Question

Can you help me solve this so that I can better understand?

Compare the monthly payments and total loan costs for the following pairs of loan options. Assume that both loans are fixed rate and have the same closing costs. You need a $50,000 loan. Option 1: a 30-year loan at an APR of 5.55%. Option 2: a 15-year loan at an APR of 5.15%. Find the monthly payment for each option. The monthly payment for option 1 is $- The monthly payment for option 2 is 5 (Do not round until the final answer. Then round to the nearest cent as needed.)

Answer 1

Option 1:

Amount of loan = $50,000

Annual interest rate = 5.55%
Monthly interest rate = 5.55% / 12
Monthly interest rate = 0.4625%

Period = 30 years or 360 months

Let monthly payment be $x

$50,000 = $x/1.004625 + $x/1.004625^2 + …. + $x/1.004625^359 + $x/1.004625^360
$50,000 = $x * (1 - (1/1.004625)^360) / 0.004625
$50,000 = $x * 175.152809
$x = $285.47

Monthly payment = $285.47

Option 2:

Amount of loan = $50,000

Annual interest rate = 5.15%
Monthly interest rate = 5.15% / 12
Monthly interest rate = 0.42917%

Period = 15 years or 180 months

Let monthly payment be $x

$50,000 = $x/1.0042917 + $x/1.0042917^2 + …. + $x/1.0042917^179 + $x/1.0042917^180
$50,000 = $x * (1 - (1/1.0042917)^180) / 0.0042917
$50,000 = $x * 125.214189
$x = $399.32

Monthly payment = $399.32