Hands-on technical careers you can enter through apprenticeship: troubleshooting, safety awareness, the routes in, and the trade math you will use every day.

Foundations for Skilled Technical STEM Careers

## Why it matters

In the US trades, most measurements are still in **inches and fractions of an inch**. Electricians measure box heights and conduit runs; mechanics measure shaft lengths, shims and bolt sizes; everyone reads a tape measure many times a day. A mistake of 1/8 inch can mean a bracket that does not line up or a part that does not fit. Fraction fluency is also a big part of apprenticeship aptitude tests.

## Core idea: reading a tape measure

On a standard imperial tape, each inch is divided into 16 equal parts. The marks get shorter as the fractions get smaller:

- The **longest** mark in the middle of each inch is **1/2**.
- The next longest marks are **1/4** and **3/4**.
- Then **1/8, 3/8, 5/8, 7/8**.
- The shortest marks are the **sixteenths**: 1/16, 3/16, 5/16 and so on.

Count the small marks from the last whole inch. If the measurement is 9 inches and 11 small marks, it is **9 11/16 in**. If the count is even, simplify: 10/16 = 5/8, so 9 and 10 marks is **9 5/8 in**.

Most tapes also mark every 16 inches (a common stud spacing) and show feet: 7' 4 1/2" means 7 feet 4 and a half inches, or 7 x 12 + 4.5 = **88.5 inches**.

## Adding and subtracting fractions

To add fractions, they need a **common denominator** (the same bottom number). Tape fractions make this easy because 2, 4, 8 and 16 all divide into 16.

**Example 1: 3/8 + 5/16**

- Convert 3/8 to sixteenths: multiply top and bottom by 2, giving 6/16.
- Add: 6/16 + 5/16 = **11/16**.

**Example 2: two pieces, 5 3/8 in and 2 7/16 in, laid end to end**

- Add the whole inches: 5 + 2 = 7.
- Add the fractions: 3/8 = 6/16, and 6/16 + 7/16 = 13/16.
- Total: **7 13/16 in**.

**Example 3: 7/8 + 3/4**

- 7/8 stays 7/8; 3/4 = 6/8.
- 7/8 + 6/8 = 13/8, which is more than one whole inch.
- 13/8 = 1 5/8 (because 8/8 = 1, with 5/8 left over).

**The general rule** (works for any fractions): a/b + c/d = (a x d + c x b) / (b x d), then simplify by dividing top and bottom by their greatest common divisor. For 1/2 + 1/4: (1 x 4 + 1 x 2) / (2 x 4) = 6/8 = **3/4**.

**Subtraction** works the same way. A 96 in stick minus a 35 5/8 in cut: 96 = 95 8/8, and 95 8/8 - 35 5/8 = **60 3/8 in** (ignoring the width of the saw cut, the "kerf", which you should allow for in real work).

## Converting decimals and metric

Calculators and digital calipers give decimals. To turn a decimal inch into the nearest sixteenth, multiply by 16 and round:

- 0.30 in x 16 = 4.8, which rounds to 5, so **5/16 in**.
- 2.69 in: the fraction part 0.69 x 16 = 11.04, so **2 11/16 in**.

Common equivalents worth memorising: 1/8 = 0.125, 1/4 = 0.25, 3/8 = 0.375, 1/2 = 0.5, 5/8 = 0.625, 3/4 = 0.75, 7/8 = 0.875.

Metric: **1 inch = 25.4 mm** exactly. Machinery from many manufacturers uses metric bolts, so mechanics switch often: 3/4 in = 0.75 x 25.4 = 19.05 mm, which is why a 19 mm wrench nearly fits a 3/4 in bolt (but use the correct one: "nearly" strips heads).

## Worked example: centring a box

An electrician needs to centre a light box on a wall that is 11' 5 1/4" wide.

1. Convert to inches: 11 x 12 + 5 1/4 = 137 1/4 in.
2. Halve it: 137 1/4 / 2. Half of 137 is 68 1/2; half of 1/4 is 1/8. Total: 68 1/2 + 1/8 = 68 4/8 + 1/8 = **68 5/8 in**.
3. Mark 68 5/8 in from either end, and check from the other end.

## Common mistakes

- **Miscounting marks.** Count from the last whole inch, and double-check by reading the next larger mark (for example, 10/16 should sit on the 5/8 mark).
- **Adding denominators.** 1/2 + 1/4 is not 2/6. Convert to a common denominator first.
- **Forgetting to simplify.** 12/16 should be written 3/4.
- **Mixing feet and inches.** Convert everything to inches before adding.
- **Hooking errors.** The tape's end hook slides slightly to account for its own thickness on push and pull measurements. Do not "fix" it; keep the hook clean and undamaged.

