In a nutshell, and going from memory that hasn't been exercised on this topic since, here's what I learned back then:
- Each plane has a datum [originally I recalled this as the center of gravity (CG), but the CG changes based on the moments whereas the datum is a fixed reference point] that is defined by the manufacturer in a book that comes with the plane; that book might be the Pilot's Operating Handbook (POH). Where this is actually physically located is not of practical importance; you just need the numbers (described below) that are related to it. The point is that doing the weight and balance ensures that the plane is (1) not carrying too much weight and thus will fly, and normally at that, and (2) the weight is properly balanced, not introducing any strange rotational forces (we all remember doing "moment about an axis" calculations in high school physics, right?).
- The book also gives some basic numbers to work with, such as the arm (distance from the datum) for each row of seats and the luggage compartment (i.e., the places where weights will be located) and the empty weight, arm and moment (which might have been updated since the book was published if modifications to the plane have been made).
- You draw a matrix that has weight, arm, and moment (which is weight x arm) on one axis and then the items/locations contributing weight on the other axis (such as the empty weight, front seat, rear seat, luggage, fuel, and oil).
- The arm for each weight contributor is known and will be the same from calculation to calculation (unless the plane is reconfigured, I guess -- you could have the back seat moved or removed, right?). Both passengers in the front seats combined make the weight contributor corresponding to the front seat arm; same for the back seat. All luggage in the luggage compartment combined will be entered for that arm.
- The empty weight, arm and moment are known and will be the same from calculation to calculation, unless the plane is modified, in which case you just have new numbers from that point forward. Things like swapping out instruments may require the empty weight to be reassessed.
- Fuel and oil have known weights per gallon or quart. Let's call those multipliers, which will be the same from calc to calc. For each calculation, then, you just need to multiply the actual quantity (such as 30 gallons of fuel or 5 quarts of oil) times the multiplier to get the actual weights for the flight you're planning.
- Estimate high. When taking your aunt in the back seat, you politely ask her for her weight and secretly assume she's answering the way we all do -- we lie to be skinnier. Include a fudge-factor to be on the safe side.
- Multiply the weights times the arms and enter those as the moments for each weight contributor.
- Sum all of the weights to find a total weight. Sum all the moments to find a total moment. In the POH, find the weight and balance chart and see whether the total weight and total moment from your matrix intersect in the envelope for allowable load.
- If so, go have fun already!
- If not, try moving things around -- heavier stuff up front will reduce the overall moment; or you can try planning to take less fuel (and thus have a shorter flight), but consider the situation of arriving at the airport and finding your plane full; or something has to be left behind.
When Husband was teaching me, we used a whiteboard and went through several potentially real-world scenarios with our plane, such as "how much luggage can we carry if we want to take a 4-hour flight with the 90-lb Dog in the back seat?" or "can I go too if Brother and Brother's Girlfriend want Husband to take them up for a local aerial tour?"
That's all I remember right now. When next I have to do an actual weight and balance, I'll review this and see what I left out.
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