Use PMT equals StartFraction Upper P left parenthesis StartFraction r Over n EndFraction right parenthesis Over left bracket 1 minus left parenthesis 1 plus StartFraction r Over n EndFraction right parenthesis Superscript negative nt right bracket EndFraction
to determine the regular payment amount, rounded to the nearest dollar. In terms of paying less in interest, which is more economical for a $170 comma 000
mortgage: a 30-year fixed-rate at 7.5
%
or a 20-year fixed-rate at 7
%?
How much is saved in interest?
To find out which option is more economical, we need to compute the regular payment amount for both mortgages using the provided formula and then find out how much interest is paid over the term of the loan.
1. For a 30-year fixed-rate mortgage at 7.5 %:
In this case:
P = $170,000
r = 7.5 / 100 = 0.075 (annual interest rate in decimal)
n = 12 (payments per year)
t = 30 (number of years)
PMT = P * ((r/n) / (1 - (1 + (r/n))^(-n*t)))
PMT = 170,000 * ((0.075/12) / (1 - (1 + (0.075/12))^(-12*30)))
PMT = 1,187.86
Round to the nearest dollar: PMT = $1,188
Total amount paid over 30 years = PMT * n*t = 1,188 * 12 * 30 = $427,680
Total interest paid = Total amount paid - Initial loan amount = $427,680 - $170,000 = $257,680
2. For a 20-year fixed-rate mortgage at 7 %:
In this case:
P = $170,000
r = 7 / 100 = 0.07 (annual interest rate in decimal)
n = 12 (payments per year)
t = 20 (number of years)
PMT = P * ((r/n) / (1 - (1 + (r/n))^(-n*t)))
PMT = 170,000 * ((0.07/12) / (1 - (1 + (0.07/12))^(-12*20)))
PMT = 1,316.36
Round to the nearest dollar: PMT = $1,316
Total amount paid over 20 years = PMT * n*t = 1,316 * 12 * 20 = $315,840
Total interest paid = Total amount paid - Initial loan amount = $315,840 - $170,000 = $145,840
Comparing the interest paid for both loans, the 20-year fixed-rate mortgage at 7% is more economical as it results in less interest being paid ($145,840) compared to the 30-year fixed-rate mortgage at 7.5% where the total interest paid is $257,680.
The amount saved in interest with the 20-year fixed-rate mortgage, compared to the 30-year fixed-rate mortgage, is $257,680 - $145,840 = $111,840.
To determine the regular payment amount using the PMT (Payment) formula, we will take the following values into consideration:
P = Principal amount (mortgage value) = $170,000
r = Annual interest rate (in decimal form)
n = Number of payments per year
t = Total number of years
First, let's calculate the regular payment amount for each mortgage option and compare them to see which is more economical in terms of paying less interest:
A. 30-year fixed-rate mortgage at 7.5%:
r = 7.5% = 0.075 (in decimal form)
n = 12 (monthly payments)
t = 30 (years)
Using the PMT formula: PMT = [P * (r/n)] / [1 - (1 + r/n)^(-nt)]
PMT = [170000 * (0.075/12)] / [1 - (1 + 0.075/12)^(-30*12)]
Calculate the PMT to the nearest dollar.
B. 20-year fixed-rate mortgage at 7%:
r = 7% = 0.07 (in decimal form)
n = 12 (monthly payments)
t = 20 (years)
Using the PMT formula: PMT = [P * (r/n)] / [1 - (1 + r/n)^(-nt)]
PMT = [170000 * (0.07/12)] / [1 - (1 + 0.07/12)^(-20*12)]
Calculate the PMT to the nearest dollar.
To determine which option is more economical in terms of paying less interest, compare the PMT values calculated for both options. The option with the lower PMT will result in paying less interest over the term of the mortgage.
To calculate the amount saved in interest, subtract the total interest paid over the term of the mortgage for each option. Multiply the PMT by the total number of payments and subtract the principal amount to find the total interest paid.
Please provide the values for r and n in decimal form (e.g., 7.5% = 0.075, 20 payments per year = 12).