Every aviation maintenance technician (AMT) regularly encounters mathematical expressions when calculating torque values, fuel quantities, electrical loads, gear ratios, and structural dimensions. A single arithmetic mistake in these calculations can lead to incorrect rigging, improper fuel loading, or a dangerously over-torqued fastener. That is why understanding the order of operations—the universally agreed-upon sequence in which mathematical operations must be performed inside any expression—is an absolute foundational skill for anyone pursuing an FAA Airframe or Powerplant certificate.
The order of operations is commonly remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. This rule ensures that every technician, engineer, and inspector who encounters the same mathematical expression arrives at exactly the same answer. In aviation, where tolerances can be measured in thousandths of an inch and pressure values must be precise, that consistency is not optional—it is essential.
How the Order of Operations Works
When you encounter a mathematical expression containing multiple operations, you do not simply read left to right and calculate as you go. Instead, you work through the expression in a strict hierarchy. Here is what each level of PEMDAS means in practice:
Parentheses (and Brackets)
Any operation enclosed inside parentheses ( ), brackets [ ], or braces { } is resolved first, working from the innermost set outward. Parentheses are the most powerful tool in a mathematical expression because they override every other rule. In maintenance documentation, parentheses often appear in formulas for Ohm's Law combinations, area calculations for structural panels, or unit conversion chains. For example, in the expression (3 + 5) × 4, you add 3 and 5 first to get 8, then multiply by 4 to get 32. Without the parentheses—3 + 5 × 4—multiplication comes before addition, so the result is 3 + 20 = 23. Two different answers from one small formatting difference illustrates why reading technical formulas carefully is a safety habit, not just a study tip.
Exponents (Powers and Roots)
After all parenthetical groups are resolved, you evaluate exponents—powers and roots. Exponents appear frequently in aviation maintenance contexts. The area of a circle, used when calculating piston displacement or hydraulic actuator force, involves squaring the radius: A = πr². Electrical power calculations use squared values: P = I²R and P = E²/R. Square roots appear in airspeed relationships and in some structural stress formulas. A technician who applies the exponent step out of order will generate a completely wrong answer. For instance, in 2 + 3², you must square the 3 first to get 9, then add 2 for a result of 11. Doing it left to right yields 25—a 127% error.
Multiplication and Division
Multiplication and division occupy the same level of priority and are performed left to right in the order they appear in the expression. This is a critical detail: these two operations share equal rank; neither always comes first. In the expression 12 ÷ 4 × 3, you work left to right: 12 ÷ 4 = 3, then 3 × 3 = 9. Reversing the order—multiplying first—would give 12 ÷ 12 = 1, which is completely wrong. In maintenance math, you encounter this tier constantly: computing gear ratios, converting units (pounds per square inch to pounds per square foot), calculating fuel flow rates, or determining wire gauge ampacity from current and resistance values.
Addition and Subtraction
Finally, addition and subtraction are resolved, again working left to right, sharing equal priority just as multiplication and division do. These operations are straightforward but frequently the site of errors when technicians skip ahead or try to combine steps mentally. In complex multi-operation expressions, rushing to add or subtract before completing the multiplication and division steps is one of the most common mistakes seen on FAA written examinations.
Worked Examples in a Maintenance Context
Seeing PEMDAS applied to realistic maintenance problems cements understanding far better than abstract examples. Consider the following scenarios:
- Electrical circuit calculation: A simple circuit has a 24-volt source powering two resistors in series: 6 ohms and 2 ohms. Total current is I = E ÷ (R1 + R2) = 24 ÷ (6 + 2) = 24 ÷ 8 = 3 amperes. The parentheses force the addition of resistances before the division. Without them, dividing 24 by 6 first would yield 4, then adding 2 gives 6—a completely wrong current value that could cause a technician to select an undersized wire or incorrect circuit breaker.
- Piston displacement: Displacement per cylinder is calculated as V = π × r² × stroke. If radius is 2 inches and stroke is 3 inches, the correct sequence is: square the radius first (2² = 4), then multiply by π (≈3.1416), then multiply by stroke: 3.1416 × 4 × 3 ≈ 37.70 cubic inches. A technician who adds stroke to the radius before squaring will compute an entirely different and erroneous displacement.
