Question

If the car I purchased has payments of $350 per month for 60 months, and the financed amount totaled $19,339, what is the ANNUAL interest rate for the loan? a. 3.49% b. 27.4% c. 3.29% d. 0.274% please show work.

Answer 1

Solution:
	Answer is c. 3.29%
Working Notes:
	Since , payments for car is made per month, the totaled financed amount is the present value of all the payments during the loan period Which is calculated by calculating present value of annuity of the per month payments
	Present value of annuity is the total financed amount = $19,339
	present value of annuity = Px ( 1-(1 /(1 + i)^n))/ i
	P=monthly payment = $350
	i= interest rate per period = r/12	[ Since payment is per month and interest rate is expressed in per annum ]
	n= no. Of period =60 months
	PV of annuity= 350 x (1 -(1/(1+r/12)^60))/r/12
Now
	put value of r from given choices
a.	3.49%
	PV of annuity= 350 x (1 -(1/(1+r/12)^60))/r/12
	=350 x (1-(1/(1+(3.49%/12))^60))/(3.49%/12)
	=19,244.21484
Which is not equal to loan amount = $19,339
b.	27.40%
	PV of annuity= 350 x (1 -(1/(1+r/12)^60))/r/12
	=350 x (1-(1/(1+(27.40%/12))^60))/(27.40%/12)
	=11,372.92
Which is not equal to loan amount = $19,339
c.	3.29%
	PV of annuity= 350 x (1 -(1/(1+r/12)^60))/r/12
	=350 x (1-(1/(1+(3.29%/12))^60))/(3.29%/12)
	19339.31146
	=19,339
Which is equal to loan amount = $19,339
d.	0.274%
	PV of annuity= 350 x (1 -(1/(1+r/12)^60))/r/12
	=350 x (1-(1/(1+(0.274%/12))^60))/(0.2740%/12)
	=20,854.44015
	=20,854
Which is not equal to loan amount = $19,339
Please feel free to ask if anything about above solution in comment section of the question.