## Try it

Open the **Do** tab. A small JavaScript fraction calculator is already written except for one line: the numerator when two fractions are added. Fill it in, then use the calculator to check the examples from this lesson.

## Quick recap

- A standard tape divides each inch into 16ths; simplify even counts.
- To add fractions, use a common denominator: a/b + c/d = (ad + cb) / bd, then simplify.
- Decimal to nearest sixteenth: multiply by 16 and round.
- 1 in = 25.4 mm exactly.


Read a tape measure and add, subtract and convert fractional measurements.

Fractions and measurements

Learn how to Read a Tape Measure - Fraction - Measuring and Marking Lesson Series -

Adding Fractions with Unlike Denominators | Math with Mr. J

## Why it matters

Trade work runs on a handful of formulas. Electricians use **Ohm's law** and the **power formula** to size circuits, choose breakers and troubleshoot. Mechanics use **area** to work out the force from a hydraulic cylinder or how much material covers a surface. You do not need to love algebra, but you do need to substitute numbers correctly, keep units straight, and rearrange a formula to find the value you are missing.

## Core idea 1: Ohm's law

Ohm's law links three quantities in a circuit:

- **Voltage (V)**, measured in volts: the electrical "push".
- **Current (I)**, measured in amperes (amps, A): how much charge flows per second.
- **Resistance (R)**, measured in ohms: how much the circuit opposes the flow.

The formula is **V = I x R**. Rearranged:

- **I = V / R** (current = voltage divided by resistance)
- **R = V / I** (resistance = voltage divided by current)

A memory aid is the triangle with V on top and I and R on the bottom: cover the one you want, and the other two show you whether to multiply or divide.

**Example:** a 120 V heater element has a resistance of 10 ohms. I = 120 / 10 = **12 A**.

**Example:** a 12 V battery is connected to a 4 ohm lamp. I = 12 / 4 = **3 A**.

## Core idea 2: the power formula

Power (P) is the rate at which energy is used, in **watts (W)**: **P = V x I**. Rearranged: **I = P / V**.

**Example:** a 1,440 W space heater on a 120 V circuit draws I = 1440 / 120 = **12 A**. On a 15 A circuit that is a big share of the capacity. For loads that run for hours (called continuous loads), the electrical code generally limits them to about 80% of the circuit rating, which for 15 A is 12 A. You will learn the exact code rules, and how your state adopts them, in your apprenticeship classes.

Combining the two formulas gives handy variations such as **P = I^2 x R** and **P = V^2 / R**. For the heater: P = 120^2 / 10 = 14,400 / 10 = 1,440 W, the same answer.

These numbers describe what an electrician calculates *on paper*. Measuring voltage or current on real circuits requires the right meter, training and supervision.

## Core idea 3: area formulas

- **Rectangle:** A = length x width. A room 12 ft x 14 ft has A = **168 sq ft**.
- **Triangle:** A = 1/2 x base x height.
- **Circle:** A = pi x r^2, or, using the diameter, A = pi x d^2 / 4. A circle 2 in across has A = 3.1416 x 1^2 = **3.14 sq in**.

**Units matter.** 1 sq ft = 12 x 12 = **144 sq in**, not 12. A 3 ft x 2 ft panel is 6 sq ft or 864 sq in.

## Worked example: hydraulic cylinder force (mechanic)

The force a hydraulic cylinder pushes with is **Force = Pressure x Area**.

- A cylinder has a 2 in bore (inside diameter). Area = pi x 2^2 / 4 = **3.14 sq in**.
- The system pressure is 1,000 psi (pounds per square inch).
- Force = 1,000 x 3.14 = about **3,142 lb**.

If the bore were 4 in, the area would be pi x 16 / 4 = 12.57 sq in, and the force about 12,566 lb: doubling the diameter **quadruples** the force. This is why a small hydraulic cylinder can lift so much, and why stored hydraulic pressure must be released and locked out before maintenance.

## Worked example: lighting a room (electrician)

A workshop is 20 ft x 30 ft. The lighting design calls for about 1 watt of LED lighting per square foot.

1. Area = 20 x 30 = 600 sq ft.
2. Lighting load = 600 x 1 = 600 W.
3. Current at 120 V = 600 / 120 = **5 A**.

The real design follows the drawings and code, but this quick check tells you the plan is reasonable.

