An echo aeroplane is loaded at take-off as follows:
Gross weight- 2659 KG
Total moment index- 706.4
Useable fuel in tge main tanks only 40 US Gal

The minimum amount of weight that must ve shifted from rear to the nose compartment to ensure the air craft remains in balance at all stages of flight closest to:
select the answer:
A) 8KG
5KG
C) Nil
D) 15 KG

My answer was 15 KG but my instructor marked it wrong.
2659Kg---706.4index
CG=2657 which is balance and I can go for C nil..but to continue the flow chart..CG shift is 23 multiply PW 2659 then ÷ difference in arms 4500= i got 13.59 KG to be shifted from the rear compartment to the nose.
He said 8KG is the correct answer..for me i'll go for D 15 KG/ C nil..because the air craft is balance.

You've missed a couple of points along the way, I fear.

First, you need to check the CG for both ZFW and TOW. You have checked for TOW and then calculated the weight to shift from the FRONT to REAR to get the CG back to the aft limit. Not what the question was asking.

What you need to do is

(a) calculate the ZFW data by taking off the fuel from the TOW data
(b) this shows that the critical loading is ZFW where you are outside the aft limit by 13.6 mm
(c) you don't need to worry about the TOW load case - it's OK, but you do need to move the CG forward for the ZFW case
(d) when you do this you find that 8 kg puts you just in front of the aft limit so that is the minimum weight change required as per the question

Now, if you were out and about flying and you chose to move 15 kg, that's fine.

The problem is with your instructor. When he found you had a different answer, why didn't he check your working and explain what was going on ? That would have identified the error in strategy as well as building your confidence in these problems.

If it were me, I'd sack the instructor as he doesn't appear to be of much value to you.

Now it's simple and clear to understand..Thank you very much sir for your help