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As I'm studying PPL theories, I've encountered with this kind of a question below and it would be great to understand the principle behind the calculation of the 'CG'

Calculated take-off mass = 2300 lbs, calculated CG = 95.75 in, fuel burn = 170 lbs on station 87.00 in. Where is the CG situated after the landing?

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Not too much theory, just simple math. You find the spot where the sum of all moments (mass x distance) = zero. Where you would balance the plane on a see-saw, that's your CG. You start out with 2300 lbs at 95.75 inches and remove 170 lbs at 87.00 inches.

First, find the difference in CG position, then remove the moment of the burned fuel.

$95.75in - 87.00in = 8.75in$

$170lb * 8.75in = 1487.5inlb$

The new weight is 2130 lbs. The new CG is located further aft, because the fuel station is fwd of the original CG (therefore, the front is lighter after burning the fuel).

$\frac{1487.5inlb}{2130lb} = 0.70in$

So new CG is at 96.45 inches.

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An equivalent, but more intuitive way for my brain...

The total moment of the aircraft at gross

$2300 lb * 95.75 in = 220025 in lb$

The moment of the burned fuel (negative)

$-170 lb * 87 in = -14790 in lb$

Combine them (add)

$205435 in lb$

Divide by the final weight $2300 lb -170 lb = 2130 lb$

$205435 in lb / 2130 lb = 96.45 in$

The location of the CG after fuel burn.

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