Private Pilot Weight and Balance — Practice Problems with Worked Solutions

Weight and balance has started getting harder to follow with all the vague study guides and half-finished examples flying around. As someone who froze solid during a mock oral exam on this exact topic, I sat down and learned doing these calculations the hard way. My CFI asked me to walk through a complete weight and balance for our Cessna 172 — actual moment arms, actual numbers, real CG envelope chart sitting right there on the table — and I stalled out halfway through. Turns out studying the concept and doing the arithmetic are two completely different things. That was a $180 lesson. What follows are fully worked scenarios using Cessna 172-specific numbers, the kind your examiner will actually hand you on checkride day.

Weight and Balance Basics — What the Checkride Actually Tests

Your examiner doesn’t care if you can recite the definition of center of gravity. They already assume you know it. What they’re watching for is whether you can execute the calculation correctly — under mild pressure, with a chart in front of you — and then interpret that result without making a catastrophic call.

There are four specific things every examiner checks:

1. Can you locate the correct arm for each station in the Pilot’s Operating Handbook?
2. Can you calculate the moment for each item — weight multiplied by arm equals moment, every time, no exceptions?
3. Can you sum the total weight and total moment correctly?
4. Can you plot the resulting CG location on the envelope chart and declare the aircraft within or out of limits?

The pass/fail line is sharp. Declare an out-of-limits aircraft airworthy and you’re done — not a ding, not a note in the debrief, an outright failure. Examiners take this seriously because the consequences of getting it wrong aren’t academic. Worth flagging before going further. But the point stands wherever it lands: there is no “close enough” on weight and balance.

Skip past the mistake I made. Understand every step before you walk into that room.

Cessna 172 Weight and Balance — The Key Numbers

But what is a reference arm, exactly? In essence, it’s a fixed distance measured from the aircraft’s datum point to a specific station — a seat, a tank, a baggage shelf. But it’s much more than that. It’s the multiplier that converts a simple weight into a moment, which is what actually tells you where the airplane wants to rotate.

Frustrated by inconsistencies across model years, I pulled through dozens of Cessna 172 POHs — N, P, R, S, and SP variants — and settled on a consistent set of arms that show up repeatedly in FAA knowledge test problems and most training aircraft documents. Use these to practice. Use your specific aircraft’s Weight and Balance Record for the actual checkride.

• Front seats (pilot and right-seat passenger): approximately 37 inches aft of datum
• Rear seats: approximately 73 inches aft of datum
• Baggage area: approximately 95 inches aft of datum
• Fuel (wing tanks, 172S): approximately 48 inches aft of datum
• Oil: approximately negative 14 inches — forward of datum, so it subtracts from your total moment

Standard weights worth memorizing: aviation gasoline (100LL) runs 6 lbs per gallon. The 172S carries 56 gallons total, 53 usable. Max gross weight on the 172S is 2,550 lbs — earlier models like the 172N cap at 2,300 lbs. The empty weight and its CG arm live in the aircraft’s Weight and Balance Record, not a generic POH table, and that number changes every time a mechanic touches the equipment list. Always pull the current document.

Practice Problem 1 — Standard Training Flight

The Scenario

You’re flying a Cessna 172S. The aircraft’s Weight and Balance Record shows an empty weight of 1,734 lbs with a CG arm of 40.5 inches. Loading is as follows:

• Pilot (front left): 170 lbs
• Passenger (front right): 145 lbs
• Two rear passengers: 155 lbs each
• Baggage: 25 lbs
• Fuel: 38 usable gallons

Step-by-Step Solution

First, convert fuel to pounds: 38 gallons × 6 lbs/gallon = 228 lbs of fuel.

Now build the table — weight times arm equals moment, every station, no skipping:

• Empty aircraft: 1,734 lbs × 40.5 in = 70,227 in-lbs
• Front seats (combined): 315 lbs × 37 in = 11,655 in-lbs
• Rear seats (combined): 310 lbs × 73 in = 22,630 in-lbs
• Baggage: 25 lbs × 95 in = 2,375 in-lbs
• Fuel: 228 lbs × 48 in = 10,944 in-lbs

Total weight: 1,734 + 315 + 310 + 25 + 228 = 2,612 lbs

Stop right there. The 172S max gross is 2,550 lbs. You’re 62 lbs over before you’ve touched the CG calculation — and your examiner expects you to catch this immediately, out loud, without being prompted.

