Question

(6) Christine Sohn bought a BMW when she came to LA as a purchased by taking a loan that was to be paid off in 20 equal, quarterly payments. The interest rate on the loan was 12 % per year with quarterly compounding. After four years, at the time that Christine made her 16th payment, she got married and sold the BMW to her friend Jane Jane made arrangements with Christine's bank to refinance the loan and to pay Christine's unpaid balance by making 16 equal quarterly payments at the same interest rate that Christine was paying. After 34 years, at the time that Jane made her 13th payment, Jane flunked out and sold the car to Ally. Ally paid the bank $3,000 cash to pay the loan balance. What was the amount of Christine's loan to purchase the BMW when it was new? County Judge. The BMW was

Answer 1

Suppose, initially Christine purchased the new BMW car by taking a loan of $ L . The load was to be paid off by her in 20 equal quarterly payments along with an interest rate of 12% per year with quarterly compounding . Thus in each quarter, the interest charged could be calculated by dividing the annual rate of interest(12%) by the number of quarters in a year (i.e., 4). So the total amount of loan inclusive of interests charged that was to be paid by Christine to the Bank is given by :

$ L [1 + (12/4)$\times$(1/100)]²⁰ =$ L[1+0.03]²⁰ = $ L$\times$ 1.03²⁰ = $ (1.806) L

Now since Christine planned to pay this entire amount in 20 equal quarterly payments, she must have decided to pay

$ (1.806) L / 20 = $ (36.12) L .

But she made her quarterly payments only for 16 times before selling the car to her friend Jane. So Jane needed to pay the remaining loan amount to the bank that is the payment due for the next 2 installments , which equals to :

$ (36.12) L $\times$ 4 = $ (144.48) L .

Now , Jane decides to pay the unpaid balance by making 16 quarterly payments at the same interest rate 12% compounded quarterly. The only difference is that now her interest would be calculated solely on the amount that she needs to pay (unpaid by Christine)

. In total 16 installments, she needs to pay an amount inclusive of interests equal to :

$(144.48) L $\times$ (1+0.03)²⁰ = $ (144.48) L $\times$ 1.806 = $ (260.93) L

Now since Jane pays this amount in 16 equal installments, she pays in each installment an amount equal to :

$ (260.93)L /16 = $ (16.308) L

But she only pays up to her 13th installment. Remaining 3 payments left unpaid when she sold the car to Ally. So to repay the entire loan balance, Ally needs to pay : $ (16.308) L $\times$ 3 = $ (48.924) L

We know that Ally ultimately pays $ 3000 . Then we can deduce that,

$ (48.924) L $ 3000

or. L = 3000 / 48.924 = 146772

Thus ,the amount of Christine's loan to purchase was $ 146772 when it was new.