10.1 Fractions, Decimals, Percentages & Ratio/Proportion

10.1 Fractions, Decimals, Percentages & Ratio/Proportion

Mathematics is the universal language and foundational tool of aviation maintenance. Every critical task performed by an Aviation Maintenance Technician (AMT)—from measuring engine crankshaft bearing journals with a vernier micrometer, to laying out fastener pitch on an aluminum fuselage skin, to calculating electrical load shedding on a 28-volt DC bus—relies upon rapid and flawless mathematical execution. An error in calculating a bend allowance, misinterpreting a fraction on an engineering blueprint, or improperly converting a torque value can result in catastrophic structural failure, flight control jamming, or engine in-flight shutdown. The Federal Aviation Administration (FAA) tests mathematical competencies thoroughly under FAA-H-8083-30B (Aviation Maintenance Technician Handbook — General) and the Aviation Mechanic General Practical Test Standards (PTS) / Airman Certification Standards (ACS).

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1. Common Fractions, Types, and Reduction Principles

A fraction represents an exact portion or division of a whole unit. It is expressed as one integer placed over another, separated by a horizontal fraction bar (vinculum) or a diagonal slash: $\text{Fraction} = \frac{\text{Numerator}}{\text{Denominator}}$

• Numerator (Top Number): Represents the number of equal fractional parts being considered or measured.
• Denominator (Bottom Number): Represents the total number of equal parts into which the whole integer is divided. The denominator can never be zero ($0$), as division by zero is mathematically undefined.

                          FRACTION ANATOMY & STRUCTURE
                          
                              Numerator (3 Parts Considered)
                                    ▲
                                    │
                                  ┌───┐
                                  │ 3 │
                                  └───┘
                                  ─────  <--- Fraction Bar (Division Operator)
                                  ┌───┐
                                  │ 8 │
                                  └───┘
                                    │
                                    ▼
                             Denominator (8 Total Equal Parts in Whole Inch)

Classifications of Fractions in Aviation Maintenance

1. Proper Fraction: A fraction whose numerator is strictly less than its denominator (e.g., $1/4\text{ in}$, $3/8\text{ in}$, $15/16\text{ in}$, $23/64\text{ in}$). The absolute value of a proper fraction is always less than one ($< 1.0$).
2. Improper Fraction: A fraction whose numerator is greater than or equal to its denominator (e.g., $5/4\text{ in}$, $9/8\text{ in}$, $33/32\text{ in}$, $16/16\text{ in}$). An improper fraction represents a quantity equal to or greater than one ($\ge 1.0$).
3. Mixed Number: A combination of a whole integer and a proper fraction (e.g., $1\frac{3}{8}\text{ in}$, $2\frac{5}{16}\text{ in}$, $4\frac{1}{2}\text{ gal}$). Mixed numbers are standard when specifying aircraft hardware lengths and fluid capacities.
4. Equivalent Fractions: Fractions that represent the identical physical value or dimension even though their numerators and denominators differ (e.g., $2/4 = 4/8 = 8/16 = 16/32 = 32/64 = 1/2$).

Converting Mixed Numbers and Improper Fractions

• Converting a Mixed Number to an Improper Fraction: Multiply the whole integer by the denominator, add the numerator, and place the resulting sum over the original denominator: $\text{Value} = W\frac{N}{D} = \frac{(W \times D) + N}{D}$ $\text{Example: Convert } 3\frac{5}{16}\text{ to an improper fraction: } \frac{(3 \times 16) + 5}{16} = \frac{48 + 5}{16} = \frac{53}{16}$
• Converting an Improper Fraction to a Mixed Number: Divide the numerator by the denominator. The whole quotient becomes the integer, and the remainder becomes the new numerator over the original denominator: $\text{Example: Convert } \frac{45}{8}\text{ to a mixed number: } 45 \div 8 = 5 \text{ with a remainder of } 5 \implies 5\frac{5}{8}$

Finding the Least Common Denominator (LCD)

To add or subtract fractions, they must share a common denominator. The Least Common Denominator (LCD)—also called the Lowest Common Denominator—is the smallest positive integer that is evenly divisible by all denominators in the set.

