The three basic quantities
Weight is the load. Arm is its distance from the reference datum. Moment is the turning effect created by that weight at that arm.
Center of gravity = Total moment ÷ Total weight
Use a disciplined table
List every item in the same units, calculate each moment, and total the weight and moment columns separately. Only then calculate center of gravity. If the loading changes, work with the change instead of rebuilding the entire problem when the data permits.
Adding or removing weight
Adding weight moves the center of gravity toward the added item. Removing weight moves it away from the removed item. This directional check can reveal an arithmetic error before the final answer is accepted.
Moving weight
When a load is moved inside the aircraft, total weight does not change. The change in moment depends on the weight moved and the distance moved.
Operational meaning
An excessively forward center of gravity can increase control forces and reduce the ability to rotate or flare. An excessively aft center of gravity can reduce longitudinal stability and make stall recovery more difficult. Total weight also affects takeoff distance, climb, ceiling, landing distance, and structural margins.