Use PMT \( =\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} \) to determine the regular payment amount, rounded to the nearest dollar. The price of a small cabin is \( \$ 45,000 \). The bank requires a \( 5 \% \) down payment. The buyer is offered two mortgage options: 20-year fixed at \( 10 \% \) or 30 -year fixed at 10\%. Calculate the amount of interest paid for each option. How much does the buyer save in interest with the 20-year option? Find the monthly payment for the 20-year option. \( \$ \square \) (Round to the nearest dollar as needed.)
Solución de inteligencia artificial de Upstudy
Responder
Solución

¡Inicia sesión para desbloquear respuestas gratis!
Una plataforma de aprendizaje en la que confían millones de estudiantes y profesores reales.

Respondido por UpStudy AI y revisado por un tutor profesional
The Deep Dive
To determine the monthly payment for the 20-year mortgage option using the PMT formula, we first need to calculate the down payment and mortgage amount. The price of the cabin is \( \$45,000 \). The down payment at \( 5\% \) is: \[ \text{Down Payment} = 45000 \times 0.05 = 2250 \] The mortgage amount (loan principal) after the down payment is: \[ \text{Mortgage Amount} = 45000 - 2250 = 42750 \] For a 20-year fixed mortgage at \( 10\% \) (or \( 0.10 \)), with \( n = 12 \) (monthly payments), and \( t = 20 \): \[ PMT = \frac{42750 \left(\frac{0.10}{12}\right)}{1 - \left(1 + \frac{0.10}{12}\right)^{-12 \times 20}} \] Calculating the values: \[ PMT = \frac{42750 \cdot \left(\frac{0.10}{12}\right)}{1 - \left(1+\frac{0.10}{12}\right)^{-240}} \] First, calculate \( \frac{0.10}{12} = 0.0083333 \), then: \[ PMT = \frac{42750 \cdot 0.0083333}{1 - (1 + 0.0083333)^{-240}} \] Calculating \( (1 + 0.0083333)^{-240} \): \[ (1 + 0.0083333)^{-240} \approx 0.122042 \] Thus, \[ PMT \approx \frac{42750 \cdot 0.0083333}{1 - 0.122042} = \frac{356.25}{0.877958} \approx 405.25 \] Rounding to the nearest dollar, the monthly payment is: \[ PMT \approx \$405 \] To find the total interest paid over the mortgage period, calculate the total payments minus the loan amount: \[ \text{Total Payments} = PMT \times \text{total months} = 405 \times 240 = 97200 \] \[ \text{Total Interest} = \text{Total Payments} - \text{Loan Amount} = 97200 - 42750 = 54450 \] Now comparing it with the 30-year mortgage option: Using the same steps for a 30-year mortgage: \[ PMT = \frac{42750 \left(\frac{0.10}{12}\right)}{1 - \left(1 + \frac{0.10}{12}\right)^{-12 \times 30}} \] Calculate \( t = 30 \): Calculating \( (1 + 0.0083333)^{-360} \) gives approximately \( 0.0299614 \): \[ PMT \approx \frac{42750 \cdot 0.0083333}{1 - 0.0299614} \approx \frac{356.25}{0.9700386} \approx 367.17 \] Rounding gives \( PMT \approx \$367 \). Finding total interest for the 30-year option: \[ \text{Total Payments} = 367 \times 360 = 132120 \] \[ \text{Total Interest} = 132120 - 42750 = 89400 \] Finally, the savings in interest with the 20-year option compared to the 30-year option: \[ \text{Savings in Interest} = 89400 - 54450 = 34950 \] In conclusion: - The monthly payment for the 20-year option is \( \boxed{405} \). - The buyer saves \( \boxed{34950} \) in interest with the 20-year option.
