To understand how to answer these load shift questions it is best to start with a little bit of common sense logic.
Shifting a load from one point to another in an aircraft will not change the total mass of the aircraft, but will change the total moment. This change in total moment will cause the C of G to move in the direction of the load shift.
To solve this problem use
M to represent the total mass of the aircraft
D to represent the distance (+ or -) that the C of G moves.
m to represent the mass of the load that is shifted.
d to represent the distance (+ or -) that the load is shifted.
Then the change in total moment is the total mass x the distance the C of G moves (which is M x D)
And the moment change caused by shifting the load is the mass of the load x the distance it is shifted (m x d)
The total moment change (M x D) will be equal to the moment change caused by the load shift (m x d).
So we have the simple equation MD = md
We want to know the mass (m) so we divide both sides of the equation by d to give
(MD)/d = m
The question states that
Holds are at +94" and +210"
The CG limits +156" (forward limit) and aft +165" (aft limit)
The total mass is 10650lbs
The CG position is at +172"
To calculate D and d we must use the equation
Distance moved = new location - initial location
We want to get the C of G from +172" to +165" so the required distance D = +165 - (+172) which is -7".
The load is to be moved from +210" to +94" so the distance
d = +94 - (+210) which is d = -116"
Inserting these values into our equation gives
m = (10650 lb x (-7")) / -116"
This gives m = 642.672 lb.
The important thing in this kind of problem is to ensure that you include the signs. It made no difference in this particular question, but it would have done had you been calculating distances forward or aft.
I also find that things work best if you start with a diagram and a simple statement of what is happening, before jumping into the equations.