Question 1207414
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A car was purchased for $1500 down and payments of $265 at the end of each month for four years. 
Interest is 9% compounded quarterly.
(a) What was the purchase price of the car?
(b) How much interest will be paid?
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<pre>
The purchase price in this problem is $1500 plus the loaned amount.


About the loan, we know that the payments are $265 at the end of each month 
for 4 years at the annual interest rate of 9% compounded quarterly.


It is equivalent to (it works as) quarterly payments of  3*265 = 795 dollars 
at the end of each quarter compounded at effective rate r = 0.09/4 per quarter.


Use the formula for periodic payments for the loan at given conditions.
The formula is

    Q = {{{L*(r/(1-(1+r)^(-n)))}}},    (1)


where L is the loan amount; r = {{{0.09/4}}} is the effective interest rate per quarter;
n is the number of payments (same as the number of quarters, n = 4*4 = 16); 
Q is the quarterly payment of $795.


Substitute these values into the formula and get an equation for quarterly payments

    795 = {{{L*(((0.09/4))/(1-(1+0.09/4)^(-16)))}}}.    (2)


We can calculate the coefficient (the factor in the formula) separately

    {{{(((0.09/4))/(1-(1+0.09/4)^(-16)))}}} = 0.07511663.


Now from (1) we find L

    L = {{{795/0.07511663}}} = 10583.54 dollars (rounded to closest cent).


Thus the purchase price for the car was  $10583.54 + $1500 = $12083.54.


It is the <U>ANSWER</U> to question (a).



The amount paid back for this loan was  16 payments by 795 dollars = 16*795 = 12720 dollars.

The interest paid for the loan is the difference $12720 - $10583.54 = $2136.46


It is the <U>ANSWER</U> to question (b).
</pre>

Solved.


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The solution in the post by &nbsp;@MathLover1 is &nbsp;INCORRECT &nbsp;and &nbsp;IRRELEVANT.


It is enough to point that in her solution she combines monthly payments with quarterly rates,
which is the nonsense out of any elementary logic.


Also, &nbsp;she uses the formula for the future value of an annuity, &nbsp;which is &nbsp;IRRELEVANT, 
since in this problem we should use a mathematical model for a loan.


Madam does not know the subject, &nbsp;can not distinct an annuity from a loan, 
wrongly uses the annuity formula instead of the loan formula and does not understand 
meaning of parameters in her formula.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;179% out.



Be aware and ignore her post, &nbsp;for the safety of your mind.