_B = \frac{15000 \left(\frac{0.058}{12}\right)}{1 - \left(1 + \frac{0.058}{12}\right)^{-60}} \approx 288.60 \]

Paso 4: Calcular el interés total para el Préstamo B

\[ \text{Interés Total}_B = (\text{PMT}_B \times 60) - 15000 \approx 2315.95 \]

Paso 5: Comparar los préstamos

Comparamos los intereses totales para determinar cuál préstamo es más económico:

  • \(\text{Interés Total}_A \approx 1403.39\)
  • \(\text{Interés Total}_B \approx 2315.95\)

El Préstamo A tiene un interés total menor.

Respuesta Final

El préstamo más económico es el Préstamo A. La respuesta es \(\boxed{\text{A}}\).

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