Weight & Balance

The weight shift formula, and the two that go with it

As of August 15, 2026, PHAK Chapter 10 states the weight shift formula as a proportion — weight shifted ÷ total weight = ΔCG ÷ distance the weight is shifted — on page 10-10, with the weight-addition and weight-removal equations on page 10-11. All three are proportions, so each solves for whichever term you are missing. What they never tell you is the direction, and that is where the arithmetic usually goes wrong.

See all three equations Drill weight and balance problems

Equations in PHAK Ch. 10

Weight shift proportion

PHAK 10-10

Add / remove + sign rule

PHAK 10-11

Primary-source check: Pilot’s Handbook of Aeronautical Knowledge, Chapter 10 — Weight and Balance (FAA-H-8083-25C, pages 10-10 and 10-11), with the fuller treatment in the FAA Aircraft Weight and Balance Handbook (FAA-H-8083-1B). Last verified: 2026-08-15; date_retrieved: 2026-08-15. Every equation and every number in the worked examples below is transcribed from the FAA chapter PDF at the page named in its row — including the FAA’s own rounding.

The Three Equations

Which equation, when, and what it solves for

Pick by what happens to the total weight. If it stays the same, the weight moved. If it changed, something was loaded or offloaded — and the denominator becomes the new total weight.

Case	Use it when	Equation	Solves for	Source
Weight shift	Weight moves from one station to another. Total weight is unchanged.	Weight shifted ÷ Total weight = ΔCG ÷ Distance weight is shifted	ΔCG, or the weight you must move to hit a target CG.	PHAK 10-10
Weight addition	Cargo, baggage or a passenger is loaded. Total weight increases.	Added weight ÷ New total weight = ΔCG ÷ Distance between weight and old CG	ΔCG, which is then applied to the old CG.	PHAK 10-11
Weight removal	Cargo is offloaded, or fuel is burned off a station away from the CG.	Weight removed ÷ New total weight = ΔCG ÷ Distance between weight and old CG	ΔCG, which is then applied to the old CG.	PHAK 10-11

The FAA’s Own Numbers

Four worked examples, straight out of Chapter 10

These are the handbook’s examples, not re-derived ones — so you can hold this page beside the PDF and check every line, rounding included.

Shift 100 lb aft — what does the CG do?

PHAK 10-10

Weight shift

Given: Total weight 8,000 lb · CG station 77.0 in · 100 lb moved 120 in aft
Set up: 100 ÷ 8,000 = ΔCG ÷ 120
Result: ΔCG = 1.5 in aft
Handbook answer: New CG = 77 + 1.5 = 78.5 in aft of datum

Solve the same equation backwards — how much must I move?

PHAK 10-10

Weight shift

Given: Total weight 7,800 lb · CG 81.5 in · aft limit 80.5 in · cargo moves station 150 → station 30
Set up: Weight to be shifted ÷ 7,800 = 1.0 in ÷ 120 in
Result: Weight to be shifted = 65 lb
Handbook answer: Moving 65 lb forward 120 in brings the CG to exactly the 80.5 in aft limit

Add 140 lb of baggage at station 150

PHAK 10-11

Weight addition

Given: Total weight 6,860 lb · CG station 80.0 in · add 140 lb at station 150
Set up: 140 ÷ (6,860 + 140) = ΔCG ÷ (150 − 80) → 140 ÷ 7,000 = ΔCG ÷ 70
Result: ΔCG = 1.4 in aft
Handbook answer: New CG = 80 + 1.4 = 81.4 in

Remove 100 lb from station 150

PHAK 10-11

Weight removal

Given: Total weight 6,100 lb · CG station 80.0 in · remove 100 lb from station 150
Set up: 100 ÷ (6,100 − 100) = ΔCG ÷ (150 − 80) → 100 ÷ 6,000 = ΔCG ÷ 70
Result: ΔCG = 1.2 in forward
Handbook answer: New CG = 80 − 1.2 = 78.8 in

Where It Goes Wrong

Four mistakes the equations cannot protect you from

Every one of these produces a plausible number. That is what makes them expensive on a written test and worse in an aircraft.

