Trades Math Study Guide

Master the essential math skills for Canadian apprenticeship exams. Clear explanations, worked examples, and practice questions.

1. Whole Numbers & Basic Operations

Foundation skills for all trade calculations — managing large numbers, applying the correct order of operations, and rounding accurately.

Key Concept: BEDMAS (Brackets, Exponents, Division/Multiplication, Addition/Subtraction) determines the order you solve equations.
BEDMAS: B = Brackets, E = Exponents, D/M = Division/Multiplication (left to right), A/S = Addition/Subtraction (left to right)

Example 1: Order of Operations

Problem: Solve 8 + 12 ÷ 3 × 2 - 5
Step 1: Division and multiplication first (left to right): 12 ÷ 3 = 4
Step 2: Continue: 4 × 2 = 8
Step 3: Addition and subtraction (left to right): 8 + 8 = 16, then 16 - 5 = 11
Answer: 11

Example 2: Rounding Large Numbers

Problem: A contractor estimates material costs at $47,638.72. Round to the nearest hundred.
Step 1: Look at the tens digit: 3 (less than 5)
Step 2: Round down the hundreds place
Answer: $47,600

Example 3: Multiplication of Large Numbers

Problem: A flooring installer needs 24 sheets of plywood at 48 square feet per sheet. Total area?
Step 1: Multiply 24 × 48
Step 2: 24 × 48 = 24 × (50 - 2) = 1,200 - 48 = 1,152
Answer: 1,152 square feet

Try It

Q1: Solve 15 + 20 ÷ 5 × 3 - 4
Step 1: Division first: 20 ÷ 5 = 4
Step 2: Multiply: 4 × 3 = 12
Step 3: Add and subtract left to right: 15 + 12 - 4 = 23
Answer: 23
Q2: A carpenter works 35 hours per week for 16 weeks. At $32/hour, what's the total pay?
Step 1: Total hours = 35 × 16 = 560 hours
Step 2: Total pay = 560 × $32 = $17,920
Answer: $17,920

2. Fractions, Decimals & Percentages

Essential for material calculations, waste calculations, markup pricing, and efficiency ratings common in apprenticeship exams.

Key Concept: Fractions describe parts of a whole. Converting between fractions, decimals, and percentages is critical for trade measurements.
Fraction to Decimal: Numerator ÷ Denominator | Decimal to Percentage: Multiply by 100 | Percentage to Decimal: Divide by 100

Example 1: Adding Fractions

Problem: A plumber has a 3/8" pipe and needs to add 5/8". Total diameter?
Step 1: Denominators are the same (8), so add numerators: 3 + 5 = 8
Step 2: 8/8 = 1 inch
Answer: 1 inch

Example 2: Percentage Calculations

Problem: A supplier offers a 15% discount on a $240 order. What's the new cost?
Step 1: Calculate discount: 15% of $240 = 0.15 × $240 = $36
Step 2: New cost: $240 - $36 = $204
Answer: $204

Example 3: Decimal Conversion

Problem: Convert 7/16" to decimal inches.
Step 1: Divide 7 ÷ 16 = 0.4375
Answer: 0.4375"

Try It

Q1: Subtract 1/4 from 7/8. Express as a fraction in lowest terms.
Step 1: Convert 1/4 to eighths: 1/4 = 2/8
Step 2: 7/8 - 2/8 = 5/8
Answer: 5/8
Q2: A job site wastes 8% of 150 meters of copper wire. How much usable wire remains?
Step 1: Waste = 8% of 150 = 0.08 × 150 = 12 meters
Step 2: Usable = 150 - 12 = 138 meters
Answer: 138 meters

3. Ratios & Proportions

Used in gear ratios, mixture ratios (concrete, paint), scale drawings, and slope calculations — all common in trade work.