• Method of Prime Factorization: Break down each denominator into its prime factors, then multiply the highest power of each prime factor present.
• Aviation Binary Sizing Rule: In American aviation maintenance, standard hardware (AN/MS bolts, sheet metal thicknesses, twist drills) utilizes binary fractional denominators ($2, 4, 8, 16, 32, 64$). When working with standard fractional hardware, the LCD is always simply the largest denominator in the group, provided all other denominators are factors of it (e.g., for $1/4$, $3/8$, $5/16$, and $21/64$, the LCD is $64$).

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2. Fractional Arithmetic in Aviation Maintenance

Addition and Subtraction of Fractions

1. Convert all mixed numbers into improper fractions.
2. Find the LCD for all fractions in the equation.
3. Multiply the numerator and denominator of each fraction by the factor required to produce the LCD.
4. Add or subtract the numerators while keeping the common denominator unchanged.
5. Reduce the final fraction to its lowest irreducible terms or convert to a mixed number.

$\text{General Formula: } \frac{a}{d} \pm \frac{b}{d} = \frac{a \pm b}{d}$

Multiplication of Fractions

Multiplication does not require a common denominator. In aviation, multiplying fractions determines proportional sheet metal bend allowances, gear reductions, and hydraulic volume displacements:

1. Convert all mixed numbers into improper fractions.
2. Simplify by cross-cancelling common numerical factors between any numerator and any denominator.
3. Multiply all numerators straight across to obtain the product numerator.
4. Multiply all denominators straight across to obtain the product denominator.
5. Reduce the final result to lowest terms.

$\text{General Formula: } \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$ $\text{Example: } 2\frac{1}{4} \times \frac{8}{15} = \frac{9}{4} \times \frac{8}{15} = \frac{3 \times 2}{1 \times 5} = \frac{6}{5} = 1\frac{1}{5}$

Division of Fractions and the Reciprocal Rule

Dividing by a fraction is identical to multiplying by its reciprocal (the inverted fraction where numerator and denominator swap places):

1. Convert all mixed numbers to improper fractions.
2. Identify the divisor (the fraction doing the dividing, following the $\div$ sign).
3. Invert the divisor (take its reciprocal: $\frac{a}{b} \implies \frac{b}{a}$).
4. Change the division operator to multiplication.
5. Cross-cancel and multiply straight across.

$\text{General Formula: } \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c}$ $\text{Example: } 3\frac{3}{8} \div \frac{3}{16} = \frac{27}{8} \div \frac{3}{16} = \frac{27}{8} \times \frac{16}{3} = \frac{9 \times 2}{1 \times 1} = 18$

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3. Decimals, Precision Measurement & Conversion Standards

Decimal notation represents fractions whose denominators are positive powers of ten ($10, 100, 1,000, 10,000$). In aircraft manufacturing and engine overhaul, decimal dimensions dominate engineering drawings and micrometers.

                      DECIMAL PLACE VALUE ARCHITECTURE
  ┌────────────────────────────────────────────────────────────────────────┐
  │  Hundreds │ Tens │ Units │ . │ Tenths │ Hundredths │ Thousandths │ Ten-Thousandths │
  │    100    │  10  │   1   │ . │  1/10  │   1/100    │   1/1,000   │    1/10,000     │
  │     1     │   2  │   5   │ . │   0    │     6      │      2      │        5        │
  └────────────────────────────────────────────────────────────────────────┘
                         Value: 125.0625 inches (125-1/16")

Decimal Arithmetic Operations

• Addition & Subtraction: All numbers must be aligned strictly by their decimal points in vertical columns before performing operations. Zeros may be added as place-holders to maintain uniform column width.
• Multiplication: Multiply the numbers as if they were whole integers without regard to decimal points. Count the total number of decimal places in all factors combined; place the decimal point in the product so that it has the identical total number of decimal digits.
• Division: If the divisor contains a decimal, shift its decimal point completely to the right to make it a whole integer. Shift the decimal point in the dividend (number being divided) the exact same number of places to the right. Place the decimal point in the quotient directly above the relocated decimal in the dividend.

