3.5 Maintenance Math and Unit Discipline

Arithmetic Discipline: Order of Operations and Rounding

Maintenance math fails most often not from hard formulas but from sloppy process. Follow order of operations - parentheses, then exponents, then multiplication and division left to right, then addition and subtraction (PEMDAS). Evaluating 2 + 3 x 4 as 20 instead of 14 is a classic trap.

Signed numbers matter in W&B (forward arms are negative) and temperature work: a negative times a negative is positive, and subtracting a negative adds. Round only at the very end and to the precision the task needs; rounding mid-calculation compounds error. Scientific notation (writing 6,700 as 6.7 x 10^3) keeps large and small numbers manageable - move the decimal and adjust the exponent, adding exponents when multiplying powers of ten. Express fractions and decimals interchangeably; many aircraft dimensions are fractional (1/16 in) while instruments read decimals.

The recommended workflow is identify the formula, convert units first, solve, round last, then sanity-check the magnitude against reality.

Worked conversion: because 1 in = 25.4 mm exactly, a 3.500-in bore measured in metric tooling is 3.500 x 25.4 = 88.9 mm; reversing it, an 88.9 mm reading divided by 25.4 returns 3.500 in. Worked displacement: total engine displacement is the single-cylinder swept volume (pi x r^2 x stroke) times the number of cylinders, so a four-cylinder engine with a 5.125-in bore (r = 2.5625 in) and 4.375-in stroke displaces pi x 2.5625^2 x 4.375 x 4 = about 361 cubic inches.

Geometry: Areas and Volumes

Mechanics compute areas (wing area, sheet-metal patch coverage) and volumes (fuel tanks, baggage bays):

Worked volume example: a cylindrical tank 10 in in diameter and 20 in long holds pi x r^2 x h = 3.1416 x (5)^2 x 20 = 3.1416 x 25 x 20 = 1,570.8 cubic inches. Converting to gallons (231 in^3 per US gallon): 1,570.8 / 231 = 6.8 gallons. Two common errors are using the diameter instead of the radius (radius is half the diameter) and forgetting to square the radius. For a circle, doubling the radius quadruples the area because the radius is squared - a useful reasonableness check.

Ratios, Proportions, and Percentages

A ratio compares two quantities (for example a compression ratio of 8:1 means cylinder volume at bottom dead center is eight times the volume at top dead center). A proportion sets two ratios equal and is solved by cross-multiplying: if 2 oz of additive treats 10 gal, then x oz treats 25 gal gives x = (2 x 25) / 10 = 5 oz. Gearing, mixing, and scale-drawing problems are all proportions.

A percentage is simply a ratio expressed per hundred, so you convert by multiplying a decimal by 100 (0.05 = 5%) or dividing a percent by 100 to use it. For example, an aircraft loaded to 2,070 lb against a 2,300 lb maximum gross weight is at 2,070 / 2,300 = 90%, leaving 230 lb (10%) of margin. Percent-of-change problems (wear, growth) divide the change by the original value. Keep the base consistent - taking a percentage of the wrong total is a frequent mistake.

Units, Conversions, and Aircraft Hardware Sizing

Unit discipline prevents answers that are numerically right but dimensionally wrong:

AN and AC hardware sizes follow a dash-number convention: fluid-line fittings and tubing are sized in sixteenths of an inch by their dash number, so a -6 fitting is 6/16 = 3/8 in tube outside diameter. AN bolt diameters likewise use sixteenths (an AN4 bolt is 4/16 = 1/4 in), and the bolt's grip/length is given by a second number. Torque conversions matter when a wrench reads inch-pounds but the spec is foot-pounds (1 ft-lb = 12 in-lb), and applying foot-pound numbers to an inch-pound spec over-torques fasteners twelve-fold.

When a problem mixes metric and US units, convert everything to one system before computing, then convert the final answer back if needed.

Compression Ratio and Reasonableness Checks

Compression ratio, a recurring example, is the ratio of cylinder volume with the piston at bottom dead center to the volume at top dead center: if total volume is 80 in^3 and clearance volume is 10 in^3, the ratio is 80 / 10 = 8:1. Ratios have no units because the units cancel. Finally, build the habit of a reasonableness check: estimate the answer's order of magnitude before trusting the calculator. A fuel tank that computes to 600 gallons in a light aircraft, or a torque of 3 in-lb on a main bolt, signals a units or decimal error.

In maintenance, an arithmetic slip is not abstract - it can mean an out-of-limits part is accepted or a fastener is set to the wrong torque, so the discipline of identify-convert-solve-round-check is itself a safety practice.