A suv may be purchased for a delivery price $27,340 with a 10% down payment and 36 monthly payments of $745. 74. The vehicle may also be leased for $433.67 a month for 36 months. A down payment of $1,100 is required, and the suv is assumed to be worth $14,200 at the end of the lease.

The question is not complete but it will be assumed to find the better of the two options,

A. $745.74 monthly for 36 months with 10% down and no residual value, or

B. $433.67 monthly for 36 months with $1100 down and $14200 residual value.

It is true that car dealers make it difficult to compare prices, even with a calculator and/or compound interest reference book. The challenge here is we need to compare the effective monthly interest, i, charged in each case, which is not usually listed in books, nor given by an explicit formula.

We will proceed anyway, and take up the challenge.

Case A: 745.74 for 36 months with 10% down and no residual value.

Future value of loan = (27340*0.9) (1+i) ^36

Future value of payments = 745.74 ((1+i) ^36-1) / i

Equate the two and solve for i:

(27340*0.9) (1+i) ^36 = 745.74 ((1+i) ^36-1) / i

We can solve by trial and error, which takes a certain time, or we can use Newton's method, with an initial value of i=0.004 (by graphing), where

let f (i) = (27340*0.9) (1+i) ^36 - 745.74 ((1+i) ^36-1) / i

then f' (i) = (745.74 * ((x+1) ^36-1)) / x^2 - (26846.64 * (x+1) ^35) / x+885816.0 * (x+1) ^35

i1=i0-f (i0) / f' (i0)

after two iterations, we obtain i=0.0047888

Case B:

$433.67 monthly for 36 months with $1100 down and $14200 residual value.

Here, the future value, equated to zero,

g (x) = (27340-1100) * (1+i) ^36 - 433.67 ((1+i) ^36-1) / i = 0

Solving similarly by Newton's method, we have

i=0.004825936

This means that Option A is slightly more advantageous, with a lower interest rate (and also less hassle for the maintenance of the car during three years).