Key Concept: A ratio compares two quantities. A proportion shows two equal ratios. Use cross-multiplication to solve for unknowns.
If a/b = c/d, then a × d = b × c (Cross-Multiplication)

Example 1: Gear Ratios

Problem: A gear system has a 3:2 ratio. If the input gear makes 120 revolutions, how many does the output gear make?
Step 1: Set up the proportion: 3/2 = 120/x
Step 2: Cross-multiply: 3x = 2 × 120 = 240
Step 3: Solve: x = 240 ÷ 3 = 80
Answer: 80 revolutions

Example 2: Concrete Mixture Ratio

Problem: Concrete requires a 1:2:3 ratio (cement:sand:aggregate). How much sand for 2 bags of cement?
Step 1: The ratio is 1:2, so for 2 cement, you need 2 × 2 = 4 sand
Answer: 4 units of sand

Example 3: Scale Drawing

Problem: A floor plan uses a scale of 1:50. A room measures 6 cm on the drawing. What's the actual length?
Step 1: Set up: 1/50 = 6/x
Step 2: Cross-multiply: x = 6 × 50 = 300 cm = 3 meters
Answer: 3 meters

Try It

Q1: A 4:1 hydraulic pressure ratio means if input pressure is 50 psi, what's the output pressure?
Step 1: Set up: 4/1 = x/50
Step 2: Cross-multiply: x = 4 × 50 = 200 psi
Answer: 200 psi
Q2: Paint thinner is mixed 3 parts paint to 1 part thinner. For 12 liters of paint, how much thinner?
Step 1: Set up ratio: 3 paint : 1 thinner
Step 2: 3/1 = 12/x
Step 3: x = 12 ÷ 3 = 4 liters
Answer: 4 liters of thinner

4. Unit Conversions

Converting between imperial and metric, length, weight, volume, pressure, and temperature is essential for all trades work in Canada.

Key Conversions: 1" = 25.4 mm | 1 foot = 0.3048 m | 1 lb = 0.453 kg | 1 gal = 3.785 L | °C = (°F - 32) × 5/9

Example 1: Inches to Millimeters

Problem: Convert 3.5 inches to millimeters.
Step 1: Use conversion factor: 1" = 25.4 mm
Step 2: 3.5 × 25.4 = 88.9 mm
Answer: 88.9 mm

Example 2: Pounds to Kilograms

Problem: A load cell reads 550 lbs. What's the weight in kilograms?
Step 1: Use conversion: 1 lb = 0.453 kg
Step 2: 550 × 0.453 = 249.15 kg
Answer: 249.15 kg

Example 3: Temperature Conversion

Problem: Convert 77°F to Celsius.
Step 1: Use formula: °C = (°F - 32) × 5/9
Step 2: °C = (77 - 32) × 5/9 = 45 × 5/9 = 25°C
Answer: 25°C

Try It

Q1: Convert 240 gallons to liters.
Step 1: Use conversion: 1 gal = 3.785 L
Step 2: 240 × 3.785 = 908.4 L
Answer: 908.4 liters
Q2: A water tank holds 150 cubic feet. Convert to cubic meters (1 ft³ = 0.0283 m³).
Step 1: Use conversion factor: 1 ft³ = 0.0283 m³
Step 2: 150 × 0.0283 = 4.245 m³
Answer: 4.245 m³

5. Algebra Basics

Solving for unknown values and rearranging formulas are core to troubleshooting and trade calculations (voltage, power, heat, etc.).

Key Concept: Whatever you do to one side of an equation, do to the other. This keeps it balanced and lets you solve for unknown variables.
V = IR (Voltage = Current × Resistance) | P = IV (Power = Current × Voltage) | Q = mcΔT (Heat = mass × specific heat × temperature change)

Example 1: Solving for X

Problem: Solve 3x + 7 = 22
Step 1: Subtract 7 from both sides: 3x = 22 - 7 = 15
Step 2: Divide both sides by 3: x = 15 ÷ 3 = 5
Answer: x = 5

Example 2: Rearranging Ohm's Law

Problem: Given V = IR and V = 240 volts, I = 20 amps. Find R.
Step 1: Start with V = IR
Step 2: Divide both sides by I: R = V/I
Step 3: Substitute: R = 240/20 = 12 ohms
Answer: R = 12 ohms

