SOLUTION: A contract valued at $81500.00 requires payment of $2430.00 at the beginning of every 3 months. If interest is 6.5% compounded quarterly, calculate the term of this contract (round

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Question 1166208: A contract valued at $81500.00 requires payment of $2430.00 at the beginning of every 3 months. If interest is 6.5% compounded quarterly, calculate the term of this contract (round to years and months).
Answer by ikleyn(52778) About Me  (Show Source):
You can put this solution on YOUR website!
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A contract valued at $81500.00 requires payment of $2430.00 at the beginning of every 3 months.
If interest is 6.5% compounded quarterly, calculate the term of this contract (round to years and months).
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This problem is about a sinking fund compounded quarterly.

The initial/starting value of the fund is $81,500. The fund makes outpayments
of $2430 quarterly and is compounded quarterly at the annual rate of 6.5%.


They want to know the term of this contract, i.e. the number of quarters.


The feature of this fund is that it makes outpayments at the beginning of each quarter,
while compounding are made at the end of each quarter.


For such a fund, the formula connecting the starting value, the periodical regular 
outpayment value, the number of payments and the rate of compounding is 


    A = W%2A%281%2Br%2Fm%29%2A%28%281+-+%281%2Br%2Fm%29%5E%28-n%29%29%2F%28%28r%2Fm%29%29%29,    (1)


where A is the starting amount, W is the regular quarterly outpayment amount, 
r is the annual compounding rate, m is the number of outpayments per year (m= 4 in this problem), 
n is the total number of outpayments/compounding (the number of quarters, in this problem),
r%2Fm  is the effective rate of compounding per the quarter (3 months) period.


With the given data, formula (1) takes the form


    81500 = .    (2)


The unknown in this equation is 'n'.


To find 'n', simplify equation (2) step by step

    %2881500%2F%282430%2A%281%2B0.065%2F4%29%29%29%2A%280.065%2F4%29  = 1+-+%281%2B0.065%2F4%29%5E%28-n%29

    0.536295486 = 1+-+%281%2B0.065%2F4%29%5E%28-n%29

    %281%2B0.065%2F4%29%5E%28-n%29 = 1 - 0.536295486

    1%2F%281%2B0.065%2F4%29%5En = 0.463704514

    %281%2B0.065%2F4%29%5En = 1/0.463704514

    %281%2B0.065%2F4%29%5En = 2.156545754


At this point, take logarithm of both sides

    n*log(1+0.065/4) = log(2.156545754)


Express n and then calculate its value

    n = log%28%282.156545754%29%29%2Flog%28%281%2B0.065%2F4%29%29 = 47.6760064


ANSWER.  The number of quarters is about 47.676, and the number of years is close to 12.

Solved.

After completing calculations, I think that the problem is posed in a bad way.

Had the problem be posed correctly, the final number of quarters would be VERY close to an integer number.

But in any case, the logic of solution and the steps should be similar to that shown in this my post.