Karen obtained a $26,000 loan at 4.6% compounded semiannually.
a-1.
What monthly payment will repay the loan in 8 1/2 years? (Do not round intermediate calculations and round your final answer to 2 decimal places.)
Monthly payment $
a-2.
How much interest will Karen pay over the life of the loan
1. P = Po(1+r)^n.
r = 0.046/2 = 0.023 = Semi-annual
% rate.
n = 2Comp./yr. * 8.5yrs. = 17 Compounding periods.
P = 26,000(1.023)^17 = $38,270.058.
8.5yrs. = 102 Months.
38,270.058/102 = $375.20/Mo.
2. I = 38,270.06 - 26,000 = $12,270.06.
To find the monthly payment for the loan, we can use the formula for calculating the monthly payment for a loan:
M = P * (r / (1 - (1 + r)^(-n)))
Where:
M is the monthly payment
P is the principal amount (loan amount)
r is the interest rate per period (semiannual in this case)
n is the total number of periods (in this case, 8.5 years, which is equal to 17 semiannual periods)
a-1. Given:
P = $26,000
r = 4.6% per annum compounded semiannually, which means r = 0.046 / 2 = 0.023 per semiannual period
n = 17
Plugging in these values into the formula, we can calculate the monthly payment:
M = $26,000 * (0.023 / (1 - (1 + 0.023)^(-17)))
M = $26,000 * (0.023 / (1 - (1.023)^(-17)))
M ≈ $374.13 (rounded to 2 decimal places)
Therefore, the monthly payment to repay the loan in 8 1/2 years is $374.13.
a-2. To find out how much interest Karen will pay over the life of the loan, we can calculate the total amount paid and subtract the principal amount.
Total amount paid = Monthly payment * Total number of periods
Total amount paid = $374.13 * 17
Total amount paid ≈ $6,356.21
Interest paid = Total amount paid - Principal amount
Interest paid = $6,356.21 - $26,000
Interest paid ≈ $-19,643.79
Since the interest paid is negative, it means Karen would receive a total of $19,643.79 over the life of the loan, indicating that she would be paying back less money than she borrowed.