Example 3: Substituting into a Formula

Problem: Find power using P = IV where I = 15 amps and V = 120 volts.
Step 1: Substitute values: P = 15 × 120
Step 2: Calculate: P = 1,800 watts
Answer: 1,800 watts

Try It

Q1: Solve 2x - 5 = 13
Step 1: Add 5 to both sides: 2x = 13 + 5 = 18
Step 2: Divide by 2: x = 18 ÷ 2 = 9
Answer: x = 9
Q2: Rearrange P = IV to solve for I (current), then calculate I when P = 3000 watts and V = 120 volts.
Step 1: Rearrange P = IV to I = P/V
Step 2: Substitute: I = 3000 / 120 = 25 amps
Answer: I = 25 amps

6. Geometry & Measurement

Calculate perimeter, area, and volume for materials. Use the Pythagorean theorem for roof pitch, stair layout, and diagonal bracing.

Area of Rectangle: A = l × w | Area of Circle: A = πr² | Volume of Cylinder: V = πr²h | Pythagorean Theorem: a² + b² = c²
Key Concept: Perimeter is the distance around a shape. Area is the space inside. Volume is the space a 3D object takes up.

Example 1: Rectangular Area

Problem: A flooring contractor needs to cover a room 15 feet long by 12 feet wide. What area does he need?
Step 1: Use formula A = l × w
Step 2: A = 15 × 12 = 180 square feet
Answer: 180 sq ft

Example 2: Pythagorean Theorem for Roof Pitch

Problem: A rafter has a run (horizontal) of 8 feet and a rise (vertical) of 6 feet. What's the rafter length?
Step 1: Use a² + b² = c²
Step 2: 8² + 6² = c²
Step 3: 64 + 36 = 100, so c² = 100, c = 10 feet
Answer: 10 feet

Example 3: Cylinder Volume

Problem: A cylindrical tank has radius 2 feet and height 5 feet. What volume can it hold? (Use π ≈ 3.14)
Step 1: Use V = πr²h
Step 2: V = 3.14 × 2² × 5 = 3.14 × 4 × 5 = 62.8 cubic feet
Answer: 62.8 cubic feet

Try It

Q1: A circular opening has a radius of 3 inches. What's its area? (Use π ≈ 3.14)
Step 1: Use A = πr²
Step 2: A = 3.14 × 3² = 3.14 × 9 = 28.26 sq in
Answer: 28.26 square inches
Q2: A diagonal brace goes from corner to corner of a square frame that's 12 feet on each side. How long is the brace?
Step 1: Use a² + b² = c²
Step 2: 12² + 12² = c²
Step 3: 144 + 144 = 288, so c = √288 ≈ 16.97 feet
Answer: Approximately 17 feet

7. Trades-Specific Calculations

Real-world calculations: pipe offsets, bolt circles, material takeoffs, concrete volume, board feet, and wire specifications.

Key Concept: Trades calculations combine basic math with trade-specific knowledge. Always check units and round appropriately.
Concrete Volume: V = l × w × h ÷ 27 (to get cubic yards) | Board Feet: BF = (Length × Width × Thickness) ÷ 12 | Ampacity depends on wire gauge and insulation type

Example 1: Concrete Volume for a Pad

Problem: A concrete pad is 20 feet long, 15 feet wide, and 4 inches deep. How many cubic yards?
Step 1: Convert depth to feet: 4 inches = 4/12 = 0.33 feet
Step 2: Calculate cubic feet: 20 × 15 × 0.33 = 99 cubic feet
Step 3: Convert to cubic yards: 99 ÷ 27 ≈ 3.67 cubic yards
Answer: Approximately 3.7 cubic yards