What do you remove? Baggage goes first — 25 lbs out. Still over by 37 lbs. Reduce fuel by 7 gallons (42 lbs), bringing usable fuel to 31 gallons and fuel weight to 186 lbs. New total: 2,612 − 25 − 42 = 2,545 lbs. Within limits.

Recalculate with adjusted fuel and no baggage:

• Empty aircraft: 70,227 in-lbs
• Front seats: 11,655 in-lbs
• Rear seats: 22,630 in-lbs
• Baggage: 0
• Fuel: 186 lbs × 48 in = 8,928 in-lbs

Total moment: 70,227 + 11,655 + 22,630 + 8,928 = 113,440 in-lbs

CG location: 113,440 ÷ 2,545 = 44.6 inches aft of datum

Plot 2,545 lbs on the Y axis and 44.6 inches on the X axis of your CG envelope chart. On the 172S, the aft CG limit at that weight runs approximately 47.3 inches. Forward limit sits around 35.0 inches at max gross. At 44.6 inches — inside the envelope. Legal to fly.

Practice Problem 2 — Fuel Planning and CG Shift

The Scenario

Same aircraft, same adjusted loading from Problem 1 — 2,545 lbs, CG at 44.6 inches, total moment 113,440 in-lbs. You fly approximately two hours and burn 28 gallons.

Step-by-Step Solution

Fuel burned: 28 gallons × 6 lbs/gallon = 168 lbs removed from the fuel arm at 48 inches.

Moment removed: 168 lbs × 48 in = 8,064 in-lbs

Landing weight: 2,545 − 168 = 2,377 lbs

Landing moment: 113,440 − 8,064 = 105,376 in-lbs

Landing CG: 105,376 ÷ 2,377 = 44.3 inches aft of datum

The CG shifted slightly forward — expected, actually. The fuel arm at 48 inches sits aft of the aircraft’s CG, so burning fuel from that location nudges the balance point forward. Still within limits at landing. Legal.

That’s what makes this problem endearing to us student pilots: it’s not just about the numbers at departure. Some loading configurations that are perfectly legal at takeoff drift out of limits as fuel burns. Heavy rear passengers, minimal front-seat weight, full tanks — apparently a legal takeoff CG that wanders aft of limits once those wing tanks get low. The math tells you before the airplane does. Always verify both the takeoff and landing condition when anything about the loading looks unusual.

Reading the CG Envelope Chart — What Examiners Expect

The CG envelope chart is a two-axis graph — X axis shows CG location in inches aft of datum, Y axis shows gross weight in pounds. Your calculated point must land inside the shaded envelope. Simple enough in principle. Less simple at 8 AM with an examiner watching your pencil.

Common student mistake: reversing the axes and plotting weight on X, CG on Y. The chart becomes unreadable and the examiner catches it immediately. Every time.

Two boundaries define the envelope, and they mean different things:

• Forward CG limit: A structural and handling boundary. Too far forward and the nose goes heavy — elevator authority drops, rotation on takeoff requires excessive back pressure, and at the extreme, the aircraft may not rotate at all at published Vr.
• Aft CG limit: A stability boundary. Too far aft and the tail loses its ability to damp pitch movement. The airplane becomes twitchy, stall recovery degrades, and there’s less margin for pilot error in an upset — a genuinely dangerous situation, not just a paperwork problem.

While you won’t need a graduate-level aerodynamics lecture, you will need a handful of solid explanations ready to go. When an examiner asks what happens if the CG is aft of limits, “it would be out of limits” might be the worst possible answer — circular, tells them nothing. What they want: reduced longitudinal stability, increased pitch sensitivity, degraded stall recovery characteristics, and a shrinking margin for error in unusual attitude recovery. Explain the aerodynamics. That’s the difference between reciting a rule and actually understanding it, and examiners hear that difference on every single checkride.

The numbers here give you a working framework. The habit of checking every condition — takeoff through landing — is what actually keeps people safe.