The principal =$6000.
The annual rate of interest =13.36%
Therefore the rate of semi annual interest = 13.36/2=6.68% or 0.0668 per dollar.
The principal P for n compounding periods with rate of interest x for the compounding period will grow to an amount A given by: A = P(1+x)^n
Here x= 13.36%/2=6.68% 0r 0.668 per dollar. and n=3.5*2=7
The compound interest with principal for 3.5 yrs or seven semi annuals is 6000(1.0668)^(3.5*2)= 6000(1.0668^7)=$9434.79. None of the choices given are correct.The choice at c is only nearest. That is the solution.The problem for you is that correct answer has to be chosen from all the wrong answers given. To add to the confusion, there are nearly equal choices at A and B which appears correct for you, but both are not correct.There is nothing said about interest other than EAR. So why introduce different APR . So work it on EAR/2 for semiannual compounding.
To get the amount at A. 9305.55 the amount you should have an effective annual interest rate {[(9305.55/6000)^1/7]-1}*2*100%=12.94%
To get the amount at B. 9305.95 you should have the effective annual interest rate {[(9305.55/6000)^(1/7)]-1}*2*100%=12.94%.
To get the amount at c. 9466, you should have the effective annual interest rate 13.46% nearly.
You get the amount at D 10,037 ( revised by me instead of 10.037 to be realistic) , if the effective annual interest rate is 15.2543% approximately.
You worked out in two methods . (1) Taking 3.5 years and compounding period as one year , principle $6000 and EAR =13.36% and obtained a value of 9305.95. but it is not the solution of the original problem, though it tallies with the choice C. (2) Again there is a calculation of compounding semi annually with a semiannual interest 6.47 and getting an amount $9305.55. Both methods are different and not the solution for the original problem.
The solution of the original problem is as given in bold type.
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