Fraction to Decimal Conversion Table (Standard Aviation Increments)

| Fraction (in) | Decimal (in) | Standard Aviation Application / Hardware Size | | :---: | :---: | : | | $1/64$ | $0.015625$ | Minimum structural sheet metal shim increment; #80 drill bit range | | $1/32$ | $0.03125$ | Minimum safety wire diameter ($0.032\text{ in}$); $1/32\text{ in}$ cotter pins | | $3/64$ | $0.046875$ | Thin aluminum skin doubler gauge | | $1/16$ | $0.0625$ | AN3 bolt diameter step ($3/16 = 3 \times 1/16\text{ in}$); $1/16\text{ in}$ bend radius | | $3/32$ | $0.09375$ | 3/32-inch solid rivet diameter (AN470AD3 / MS20470AD3) | | $1/8$ | $0.1250$ | 1/8-inch solid rivet diameter (AN470AD4 / MS20470AD4); AN bolt length dash | | $5/32$ | $0.15625$ | 5/32-inch solid rivet diameter (AN470AD5 / MS20470AD5) | | $3/16$ | $0.1875$ | 3/16-inch solid rivet diameter (AN470AD6); AN3 bolt shank diameter | | $1/4$ | $0.2500$ | AN4 bolt diameter ($4/16\text{ in}$); standard hydraulic line OD | | $5/16$ | $0.3125$ | AN5 bolt diameter ($5/16\text{ in}$) | | $3/8$ | $0.3750$ | AN6 bolt diameter ($6/16\text{ in}$); standard rigid fuel tube OD | | $7/16$ | $0.4375$ | AN7 bolt diameter ($7/16\text{ in}$) | | $1/2$ | $0.5000$ | AN8 bolt diameter ($8/16\text{ in}$); heavy structural fitting attachment | | $5/8$ | $0.6250$ | AN10 bolt diameter ($10/16\text{ in}$) | | $3/4$ | $0.7500$ | AN12 bolt diameter ($12/16\text{ in}$) | | $1.0$ | $1.0000$ | AN16 bolt diameter ($16/16\text{ in}$) |

Rounding Rules & Significant Figures in Aviation Maintenance

In aviation maintenance manuals, precision tolerances govern part rejection:

1. Standard 5-Up Rule: If the digit immediately following the desired degree of precision is $5, 6, 7, 8,$ or $9$, round up by adding $1$ to the preceding digit. If the digit is $0, 1, 2, 3,$ or $4$, drop it without changing the preceding digit.
2. Intermediate Precision: Never round intermediate values during multi-step calculations. Retain at least 4 to 6 decimal places throughout intermediate steps, rounding only the final published result to prevent compounding round-off error.
3. Crankshaft & Bearing Tolerances: Internal engine measurements require four decimal places ($0.0001\text{ in}$). For example, an engine crankshaft main journal dimensioned as $2.3750\text{ in}$ with a wear limit of $-0.0015\text{ in}$ must be rejected if micrometer readings fall below $2.3735\text{ in}$.

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4. Percentages, Percentage Change & Maintenance Thresholds

A percentage (symbolized by %) is a fraction whose denominator is always $100$. The term originates from the Latin per centum, meaning "by the hundred." $\text{Percentage } (P) = \frac{\text{Part}}{\text{Whole}} \times 100\%$

• Converting Decimal to Percentage: Shift the decimal point two places to the right and append the % symbol ($0.85 = 85%$).
• Converting Percentage to Decimal: Remove the % symbol and shift the decimal point two places to the left ($7.5% = 0.075$).