Adding ΔCG when it should be subtracted

The equation gives you a magnitude, never a direction

PHAK 10-11 states the rule explicitly: if the CG is shifting aft, add ΔCG to the old CG; if it is shifting forward, subtract it. Decide the direction first by asking whether the weight went toward the tail or the nose, then apply the number.

Using the old total weight in the add/remove equation

The denominator is the NEW total weight

Add 140 lb to a 6,860 lb aircraft and the denominator is 7,000, not 6,860. Remove 100 lb from 6,100 and it is 6,000, not 6,100. The weight-shift equation is the exception — there the total weight genuinely does not change, so old and new are the same number.

Using the station number as the distance

Distance is measured from the old CG, not from the datum

Adding weight at station 150 to an aircraft whose CG is at 80.0 in gives a distance of 70 in, not 150 in. In the weight-shift equation the distance is instead between the two stations — 150 to 30 is 120 in.

Treating the proportional formula as an approximation

It is exact, and the moment method proves it

Work the first example the long way: 100 lb at station 150 is 15,000 in-lb, at station 30 it is 3,000 in-lb, a change of 12,000 in-lb. Add that to the original 616,000 in-lb and divide by 8,000 lb — 628,000 ÷ 8,000 = 78.5 in, the same answer the proportion gave. PHAK 10-10 runs both.

Weight and Balance FAQ

Q1What is the weight shift formula?

Weight shifted ÷ total weight = ΔCG ÷ distance the weight is shifted. PHAK Chapter 10 (page 10-10) states it as a proportion, which means you can solve it for either unknown: for the CG change when you know the weight moved, or for the weight you must move to achieve a required CG change.

Q2Where in the PHAK are the weight shift and add/remove equations?

Chapter 10, Weight and Balance. The weight-shift proportion and its two worked examples are on page 10-10; the weight-addition and weight-removal equations, the FAA’s examples for each, and the add-versus-subtract sign rule are on page 10-11. In the current edition (FAA-H-8083-25C) the chapter is titled “Weight and Balance” and the section is “Weight Addition or Removal”.

Q3What is the weight shift formula for subtracting weight?

Removing weight uses the addition equation with the new total weight reduced: weight removed ÷ new total weight = ΔCG ÷ distance between the weight and the old CG. The FAA’s example removes 100 lb from station 150 on a 6,100 lb aircraft with its CG at 80.0 in — 100 ÷ 6,000 = ΔCG ÷ 70 — giving ΔCG = 1.2 in, subtracted because the CG shifts forward: 80 − 1.2 = 78.8 in.

Q4Do I add or subtract the CG change?

Add it if the CG moves aft, subtract it if the CG moves forward — PHAK 10-11 says to decide this by mentally working out which way the CG will move for the particular weight change, before touching the arithmetic. Removing weight from behind the CG moves the CG forward; adding weight behind the CG moves it aft.

Q5What is the difference between the weight shift formula and the weight addition formula?

The denominators. In a weight shift nothing enters or leaves the aircraft, so total weight is unchanged and the distance term is the distance between the two stations. In an addition or removal the total weight changes, so the denominator is the new total weight and the distance term is measured from the old CG to the station where the weight was added or removed.

Q6How do you calculate the new CG after adding baggage?

Divide the added weight by the new total weight, multiply by the distance from the old CG to the baggage station, and apply the result to the old CG. Adding 140 lb at station 150 to a 6,860 lb aircraft with its CG at 80.0 in: 140 ÷ 7,000 × 70 = 1.4 in aft, so the new CG is 81.4 in. Then check it against the aircraft’s forward and aft CG limits for that weight.

Where this fits in the rest of the ground school

Load factor and maneuvering speed — why a CG outside limits changes the speeds you are allowed to fly.
Pilot’s Handbook of Aeronautical Knowledge — Chapter 10 covers datum, arm, moment and the CG envelope in full.
E6B flight computer — the same proportional arithmetic you will do on the written test.
Private Pilot ACS — weight and balance is a named task with its own risk-management elements.

Turn the equations into recall

Written-test weight and balance questions give you a loading and ask for a number, under time pressure. Drill them until picking the right denominator and the right sign is automatic, then the arithmetic is the easy part.

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