When weights are at multiple stations, the center of gravity is the distance from the reference datum where the entire aircraft’s weight could be considered to act. You determine it by using moments. A moment is produced by each weight times its arm (the distance from the datum). Add up all those individual moments to get the total moment. Then divide that total moment by the sum of all weights (the total weight). The result is the CG location. In other words, CG equals total moment divided by total weight. This reflects the idea that the CG is the weighted average position of all the weights. For example, if you have two weights: one of 100 units at an arm of 5 units, and another of 50 units at an arm of 15 units, the total moment is 100×5 + 50×15 = 500 + 750 = 1250. The total weight is 150. The CG is 1250 ÷ 150 ≈ 8.33 units from the datum. Dividing total weight by total moment would give an inverse quantity with units of 1/length, not a distance. Averaging weights alone ignores how far each weight is from the datum, so it doesn’t locate CG.

Maximum takeoff weight is the most the airplane is certified to weigh for a safe takeoff. It is found by adding the weight of the aircraft in its empty condition to the useful load, which is the weight of everything it’s allowed to carry—payload (people, baggage, cargo) and usable fuel. So empty weight plus useful load equals the maximum takeoff weight. The other options miss a required component or mix up what’s being added, for example excluding payload, excluding fuel, or subtracting BEW from MTOW to get only the usable load.

Distributing weight toward the mid-arms shifts the balance point toward the aircraft’s center. The center of gravity is the weighted average location of all loads, so placing more weight near the center pulls that average toward the center of the aircraft. In practice, moving weight away from the forward or aft extremes toward the middle reduces the moments those extreme weights contribute, causing the CG to move inward, toward the middle near the aircraft’s center of gravity. It won’t move toward the tail or toward the forward limit unless weight is actually moved in those directions, and it will change as soon as weight is redistributed.

The CG envelope is all about keeping the airplane controllable and within safe structural limits by restricting how it can be loaded. It maps the permissible combinations of total weight and forward or aft center of gravity, showing the regions where the airplane has enough longitudinal stability and enough elevator authority to fly safely, take off, maneuver, and land. Sticking inside this envelope ensures you won’t end up with a weight distribution that makes the aircraft nose-heavy or tail-heavy, which could lead to excessive stick forces, loss of control authority, or reaching structural limits. It also accounts for how weight shifts during phases of flight (fuel burn, payload changes) and for different configurations (such as flaps or gear down), so loading plans, fuel planning, and preflight weight/ balance calculations all aim to keep the aircraft within these boundaries. This concept isn’t about choosing engine configurations, planning fuel stops, or scheduling landing gear retraction—the envelope specifically defines safe weight and CG combinations for flight.