- Fuel quantity conversion: A tank holds 50 gallons of aviation gasoline (avgas). Avgas weighs approximately 6 pounds per gallon. If the technician needs to determine the weight added to the aircraft after topping off from 20 gallons, the expression is (50 − 20) × 6 = 30 × 6 = 180 pounds. Parentheses ensure the subtraction happens before the multiplication. Without them, following strict PEMDAS, multiplication comes first: 50 − (20 × 6) = 50 − 120 = −70 pounds, which is nonsensical in context—a clear signal that parentheses are required to express the intended meaning.
Why It Matters for AMT Work
The FAA General written test (AMG) includes multiple questions that directly test whether a candidate applies the order of operations correctly. These questions are often disguised inside real-sounding maintenance scenarios involving weight and balance moments, resistor networks, or area and volume computations. Beyond the test, the practical stakes are significant.
Aircraft maintenance data—whether in a Maintenance Manual, an Airworthiness Directive, or a Structural Repair Manual—expresses formulas in standard mathematical notation. When a formula is written without parentheses around a group of terms, the technical author assumed the reader understands PEMDAS and will apply it correctly. Misreading a torque formula or a hydraulic pressure calculation because the technician ignored operation hierarchy can result in an incorrect maintenance action. The FAA's Aviation Maintenance Technician Handbook—General (FAA-H-8083-30) establishes mathematics as a core general subject area precisely because arithmetic precision underlies virtually every other technical competency.
Key Numbers and Rules
- PEMDAS sequence: Parentheses → Exponents → Multiplication/Division (left to right, equal priority) → Addition/Subtraction (left to right, equal priority).
- Multiplication and division share one tier—always work left to right across that tier; do not assume multiplication always precedes division.
- Addition and subtraction share one tier—same rule, always left to right.
- Nested parentheses: Resolve the innermost set first, then work outward. Example: 2 × [3 + (4 − 1)] = 2 × [3 + 3] = 2 × 6 = 12.
- Implied multiplication: When a number sits directly beside parentheses with no operator shown—such as 4(3 + 2)—the multiplication is implied and still follows PEMDAS: resolve parentheses first (5), then multiply by 4 = 20.
- Fraction bars act as grouping symbols: Everything above or below a fraction bar must be evaluated as a complete group before dividing numerator by denominator.
Memory Aid
The standard mnemonic is PEMDAS, often remembered as the phrase "Please Excuse My Dear Aunt Sally." Each word maps to an operation:
- Please → Parentheses
- Excuse → Exponents
- My → Multiplication
- Dear → Division
- Aunt → Addition
- Sally → Subtraction
Remember: M and D share equal rank (left to right), and A and S share equal rank (left to right). The mnemonic orders them for memory purposes only—it does not mean multiplication is always done before division.
Common Test Traps
- Ignoring left-to-right order for multiplication and division: Many candidates assume multiplication always beats division. In 8 ÷ 2 × 4, the correct answer is 16 (left to right), not 1 (if you multiply first). FAA test questions deliberately construct these expressions to catch this error.
- Missing implied parentheses in fractions: An expression like (6 + 2) / (5 − 1) requires both the numerator and denominator to be simplified before dividing. Test questions sometimes present this without explicit parentheses, expecting you to recognize the fraction bar as a grouping symbol.
- Applying exponents too late: In expressions such as 3 + 2², some candidates add 3 + 2 first to get 5, then square it for 25. The correct approach is 2² = 4, then 3 + 4 = 7. Exponents are applied to their base only, before any surrounding addition or subtraction.
- Skipping inner parentheses in nested expressions: With nested brackets, candidates sometimes solve the outer group first. Always start at the innermost set and work outward—skipping inward or solving randomly produces wrong answers on nearly every multi-step problem.
- Confusing subtraction order with commutative properties: Subtraction is not commutative—order matters. In a chain like 10 − 3 + 2, work left to right: (10 − 3) + 2 = 9, not 10 − (3 + 2) = 5. Test questions use this trap frequently in weight-and-balance and fuel-quantity scenarios.