Module 1.1 Arithmetic
1,012 words · checked 2026-10-03 · Free
Module 1 is a short paper (16 questions for category A, 32 for B1/B2/B2L/B3 under EASA) but every later module relies on it. Expect quick calculations: fractions, percentages, ratios, unit conversions, areas and volumes. CAA venues do not allow calculators, and many Part-147 exams do not either, so practise by hand.
Knowledge levels
| Sub-module | A | B1 | B2 / B2L | B3 |
|---|---|---|---|---|
| 1.1 Arithmetic | 1 | 2 | 2 | 2 |
Level 1 = know the terms and do simple examples; level 2 = apply formulas and solve practical problems.
Terms and signs
- Sum (+), difference (−), product (×), quotient (÷).
- Integer: whole number. Prime: divisible only by 1 and itself (2, 3, 5, 7, 11 ...).
- Factor: divides exactly into a number (factors of 12: 1, 2, 3, 4, 6, 12). Multiple: the number times an integer (multiples of 4: 4, 8, 12 ...).
- HCF (highest common factor) and LCM (lowest common multiple): LCM is the lowest common denominator when adding fractions.
- Order of operations (BODMAS / BIDMAS): Brackets, Orders (powers, roots), Division and Multiplication (left to right), Addition and Subtraction (left to right).
Key fact: 6 + 4 × 3 = 18, not 30. Multiplication comes before addition. A favourite distractor is the answer you get by working left to right.
Fractions
- Proper fraction: numerator smaller than denominator (3/8). Improper: numerator ≥ denominator (11/8 = 1 3/8, a mixed number).
- Add/subtract: bring to a common denominator. 3/8 + 1/4 = 3/8 + 2/8 = 5/8.
- Multiply: numerator × numerator, denominator × denominator. 2/3 × 3/4 = 6/12 = 1/2.
- Divide: invert the divisor and multiply. 3/4 ÷ 3/8 = 3/4 × 8/3 = 2.
- Fraction to decimal: divide numerator by denominator. 5/16 = 0.3125.
Common inch fractions are worth memorising: 1/16 = 0.0625, 1/8 = 0.125, 3/16 = 0.1875, 1/4 = 0.25, 3/8 = 0.375, 1/2 = 0.5, 5/8 = 0.625, 3/4 = 0.75.
Decimals and rounding
- Decimal places count digits after the point; significant figures count from the first non-zero digit. 0.004 560 has 3 significant figures (4, 5, 6) - the trailing zero only counts if it is meaningful.
- Multiplying by 10, 100, 1 000 moves the point 1, 2, 3 places right.
Percentages
- Percent = per hundred. x % of y = x/100 × y.
- Percentage change = (new − old) / old × 100.
- Reverse percentage: after a 20 % increase a value is 120 % of the original, so original = new ÷ 1.2.
Worked example - percentage change
A tyre is inflated to 200 psi when hot and drops to 185 psi when cold. What is the percentage drop?
- Change = 200 − 185 = 15 psi
- Percentage = 15 / 200 × 100 = 7.5 %
Distractor trap: dividing by the new value (15 / 185 = 8.1 %).
Ratio and proportion
- A ratio compares quantities: 3 : 1. Simplify by dividing by the HCF (12 : 8 = 3 : 2).
- To share an amount in a ratio, add the parts. Share 60 litres in 3 : 2 → 5 parts, 1 part = 12 litres → 36 L and 24 L.
- Direct proportion: y = kx (double one, double the other). Inverse proportion: y = k/x (double one, halve the other).
Worked example - gear ratio
A driving gear has 20 teeth and meshes with a driven gear of 60 teeth. The driver turns at 1 200 rpm.
- Gear ratio = driven teeth / driver teeth = 60 / 20 = 3 : 1
- Driven speed = 1 200 × 20 / 60 = 400 rpm (speed is inversely proportional to the number of teeth)
- Torque on the driven shaft is about 3 times greater (ignoring losses).
Averages
- Mean = sum of values ÷ number of values.
- Median = middle value when sorted. Mode = most frequent value.
- Average speed = total distance ÷ total time (not the mean of two speeds).
Weights, measures and conversion factors
| Quantity | Conversion |
|---|---|
| Length | 1 in = 25.4 mm exactly; 1 ft = 0.3048 m; 1 nautical mile = 1 852 m |
| Mass | 1 kg ≈ 2.205 lb; 1 lb ≈ 0.4536 kg |
| Volume | 1 US gal ≈ 3.785 L; 1 Imperial gal ≈ 4.546 L; 1 L = 1 000 cm³ |
| Pressure | 1 bar = 100 kPa ≈ 14.5 psi; 1 psi ≈ 6.895 kPa; standard atmosphere = 1 013.25 hPa = 29.92 inHg |
| Temperature | °F = °C × 9/5 + 32; K = °C + 273.15 |
| Speed | 1 knot = 1 nautical mile per hour ≈ 1.852 km/h |
Worked example - unit conversion
Convert a bolt length of 2 3/8 in to millimetres.
- 2 3/8 in = 2.375 in
- 2.375 × 25.4 = 60.325 mm
Distractor trap: dividing by 25.4 (0.0935) - always check the answer makes sense (millimetres must be a larger number than inches).
SI prefixes
| Prefix | Symbol | Factor |
|---|---|---|
| giga | G | 10⁹ |
| mega | M | 10⁶ |
| kilo | k | 10³ |
| milli | m | 10⁻³ |
| micro | µ | 10⁻⁶ |
| nano | n | 10⁻⁹ |
| pico | p | 10⁻¹² |
Areas and volumes
| Shape | Area / volume |
|---|---|
| Rectangle | A = length × width |
| Triangle | A = ½ × base × height |
| Circle | A = πr² = πd²/4; circumference = πd |
| Trapezium | A = ½ (a + b) × h |
| Cylinder | V = πr²h |
| Sphere | V = 4/3 πr³; surface = 4πr² |
| Cone | V = 1/3 πr²h |
Worked example - piston displacement
A cylinder has a bore of 10 cm and a stroke of 12 cm. What is its swept volume?
- Area = π × d² / 4 = 3.1416 × 100 / 4 = 78.54 cm²
- Volume = area × stroke = 78.54 × 12 = 942.5 cm³
For a 4-cylinder engine the total displacement is 4 × 942.5 ≈ 3 770 cm³. Trap answer: using the diameter as the radius (4 × too large).
Squares, cubes and roots
- 12² = 144, 15² = 225, 2³ = 8, 3³ = 27, 10³ = 1 000.
- √144 = 12, ∛27 = 3, √2 ≈ 1.414, √3 ≈ 1.732.
- Doubling a diameter multiplies area by 4 and volume of a sphere by 8.
Key fact: Area scales with the square of a length; volume with the cube. Halving the radius of a pipe cuts its cross-section area to one quarter.
Exam tips
- Estimate first, then check the options: three-option questions often contain one answer with the wrong power of ten.
- Keep units in every line of a calculation.
- When a ratio is involved, ask "should the answer be bigger or smaller?" before choosing.