Example 2: Board Feet Calculation

Problem: Calculate board feet for a board 12 feet long, 8 inches wide, and 1 inch thick.
Step 1: Convert dimensions: Length = 12 ft, Width = 8 in, Thickness = 1 in
Step 2: Use formula: BF = (12 × 8 × 1) ÷ 12 = 96 ÷ 12 = 8 board feet
Answer: 8 board feet

Example 3: Pipe Offset Calculation

Problem: Two pipe sections must offset 3 inches horizontally. Using 45° elbows, what's the center-to-center distance?
Step 1: For a 45° offset, center-to-center = offset ÷ sin(45°) = offset × 1.414
Step 2: 3 × 1.414 = 4.24 inches
Answer: Approximately 4.24 inches

Try It

Q1: How many cubic yards of gravel are needed for a driveway 30 feet long, 10 feet wide, and 3 inches deep?
Step 1: Convert depth to feet: 3 in = 3/12 = 0.25 feet
Step 2: Volume in cubic feet: 30 × 10 × 0.25 = 75 cubic feet
Step 3: Convert to cubic yards: 75 ÷ 27 ≈ 2.78 cubic yards
Answer: Approximately 2.8 cubic yards
Q2: Calculate board feet for 5 boards, each 10 feet long, 6 inches wide, and 2 inches thick.
Step 1: One board: (10 × 6 × 2) ÷ 12 = 120 ÷ 12 = 10 BF
Step 2: Five boards: 10 × 5 = 50 board feet
Answer: 50 board feet

8. Reading Measurements & Tolerances

Precision is critical. Learn to read rules, tape measures, micrometers, and vernier calipers to the required accuracy. Understand tolerance notation for quality work.

Key Concept: A tape measure typically shows 1/16" divisions. Micrometers read to 0.001" and vernier calipers to 0.05 mm. Always check the scale on your tool.
Ruler divisions: 1" = 2 halves = 4 quarters = 8 eighths = 16 sixteenths | Micrometer: Read main scale + thimble scale | Tolerance: ±0.005" means ±5 thousandths of an inch

Example 1: Reading a Tape Measure

Problem: A tape measure shows a mark between 5 and 6 inches, at the 9th small division. What's the measurement?
Step 1: Each small division = 1/16 inch
Step 2: 5 inches + (9 × 1/16) = 5 + 9/16 = 5 9/16 inches
Answer: 5 9/16 inches

Example 2: Micrometer Reading

Problem: A micrometer's main scale shows 0.275" and the thimble is on the 8th mark (×0.001"). Read the total.
Step 1: Main scale: 0.275 inches
Step 2: Thimble: 8 marks × 0.001 = 0.008 inches
Step 3: Total: 0.275 + 0.008 = 0.283 inches
Answer: 0.283 inches

Example 3: Tolerance Interpretation

Problem: A part specifies 2.500" ±0.005". Is a measured value of 2.501" acceptable?
Step 1: Low limit: 2.500 - 0.005 = 2.495"
Step 2: High limit: 2.500 + 0.005 = 2.505"
Step 3: 2.501" falls within 2.495" to 2.505", so YES, it's acceptable
Answer: Yes, the part is within tolerance

Try It

Q1: A tape measure shows a mark 3 inches plus 13 sixteenths. Write as a decimal.
Step 1: 13/16 = 0.8125 inches
Step 2: 3 + 0.8125 = 3.8125 inches
Answer: 3.8125 inches or 3 13/16 inches
Q2: A part has a tolerance of 1.250" ±0.010". Is a measured value of 1.238" acceptable?
Step 1: Low limit: 1.250 - 0.010 = 1.240"
Step 2: High limit: 1.250 + 0.010 = 1.260"
Step 3: 1.238" is below 1.240", so NO, it's out of tolerance
Answer: No, reject the part

Master These Skills for Success

These eight math foundations cover 80% of what appears on Canadian apprenticeship exams. Practice regularly using real-world trade problems. Always double-check your units, round appropriately, and use the formulas provided. Work through the examples step-by-step, then test yourself with the practice questions. Success on the math exam comes from understanding the concepts and building calculation speed through repetition.