How much did the car originally cost

Question

An individual agrees to pay $6,000 per year for three years to payoff a car loan. His payments are always
made at the end of each year. If the interest rate in year 1 is 3%, in year 2 is 3.5% and in year 3 is 4%,
and if compounding is done twice a year, how much did the car originally cost? (That is, work out the
price he must have agreed to pay for the car.)

Answer 1

Hi Harjit,

 

We can calculate the Total Original Cost (C) using the Compound Interest formula for each of the three years to determine the Principal (P) for each year and then sum the three determined Principals to get the

Total Original Cost (C):

 

    Year #1:

         F = P(1 + r/p)^(t)(p)  

         [Compound Interest formula]

               F (principal + interest) = $6,000

               P (principal) = unknown

               r (annual interest rate) = 3% (0.03)

               p (periods per year) = 2 (semi-annually)

               t (years) = 1

         6000 = P(1 + 0.03/2)⁽¹⁾⁽²⁾

         6000 = P(1 + 0.015)²

         6000 = P(1.015)²

         6000 = P(1.030225)

         P = 6000/1.030225

         P = $5,823.97

 

     Year #2:

         F = P(1 + r/p)^(t)(p) 

         [Compound Interest formula]

               F (principal + interest) = $6,000
               P (principal) = unknown
               r (annual interest rate) = 3.5% (0.035)

               p (periods per year) = 2 (semi-annually)
               t (years) = 1

         6000 = P(1 + 0.035/2)⁽¹⁾⁽²⁾

         6000 = P(1 + 0.0175)²

         6000 = P(1.0175)²

         6000 = P(1.03530625)
         6000 = P(1.03530625)
         P = 6000/1.03530625
         P = $5,795.38

 

    Year #3:
         F = P(1 + r/p)^(t)(p)

         [Compound Interest formula]
               F (principal + interest) = $6,000
               P (principal) = unknown
               r (annual interest rate) = 4% (0.04)

               p (periods per year) = 2 (semi-annually)
               t (years) = 1
         6000 = P(1 + 0.04/2)⁽¹⁾⁽²⁾

         6000 = P(1 + 0.02)²

         6000 = P(1.02)²

         6000 = P(1.0404)
         P = 6000/1.0404
         P = $5,767.01

 

Sum the three determined Principals to calculate the

Total Original Cost (C):

    C = 5,823.97 + 5,795.38 + 5,767.01

    C = $17,386.36 (our problem answer)

 

As an added benefit, we can also calculate the

Total Interest (I) by subtracting the

Total Original Cost (C) from the Total Final Cost (F):

    F = $6,000 x 3 years

    F = $18,000

    I = 18,000 - 17,386.36

    I = $613.64

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan.