Weight and Balance — FAA A&P Test Questions (ACS AM.I.C)

Weight and Balance is the one General subject area that is almost purely arithmetic, and it is also the one most candidates can fix in an afternoon. Nearly every question resolves to three relationships: moment equals weight times arm, center of gravity equals total moment divided by total weight, and useful load equals maximum allowable weight minus empty weight. Around those sit definitions you have to know exactly, including datum, arm, tare weight, ballast and empty weight. The FAA tests the definitions as hard as the math, and the math is arithmetic you can do on paper if you set it up correctly.

Written by the AMTprep editorial team · Published · Last reviewed

Written against the FAA primary sources cited at the foot of this page.

What ACS AM.I.C covers

The questions in this area split cleanly between computation and vocabulary, and both halves follow FAA-H-8083-30. On the computation side, expect to find a moment by multiplying a weight by its arm, to find a center of gravity by dividing a total moment by a total weight, and to say which way the CG shifts when equipment is added or removed at a stated location. A typical stem gives an aircraft weighing 4,000 pounds with a total moment of 600,000 inch-pounds and asks where the CG lies, which is a single division. Others ask what happens to the CG when an item is removed from a station aft of it, which needs no arithmetic at all if you know the direction rule. On the vocabulary side you need the terms used on an actual weighing sheet: the datum as the imaginary plane all arms are measured from, arms forward of the datum carrying a negative sign and arms aft a positive one, tare as the chocks, blocks and leveling gear that must be subtracted from each scale reading, net weight as what remains, empty weight, maximum allowable gross weight, useful load, and ballast added to bring a CG back inside limits. Practical procedure appears as well: leveling the aircraft, positioning and zeroing the scales, and recording the result in the aircraft's weight and balance records, which 14 CFR Part 43 ties to the maintenance record entry.

Where this sits on the test

ACS AM.I.C is tested on the FAA General written test, one of 1,558 ACS-tagged questions in the General bank. Every question tagged to this area carries a worked rationale and its FAA handbook reference, so you can drill the code itself rather than the whole test.

FAA handbook references

  • FAA-H-8083-30

Three traps candidates fall into

  1. Arms measured forward of the datum are negative and arms aft of it are positive. Candidates who ignore the sign add every moment and get a CG that could not exist on the airframe. If your computed CG lands outside the physical aircraft, you almost certainly dropped a negative arm somewhere in the table.
  2. Tare is not part of the aircraft. It is everything on the scale that is not the aircraft, meaning chocks, blocks, jacks and leveling gear, and it is subtracted from each scale reading before anything else happens. Candidates treat the scale reading and the net weight as the same number and carry that error through every later step.
  3. Removing weight aft of the CG moves the CG forward, and removing weight forward of the CG moves it aft. Candidates memorize the phrase remove weight, CG moves forward with no reference to location, then miss the half of the questions where the item came from ahead of the center of gravity.

43 free sample questions from ACS AM.I.C

  1. AM.I.CTap an answer

    The center of gravity (CG) of an aircraft is best described as the point where the aircraft would

  2. AM.I.CTap an answer

    The moment of an item in a weight and balance computation is found by

  3. AM.I.CTap an answer

    An aircraft has a total weight of 4,000 pounds and a total moment of 600,000 inch-pounds. The center of gravity is located at

  4. AM.I.CTap an answer

    Useful load is determined by

  5. AM.I.CTap an answer

    An arm measured forward of the datum is given which sign in weight and balance computations?

  6. AM.I.CTap an answer

    Ballast is added to an aircraft primarily to

  7. AM.I.CTap an answer

    When weighing an aircraft, the weight registered on the scales minus the weight of chocks, blocks, and other items used to level the aircraft is called the

  8. AM.I.CTap an answer

    If a piece of equipment is removed from a location aft of the CG, the center of gravity of the aircraft will

  9. AM.I.CTap an answer

    The center of gravity range of an aircraft is best described as the

  10. AM.I.CTap an answer

    Two weights are installed aft of an aircraft's center of gravity: a 10-pound weight 40 inches aft and a 5-pound weight 20 inches aft. How far forward of the center of gravity must a 20-pound weight be placed to balance the aircraft?

  11. AM.I.CTap an answer

    Two items weighing 11 pounds and 8 pounds are placed 2 inches and 1 inch, respectively, aft of the center of gravity of an aircraft. How far forward of the center of gravity must a 30-pound weight be placed so the center of gravity is not changed?

  12. AM.I.CTap an answer

    The single point at which the total weight of a body is considered to act is the

  13. AM.I.CTap an answer

    On an aircraft, the reference edge, line, or plane from which all longitudinal dimensions are measured is called the

  14. AM.I.CTap an answer

    On an aircraft station diagram, water lines are used to locate points by their

  15. AM.I.CTap an answer

    Who is responsible for ensuring that the weighing equipment used for an aircraft weight and balance check is serviceable and in calibration before use?

  16. AM.I.CTap an answer

    How often should aircraft weighing scales be calibrated?

  17. AM.I.CTap an answer

    What is the term for the maximum weight of an aircraft, including the airframe, powerplant, required equipment, and any other items of equipment considered to be permanently installed, but excluding usable fuel, crew, payload, and oil?

  18. AM.I.CTap an answer

    The center of gravity (CG) of an aircraft is defined as the point about which

  19. AM.I.CTap an answer

    The horizontal distance from the reference datum to the center of gravity of an item is known as the item's

  20. AM.I.CTap an answer

    The center of gravity of a loaded aircraft is determined by

  21. AM.I.CTap an answer

    From which point are all weight and balance arms and CG locations measured?

  22. AM.I.CTap an answer

    Who establishes the location of the reference datum for a particular aircraft?

  23. AM.I.CTap an answer

    The useful load of an aircraft is determined by

  24. AM.I.CTap an answer

    When an aircraft is weighed, which of the following fluids must normally be drained or accounted for as residual (unusable) according to the manufacturer's weighing instructions?

  25. AM.I.CTap an answer

    The empty weight center of gravity (EWCG) of an aircraft weighed on three scales is found by dividing the

  26. AM.I.CTap an answer

    An aircraft is loaded so that its center of gravity is forward of the forward CG limit. The most likely effect on the aircraft is that it will be

  27. AM.I.CTap an answer

    An aircraft loaded so that its center of gravity is aft of the rearward limit will most likely be

  28. AM.I.CTap an answer

    The maximum weight of an aircraft, fully loaded and ready for flight, that is approved for operation is called the

  29. AM.I.CTap an answer

    When weight is added to or removed from an aircraft, the change in the location of the CG can be computed by dividing the

  30. AM.I.CTap an answer

    When the position of the center of gravity is expressed as a percentage of the mean aerodynamic chord (MAC), 0 percent MAC represents the

  31. AM.I.CTap an answer

    An aircraft has an empty weight of 1,800 pounds and a maximum gross weight of 2,500 pounds. What is the useful load?

  32. AM.I.CTap an answer

    A 50-pound radio is installed at station +30 (30 inches aft of the datum). What is the moment of this installation about the datum?

  33. AM.I.CTap an answer

    If a weight is added at a station forward of the reference datum, its moment is considered to be

  34. AM.I.CTap an answer

    An aircraft's empty weight and empty weight CG must be recomputed and the records revised when

  35. AM.I.CTap an answer

    The arm of an item is measured in

  36. AM.I.CTap an answer

    Improper loading that places an aircraft outside its CG limits primarily affects the aircraft's

  37. AM.I.CTap an answer

    What is the primary purpose of leveling an aircraft before weighing it?

  38. AM.I.CTap an answer

    When computing the weight and balance of an aircraft, the total weight multiplied by the CG distance from the datum equals the

  39. AM.I.CTap an answer

    After an aircraft is weighed and the empty weight CG determined, the technician must

  40. AM.I.CTap an answer

    The center of gravity of an empty aircraft, with only residual fluids and permanently installed equipment, is referred to as the

  41. AM.I.CTap an answer

    An aircraft with a total weight of 2,000 pounds and a total moment of 170,000 inch-pounds has a center of gravity located at

  42. AM.I.CTap an answer

    Which document must a mechanic ensure remains current and accurate when equipment changes affect an aircraft's weight and balance?

  43. AM.I.CTap an answer

    Adverse-loaded CG checks are performed to verify that an aircraft remains within its CG limits under the

The center of gravity (CG) of an aircraft is best described as the point where the aircraft would

ACS code: AM.I.C

Correct answer: balance if it were suspended

Rationale: The center of gravity is the point about which the nose-heavy and tail-heavy moments are exactly equal; it is the point at which the aircraft would balance if suspended. It is found by dividing total moment by total weight. The other options describe stall behavior and ground contact, which are unrelated to the definition of CG.

The moment of an item in a weight and balance computation is found by

ACS code: AM.I.C

Correct answer: multiplying the weight by its arm

Rationale: Moment equals weight multiplied by arm, expressed in inch-pounds. This is the basic relationship used to build a weight and balance computation, where the total moment is divided by total weight to find the CG. Dividing or subtracting the values does not yield a valid moment and would produce incorrect balance results.

An aircraft has a total weight of 4,000 pounds and a total moment of 600,000 inch-pounds. The center of gravity is located at

ACS code: AM.I.C

Correct answer: 150 inches from the datum

Rationale: The CG is found by dividing total moment by total weight: 600,000 ÷ 4,000 = 150 inches from the datum. The other answers result from incorrect arithmetic, such as using the wrong divisor, and do not satisfy the moment-divided-by-weight relationship.

Useful load is determined by

ACS code: AM.I.C

Correct answer: subtracting the empty weight from the maximum allowable weight

Rationale: Useful load is the difference between maximum allowable (gross) weight and empty weight; it represents the crew, payload, and usable fuel the aircraft can carry. Reversing the subtraction yields a negative value and adding the two weights together produces a meaningless total, so those distractors are incorrect.

An arm measured forward of the datum is given which sign in weight and balance computations?

ACS code: AM.I.C

Correct answer: Negative

Rationale: By convention, arms measured aft of the datum are positive and arms measured forward of the datum are negative. Correctly applying these signs is essential so the resulting moments add and subtract properly. Assigning a positive or zero value to a forward arm would corrupt the moment calculation.

Ballast is added to an aircraft primarily to

ACS code: AM.I.C

Correct answer: bring the center of gravity within allowable limits

Rationale: Ballast is weight added to move the CG into the allowable range, most often when the CG falls outside limits after loading or a modification. It does not raise the certificated maximum gross weight and, since it adds weight, it cannot reduce empty weight, making those distractors incorrect.

When weighing an aircraft, the weight registered on the scales minus the weight of chocks, blocks, and other items used to level the aircraft is called the

ACS code: AM.I.C

Correct answer: net weight at each weighing point

Rationale: The net weight at each weighing point is the scale reading minus the tare, where tare is the weight of chocks, blocks, and similar items on the scale. Tare itself is what is subtracted, not the resulting value, and useful load is an entirely different quantity unrelated to scale tare.

If a piece of equipment is removed from a location aft of the CG, the center of gravity of the aircraft will

ACS code: AM.I.C

Correct answer: move forward

Rationale: Removing weight from behind the CG reduces the tail-heavy moment, so the CG shifts toward the lighter end, which is forward. The CG always moves away from weight that is removed and toward weight that is added. It cannot move aft or stay the same when a tail-located item is taken out.

The center of gravity range of an aircraft is best described as the

ACS code: AM.I.C

Correct answer: distance between the forward and aft CG limits

Rationale: The CG range is the distance between the most forward and most aft allowable CG limits, within which the loaded CG must fall for safe operation. The distance from the datum to the wing is unrelated, and the difference between empty and gross weight defines useful load, not the CG range.

Two weights are installed aft of an aircraft's center of gravity: a 10-pound weight 40 inches aft and a 5-pound weight 20 inches aft. How far forward of the center of gravity must a 20-pound weight be placed to balance the aircraft?

ACS code: AM.I.C

Correct answer: 25 inches.

Rationale: Moment equals weight times arm. The two aft weights create a moment of (10 lb x 40 in) + (5 lb x 20 in) = 500 in-lb aft. To balance, the 20-pound weight must produce an equal opposing moment forward: 500 in-lb divided by 20 lb = 25 inches forward of the center of gravity.

Two items weighing 11 pounds and 8 pounds are placed 2 inches and 1 inch, respectively, aft of the center of gravity of an aircraft. How far forward of the center of gravity must a 30-pound weight be placed so the center of gravity is not changed?

ACS code: AM.I.C

Correct answer: 1 inch.

Rationale: To keep the center of gravity unchanged, the forward moment must equal the total aft moment. The aft moment is (11 × 2) + (8 × 1) = 30 inch-pounds. The 30-pound weight must produce the same moment forward, so 30 × d = 30, giving d = 1 inch.

The single point at which the total weight of a body is considered to act is the

ACS code: AM.I.C

Correct answer: center of gravity.

Rationale: The center of gravity is the single point at which the entire weight (total gravitational force) of a body can be considered to act; supporting the body there produces balance. A fulcrum is a pivot point, not the location where the body's weight acts.

On an aircraft, the reference edge, line, or plane from which all longitudinal dimensions are measured is called the

ACS code: AM.I.C

Correct answer: datum.

Rationale: The datum is the reference edge, line, or plane from which all longitudinal (fore-and-aft) station dimensions are measured. Water lines and butt lines are themselves located relative to a reference and describe vertical and lateral positions, not the origin reference itself.

On an aircraft station diagram, water lines are used to locate points by their

ACS code: AM.I.C

Correct answer: vertical height measured above a horizontal reference.

Rationale: Water lines are horizontal reference planes used to locate points by their vertical height above a chosen reference. Fore-and-aft positions are given by fuselage stations, and left/right positions by buttock (butt) lines, so the vertical reference is the water line.

Who is responsible for ensuring that the weighing equipment used for an aircraft weight and balance check is serviceable and in calibration before use?

ACS code: AM.I.C

Correct answer: The person carrying out the weight and balance procedure.

Rationale: Responsibility for confirming that the weighing equipment is serviceable and in calibration rests with the person performing the weight and balance task, because that person certifies the result. They cannot rely on the equipment manufacturer or a separate technician to assure the equipment at the point of use.

How often should aircraft weighing scales be calibrated?

ACS code: AM.I.C

Correct answer: On a periodic basis, generally once a year.

Rationale: Weighing equipment used for aircraft weight and balance must be calibrated on a routine periodic basis, typically once a year, so its accuracy remains traceable and within tolerance. A full calibration before every weighing is impractical, and assuming scales never need calibration would invalidate the recorded weights.

What is the term for the maximum weight of an aircraft, including the airframe, powerplant, required equipment, and any other items of equipment considered to be permanently installed, but excluding usable fuel, crew, payload, and oil?

ACS code: AM.I.C

Correct answer: Empty weight

Rationale: Per FAA-H-8083-1, empty weight is the weight of the airframe, engine(s), all permanently installed equipment, and unusable (residual) fuel and full operating fluids, but it excludes usable fuel, crew, and payload.

The center of gravity (CG) of an aircraft is defined as the point about which

ACS code: AM.I.C

Correct answer: the aircraft would balance if suspended

Rationale: FAA-H-8083-1 defines the center of gravity as the point at which an aircraft would balance if it were possible to suspend it at that point; it is the imaginary point where all the weight is considered to be concentrated.

The horizontal distance from the reference datum to the center of gravity of an item is known as the item's

ACS code: AM.I.C

Correct answer: arm

Rationale: FAA-H-8083-1 defines the arm as the horizontal distance, in inches, from the reference datum line to the center of gravity of an item.

The center of gravity of a loaded aircraft is determined by

ACS code: AM.I.C

Correct answer: dividing the total moment by the total weight

Rationale: FAA-H-8083-1 states that the CG location is found by dividing the total moment by the total weight of the aircraft. The result is the distance, in inches, of the CG from the datum.

From which point are all weight and balance arms and CG locations measured?

ACS code: AM.I.C

Correct answer: The reference datum

Rationale: Per FAA-H-8083-1, the reference datum is an imaginary vertical plane or line from which all horizontal measurements (arms) are taken with the aircraft in level flight attitude. The manufacturer establishes its location.

Who establishes the location of the reference datum for a particular aircraft?

ACS code: AM.I.C

Correct answer: The aircraft manufacturer

Rationale: FAA-H-8083-1 states that the reference datum is established by the aircraft manufacturer. It may be located at the nose, the firewall, the leading edge of the wing, or some other convenient location, and is identified in the aircraft specifications or Type Certificate Data Sheet.

The useful load of an aircraft is determined by

ACS code: AM.I.C

Correct answer: subtracting the empty weight from the maximum gross weight

Rationale: Per FAA-H-8083-1, useful load is found by subtracting the empty weight from the maximum allowable gross weight. It includes the pilot, passengers, usable fuel, oil, and cargo.

When an aircraft is weighed, which of the following fluids must normally be drained or accounted for as residual (unusable) according to the manufacturer's weighing instructions?

ACS code: AM.I.C

Correct answer: Usable fuel

Rationale: FAA-H-8083-1 explains that during weighing, all usable fuel is normally drained, but the residual (unusable) fuel remains and is part of the empty weight. Engine oil is included or excluded according to the type certificate data sheet; modern empty weight typically includes full oil.

The empty weight center of gravity (EWCG) of an aircraft weighed on three scales is found by dividing the

ACS code: AM.I.C

Correct answer: total moment by the total weight

Rationale: Per FAA-H-8083-1, after recording each scale's weight and computing each moment about the datum, the EWCG is found by dividing the total moment (sum of all scale moments) by the total weight (sum of all scale readings, corrected for tare).

An aircraft is loaded so that its center of gravity is forward of the forward CG limit. The most likely effect on the aircraft is that it will be

ACS code: AM.I.C

Correct answer: nose-heavy and difficult to rotate or flare

Rationale: Per FAA-H-8083-1, a forward CG makes an aircraft nose-heavy and more stable but harder to rotate and control. An excessively forward CG can make the aircraft difficult or impossible to flare for landing and can exceed elevator authority.

An aircraft loaded so that its center of gravity is aft of the rearward limit will most likely be

ACS code: AM.I.C

Correct answer: tail-heavy with reduced longitudinal stability

Rationale: FAA-H-8083-1 states that an aft CG decreases longitudinal stability and makes the aircraft tail-heavy. An excessively aft CG can lead to stall and spin recovery difficulties and reduced controllability.

The maximum weight of an aircraft, fully loaded and ready for flight, that is approved for operation is called the

ACS code: AM.I.C

Correct answer: maximum gross weight

Rationale: Per FAA-H-8083-1, maximum (allowable) gross weight is the maximum authorized weight of the aircraft and its contents as listed in the specifications or Type Certificate Data Sheet.

When weight is added to or removed from an aircraft, the change in the location of the CG can be computed by dividing the

ACS code: AM.I.C

Correct answer: change in moment by the new total weight

Rationale: FAA-H-8083-1 gives the weight-shift/addition formula: the change in CG (ΔCG) equals the change in moment divided by the new total weight. When weight is added or removed, the CG shift is the added (or removed) moment divided by the new gross weight.

When the position of the center of gravity is expressed as a percentage of the mean aerodynamic chord (MAC), 0 percent MAC represents the

ACS code: AM.I.C

Correct answer: leading edge of the mean aerodynamic chord

Rationale: Per FAA-H-8083-1, when CG is expressed in percent MAC, the leading edge of the MAC (LEMAC) is 0 percent and the trailing edge (TEMAC) is 100 percent. The CG location is computed relative to the leading edge of the MAC.

An aircraft has an empty weight of 1,800 pounds and a maximum gross weight of 2,500 pounds. What is the useful load?

ACS code: AM.I.C

Correct answer: 700 pounds

Rationale: Per FAA-H-8083-1, useful load = maximum gross weight − empty weight. 2,500 − 1,800 = 700 pounds.

A 50-pound radio is installed at station +30 (30 inches aft of the datum). What is the moment of this installation about the datum?

ACS code: AM.I.C

Correct answer: 1,500 inch-pounds

Rationale: Per FAA-H-8083-1, moment = weight × arm. 50 lb × 30 in = 1,500 in-lb. Because the item is aft of the datum, the moment is positive (+1,500 in-lb).

If a weight is added at a station forward of the reference datum, its moment is considered to be

ACS code: AM.I.C

Correct answer: negative

Rationale: Per FAA-H-8083-1, arms and moments measured forward of the datum are negative, and those aft of the datum are positive. A weight installed forward of the datum therefore produces a negative moment.

An aircraft's empty weight and empty weight CG must be recomputed and the records revised when

ACS code: AM.I.C

Correct answer: equipment is added, removed, or relocated

Rationale: Per FAA-H-8083-1, whenever equipment is added, removed, or relocated such that the empty weight or empty weight CG changes, the technician must revise the weight and balance records to reflect the new condition.

The arm of an item is measured in

ACS code: AM.I.C

Correct answer: inches

Rationale: Per FAA-H-8083-1, the arm is the horizontal distance, measured in inches, from the reference datum to the center of gravity of the item.

Improper loading that places an aircraft outside its CG limits primarily affects the aircraft's

ACS code: AM.I.C

Correct answer: stability and controllability

Rationale: FAA-H-8083-1 emphasizes that loading an aircraft outside its CG limits adversely affects stability and controllability, which can make the aircraft dangerous or impossible to control.

What is the primary purpose of leveling an aircraft before weighing it?

ACS code: AM.I.C

Correct answer: To ensure accurate horizontal arm measurements

Rationale: Per FAA-H-8083-1, the aircraft must be level both longitudinally and laterally during weighing so that the measured arms (horizontal distances from the datum to each weighing point) are accurate. An out-of-level condition introduces error into the arm measurements.

When computing the weight and balance of an aircraft, the total weight multiplied by the CG distance from the datum equals the

ACS code: AM.I.C

Correct answer: total moment

Rationale: Per FAA-H-8083-1, moment equals weight times arm; therefore the total weight multiplied by the CG distance (the total arm) equals the total moment of the loaded aircraft.

After an aircraft is weighed and the empty weight CG determined, the technician must

ACS code: AM.I.C

Correct answer: record the results in the aircraft weight and balance records

Rationale: FAA-H-8083-1 requires that the results of weighing and the empty weight CG computation be recorded in the aircraft weight and balance records, including the date, empty weight, empty weight CG, and useful load.

The center of gravity of an empty aircraft, with only residual fluids and permanently installed equipment, is referred to as the

ACS code: AM.I.C

Correct answer: empty weight center of gravity

Rationale: Per FAA-H-8083-1, the empty weight center of gravity (EWCG) is the CG of the aircraft in its empty weight condition, used as the baseline for all subsequent loading computations.

An aircraft with a total weight of 2,000 pounds and a total moment of 170,000 inch-pounds has a center of gravity located at

ACS code: AM.I.C

Correct answer: 85 inches aft of the datum

Rationale: Per FAA-H-8083-1, CG = total moment ÷ total weight. 170,000 in-lb ÷ 2,000 lb = 85 inches aft of the datum.

Which document must a mechanic ensure remains current and accurate when equipment changes affect an aircraft's weight and balance?

ACS code: AM.I.C

Correct answer: The aircraft weight and balance record

Rationale: Per FAA-H-8083-1, the aircraft's weight and balance record (the equipment list and empty weight CG record) must be kept current whenever equipment is changed so the aircraft can be loaded within its approved limits.

Adverse-loaded CG checks are performed to verify that an aircraft remains within its CG limits under the

ACS code: AM.I.C

Correct answer: most extreme forward and aft loading conditions

Rationale: Per FAA-H-8083-1, adverse-loaded CG checks determine whether the aircraft stays within limits under the most extreme (most forward and most aft) practical loading conditions. The forward check loads weight forward and the aft check loads weight aft to confirm CG stays within limits.

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Frequently asked questions

How do you calculate the center of gravity of an aircraft?
Add every weight to get the total weight, multiply each weight by its arm to get its moment, add the moments to get the total moment, then divide the total moment by the total weight. The answer is a distance from the datum in inches. An aircraft at 4,000 pounds with a 600,000 inch-pound total moment has its CG 150 inches aft of the datum.
What is the difference between useful load and payload?
Useful load is the maximum allowable weight minus the empty weight, so it is everything you are permitted to add: usable fuel, crew, passengers, cargo and any drainable fluids not counted in the empty weight. Payload is the part of the useful load left for people and cargo once the fuel is aboard. Written questions usually ask for useful load, which is a single subtraction.
Where is the datum located on an aircraft?
Wherever the manufacturer put it. The datum is an imaginary vertical plane chosen by the manufacturer and published in the type certificate data sheet and the aircraft's weight and balance records. It may sit at the nose, at the firewall, or some distance ahead of the aircraft entirely. Its position is arbitrary; what matters is that every arm on that aircraft is measured from the same plane.
Can I use a calculator on the weight and balance questions?
Testing centers generally allow a simple non-programmable calculator, and the proctor may inspect it and require the memory to be cleared before you start, so confirm the current policy with your center before test day. The numbers in these questions are chosen to divide cleanly anyway, and knowing the setup matters far more than the arithmetic itself.

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