                      PERCENTAGE RELATIONSHIP TRIANGLE
                                   [ Part ]
                                     /  \
                                    /    \
                                   /  ×   \
                       [ Rate (%) ] ────── [ Base (Whole) ]
                                   
             Part = Rate × Base  │  Rate = Part / Base  │  Base = Part / Rate

Percentage Change (Degradation & Expansion Formula)

Technicians calculate percentage change to monitor component wear, engine performance degradation, and structural deformation: $\text{Percentage Change } (\%) = \frac{|\text{New Value} - \text{Original Value}|}{\text{Original Value}} \times 100\%$

Critical Aviation Percentage Thresholds

• Aircraft Battery Capacity Degradation (14 CFR Part 23/25 / AC 43.13-1B): An aircraft lead-acid or nickel-cadmium (NiCad) main battery must deliver at least $80%$ of its rated ampere-hour (Ah) capacity during a constant-current discharge test. Any battery delivering less than $80%$ capacity is legally unairworthy and must be overhauled or replaced.
• Continuous Electrical Bus Loading: Under FAA Advisory Circular AC 43.13-1B, an engine-driven alternator or generator should not be loaded beyond $80%$ of its maximum continuous rated output under normal cruise flight conditions, preserving a $20%$ reserve margin for intermittent motor start-ups and emergency bus loads.
• Allowable Structural Corrosion Material Loss: Structural repair manuals (SRM) typically limit allowable surface corrosion grinding to a maximum of $10%$ of original skin thickness before a riveted doubler repair is required.
• Turbine Engine Exhaust Gas Temperature (EGT) Margin: Turbine overhaul schedules track EGT margin deterioration as a percentage of certified maximum take-off turbine gas temperature.

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5. Ratios, Direct and Inverse Proportions

A ratio is a mathematical comparison of two numerical values possessing identical units of measurement. It expresses how many times larger or smaller one quantity is compared to another. Ratios are written with a colon ($a:b$), as a fraction ($a/b$), or with the word "to" ($a\text{ to }b$).

A proportion is an equation stating that two individual ratios are equal: $\frac{a}{b} = \frac{c}{d} \implies a \times d = b \times c \quad \text{(Cross-Product Equality)}$

                        RATIO AND PROPORTION TYPES
  ┌────────────────────────────────────────────────────────────────────────┐
  │  DIRECT PROPORTION (y = k · x):                                        │
  │  As one variable INCREASES, the other variable INCREASES proportionally│
  │  • Fuel consumption vs. Flight Hours: (Gal_1 / Hr_1 = Gal_2 / Hr_2)    │
  │  • Blueprint Scale: 1 inch on print = 4 inches on airframe             │
  │  • Hydraulic Force Amplification: (F_1 / A_1 = F_2 / A_2)              │
  ├────────────────────────────────────────────────────────────────────────┤
  │  INVERSE PROPORTION (y = k / x  or  x_1 · y_1 = x_2 · y_2):            │
  │  As one variable INCREASES, the other variable DECREASES proportionally│
  │  • Gear Teeth vs. Rotational Speed (RPM): (N_1 · T_1 = N_2 · T_2)      │
  │  • Pulley Diameters vs. Pulley RPM: (N_1 · D_1 = N_2 · D_2)            │
  │  • Fuel-Air Mixture: 12:1 (Rich) vs. 16:1 (Lean)                       │
  └────────────────────────────────────────────────────────────────────────┘

Primary Aviation Ratios

1. Fuel-Air Ratio: The mass ratio of aviation fuel to incoming combustion air in reciprocating engines. Stoichiometric chemical perfection is $15:1$ ($15\text{ lb}$ air to $1\text{ lb}$ fuel or $0.067$). Best power mixture is approximately $12.5:1$ ($0.080$, rich), while best economy is $16:1$ ($0.0625$, lean).
2. Compression Ratio (CR): The ratio of total cylinder volume at Bottom Dead Center (BDC) compared to clearance volume at Top Dead Center (TDC), ranging from $7.0:1$ to $10.0:1$ in general aviation engines.
3. Wing Aspect Ratio ($AR$): The ratio of total wingspan squared ($b^2$) to total wing planform area ($S$): $AR = b^2 / S$. Gliders feature high aspect ratios ($>20:1$) for high lift-to-drag efficiency, while jet fighters feature low aspect ratios ($2:1$ to $4:1$) for high roll rates.
4. Gear and Pulley Mechanical Ratios: $\text{Gear Ratio} = \frac{\text{Number of Teeth on Driven Gear }(T_2)}{\text{Number of Teeth on Driving Gear }(T_1)} = \frac{\text{Driver RPM }(N_1)}{\text{Driven RPM }(N_2)}$ $\text{Pulley Speed Formula: } N_1 \times D_1 = N_2 \times D_2 \implies N_2 = \frac{N_1 \times D_1}{D_2}$ where $N_1, N_2$ are rotational speeds in RPM, and $D_1, D_2$ are pulley pitch diameters in inches.

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6. Worked Step-by-Step Calculation Examples

Example 1: Multi-Step Fraction Sizing & Sheet Metal Rivet Grip Stack-Up

Scenario: An AMT is repairing an aircraft fuselage stringer and must install an AN470AD4 (1/8-inch diameter) solid rivet through a structural stack-up consisting of three 2024-T3 aluminum sheets:

• Upper skin panel thickness: $1/16\text{ in}$
• Doubler reinforcement thickness: $3/32\text{ in}$
• Internal fuselage stringer flange thickness: $1/8\text{ in}$

Per FAA Advisory Circular AC 43.13-1B, the proper rivet length equals the total material grip thickness plus a shank protrusion allowance of $1.5$ times the rivet diameter ($1.5 \times D$) to form the shop head. Rivet lengths are designated in increments of $1/16\text{ in}$ (dash numbers).

Calculate the exact total grip thickness as a fraction and decimal, compute the required minimum rivet shank length, and select the correct AN rivet dash number.

Step-by-Step Mathematical Solution:

1. Find the Least Common Denominator (LCD) for the three sheet thicknesses ($1/16, 3/32, 1/8$): $\text{Denominators: } 16 = 2^4, \quad 32 = 2^5, \quad 8 = 2^3 \implies \text{LCD} = 32$
2. Convert each thickness to an equivalent fraction with a denominator of $32$: $\text{Upper Skin: } \frac{1}{16} = \frac{1 \times 2}{16 \times 2} = \frac{2}{32}\text{ in}$ $\text{Doubler: } \frac{3}{32} = \frac{3}{32}\text{ in}$ $\text{Stringer Flange: } \frac{1}{8} = \frac{1 \times 4}{8 \times 4} = \frac{4}{32}\text{ in}$
3. Sum the thicknesses to find total material grip: $\text{Total Grip Thickness} = \frac{2}{32} + \frac{3}{32} + \frac{4}{32} = \frac{2 + 3 + 4}{32} = \frac{9}{32}\text{ inch}$ $\text{Decimal Equivalence: } 9 \div 32 = 0.28125\text{ inch}$
4. Calculate the required shop head allowance ($1.5 \times D$) for a $1/8\text{ in}$ diameter rivet ($D = 1/8 = 4/32\text{ in}$): $\text{Head Allowance} = 1.5 \times \frac{1}{8} = \frac{3}{2} \times \frac{1}{8} = \frac{3}{16}\text{ in}$ $\text{Convert to 32nds: } \frac{3}{16} = \frac{6}{32}\text{ inch} = 0.1875\text{ inch}$
5. Compute total required rivet length: $\text{Total Length} = \text{Grip Thickness} + \text{Head Allowance} = \frac{9}{32}\text{ in} + \frac{6}{32}\text{ in} = \frac{15}{32}\text{ inch} = 0.46875\text{ inch}$
6. Determine the standard AN rivet dash length (increments of $1/16\text{ in}$): $\frac{15}{32}\text{ in} = \frac{7.5}{16}\text{ in} \implies \text{Standard available lengths are } \frac{7}{16}\text{ in} (\text{dash } -7 = 0.4375\text{ in}) \text{ and } \frac{8}{16}\text{ in} = \frac{1}{2}\text{ in} (\text{dash } -8 = 0.5000\text{ in})$ To avoid an under-length shop head below FAA minimum dimensions, the technician selects the AN470AD4-8 ($1/2\text{ in}$ length).

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Example 2: Aircraft Battery Capacity Degradation Test

Scenario: A $24\text{-volt}$ aircraft main emergency battery has a manufacturer rated capacity of $35.0\text{ Ampere-hours (Ah)}$ at the standard 1-hour discharge rate. During an annual scheduled capacity test conducted per AC 43.13-1B, the battery is discharged at a constant current of $35.0\text{ A}$. The battery terminal voltage drops to the mandatory cutoff endpoint voltage of $20.0\text{ V}$ after precisely $45.0\text{ minutes}$ ($0.75\text{ hours}$).

• Calculate the actual delivered capacity in Ampere-hours ($Ah$).
• Calculate the battery's remaining capacity as a percentage of rated capacity.
• Determine whether the battery meets the FAA airworthiness requirement for return to service.

Step-by-Step Mathematical Solution:

1. Calculate delivered capacity: $\text{Capacity } (Ah) = \text{Current } (I) \times \text{Time in Hours } (t)$ $\text{Delivered Capacity} = 35.0\text{ Amperes} \times 0.75\text{ Hours} = 26.25\text{ Ah}$
2. Calculate remaining percentage of rated capacity: $\text{Percentage Capacity} = \left(\frac{\text{Delivered Capacity}}{\text{Rated Capacity}}\right) \times 100\%$ $\text{Percentage Capacity} = \left(\frac{26.25\text{ Ah}}{35.0\text{ Ah}}\right) \times 100\% = 0.7500 \times 100\% = 75.0\%$
3. Evaluate airworthiness status against FAA standards: $\text{FAA Legal Standard: Minimum Acceptable Capacity } \ge 80.0\%$ $\text{Actual Capacity: } 75.0\% < 80.0\%$ Conclusion: The battery fails the airworthiness inspection and cannot be returned to service. It must be subjected to a full shop reconditioning deep-cycle routine or permanently condemned and replaced.

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Example 3: Pulley Drive RPM and Inverse Proportion

Scenario: An engine-driven alternator on a piston aircraft is powered from the crankshaft accessory pulley via a V-belt drive. The engine crankshaft drive pulley has an effective pitch diameter of $6.0\text{ inches}$ and rotates at an engine cruise speed of $2,400\text{ RPM}$. The alternator driven pulley has an effective pitch diameter of $2.5\text{ inches}$.

• Calculate the rotational speed (RPM) of the alternator.
• Calculate the speed step-up ratio between the alternator and the engine.

Step-by-Step Mathematical Solution:

1. Establish the inverse proportion equation for pulley drives: $N_1 \times D_1 = N_2 \times D_2$ where $N_1 = 2,400\text{ RPM}$ (Engine), $D_1 = 6.0\text{ in}$, $D_2 = 2.5\text{ in}$, and $N_2 = \text{Alternator RPM}$.
2. Solve algebraically for $N_2$: $N_2 = \frac{N_1 \times D_1}{D_2} = \frac{2,400\text{ RPM} \times 6.0\text{ in}}{2.5\text{ in}} = \frac{14,400}{2.5} = 5,760\text{ RPM}$
3. Calculate the speed ratio: $\text{Speed Ratio} = \frac{N_2}{N_1} = \frac{5,760\text{ RPM}}{2,400\text{ RPM}} = 2.4 : 1$ Conclusion: The alternator operates at $5,760\text{ RPM}$, spinning $2.4$ times faster than engine crankshaft speed.