Question

Mary Jo wants to buy a boat that is available at two dealerships. The price of the boat is the same at both dealerships. Middlefield Motors would let her make quarterly payments of 2,047.48 dollars for 9 years at a quarterly interest rate of 3.02 percent. Her first payment to Middlefield Motors would be due immediately. If Fairfax Boats would let her make equal monthly payments of 985.51 for 5 years and if her first payment to Fairfax Boats would be in 1 month, then what is the monthly interest rate that Mary Jo would be charged by Fairfax Boats? Answer as a rate in decimal format so that 12.34% would be entered as .1234 and 0.98% would be entered as .0098.

Answer 1

Step 1 - First find out the price of the boat which is same at both delarship.

We can use the present value of annuity due formula and apply it to payment terms provided by Middlefield Motors.

Present value of annuity due = P + Px{[1 - (1+r)^-(n-1)]/r}

Present value of annuity due = price of the boat = ?

P = Quarterly payment = $2047.48

r = quarterly interest rate = 3.02%

n = number of quarterly payments = 9 years * 4 = 36

Present value of annuity due = 2047.48 + 2047.48x{[1 - (1+0.0302)^-(36-1)]/0.0302}

Present value of annuity due = 2047.48 + 2047.48x{21.42462}

Present value of annuity due = 2047.48 + 2047.48x{21.42462}

Present value of annuity due = 45913.95

Price of the boat = $45,913.95

Step 2 - Find out the monthly interest rate charged by Fairfax Boats

We can use the present value of annuity formula and apply it to payment terms provided by Fairfac Boats and price derived in step 1

Present value of annuity = P x {[1 - (1+r)^-n]/r}

Present value of annuity = price of the boat = $45913.95

P = monthly payment = $985.51

r = rate of return per month = ?

n = number of monthly payments = 5 years * 12 = 60

45913.95 = 985.51 x {[1 - (1+r)^-60]/r}

46.58903 = [1 - (1+r)^-60]/r

r = 0.0087

Monthly interest rate that Mary Jo would be charged by Fairfax Boats = 0.87%