## A method for any formula

1. **Write the formula** before touching numbers.
2. **List what you know**, with units.
3. **Rearrange** to isolate the unknown.
4. **Substitute** and calculate.
5. **Sanity-check**: is the answer in the right units and a believable size? A household appliance drawing 1,200 A is a mistake.

## Common mistakes

- **Multiplying when you should divide.** Use the triangle or rearrange on paper.
- **Mixing units:** kilowatts with watts, inches with feet, diameter with radius.
- **Using the diameter in pi x r^2.** Halve it first, or use pi x d^2 / 4.
- **Rounding too early.** Keep full precision until the final answer.
- **Treating a calculation as permission.** Knowing the current a circuit should draw does not make it safe to test it live.

## Try it

Open the **Do** tab. The trade calculator has the power and area formulas written for you. Your job is one line: Ohm's law for current.

## Quick recap

- Ohm's law: V = I x R, so I = V / R and R = V / I.
- Power: P = V x I, so I = P / V.
- Area: rectangle L x W; circle pi r^2 or pi d^2 / 4; 1 sq ft = 144 sq in.
- Force from a cylinder = pressure x area.


Apply Ohm's law, the power formula and area formulas to everyday trade calculations.

Formulas on the job

Ohms Law Explained - The basics circuit theory

Electrical Formulas - Basic Electricity For Beginners

## Why it matters

Conduit, pipe and tubing rarely run in perfectly straight lines. When a run of electrical conduit needs to step around a beam, a pipe or a box, the electrician bends an **offset**: two equal bends that shift the conduit sideways and then continue in the original direction. Getting the marks right first time saves material and time, and it is one of the most visible signs of skill on a job site. The math behind it is basic right-triangle trigonometry, and the same ideas help mechanics with angles, slopes and shims.

## Core idea: the offset is a right triangle

Picture the conduit stepping over an obstacle. The two bends at angle **theta** form a right triangle:

- The **offset depth** (how far the conduit must move sideways) is the side **opposite** the angle.
- The **distance between bends** (along the conduit, from one bend mark to the other) is the **hypotenuse**.

From trigonometry, **sin(theta) = opposite / hypotenuse**. Rearranged:

> **distance between bends = offset depth / sin(theta)**

The number 1 / sin(theta) is called the **multiplier** (or cosecant). Trade tables list it so you can just multiply.

| Bend angle | Multiplier (1 / sin) | Shrink per inch of offset (trade table) |
|---|---|---|
| 10 degrees | 5.76 (often rounded to 6) | 1/16 in |
| 22.5 degrees | 2.61 | 3/16 in |
| 30 degrees | 2.00 | 1/4 in |
| 45 degrees | 1.41 | 3/8 in |
| 60 degrees | 1.15 | 1/2 in |

## Worked example 1: a 6 inch offset with 30 degree bends

- Offset depth = 6 in.
- sin(30 degrees) = 0.5, so the multiplier is 2.
- Distance between bends = 6 / 0.5 = **12 in**.

So the two bend marks go 12 inches apart.

## Worked example 2: a 10 inch offset with 45 degree bends

- sin(45 degrees) = 0.7071.
- Distance between bends = 10 / 0.7071 = 14.14 in.
- To the nearest sixteenth: 0.14 x 16 = 2.3, which rounds to 2, so 2/16 = **14 1/8 in**.

## Shrink

Because the conduit now travels diagonally for part of its length, an offset "uses up" a little extra conduit compared with a straight run. This lost length is called **shrink**. The exact geometry gives:

> **shrink = offset depth x tan(theta / 2)**

For the 6 in offset at 30 degrees: 6 x tan(15 degrees) = 6 x 0.268 = 1.61 in. The trade-table shortcut of 1/4 in per inch of offset gives 6 x 1/4 = **1 1/2 in**, close enough for most work with a hand bender. Your instructor will show how your bender and your crew apply shrink when marking a stick, because the marking conventions depend on the bender and the direction you bend.

## Choosing the angle

- **Smaller angles** (10 or 22.5 degrees) make gentler bends that are easier to pull wire through, but they need more distance between bends.
- **Larger angles** (45 or 60 degrees) fit tight spaces but add more resistance to the wire pull.

The code also limits the **total degrees of bends** between pull points (in the US, 360 degrees between pull points for most conduit types). An offset with two 30 degree bends uses 60 of those degrees.

## Converting angles for JavaScript

Calculators and programming languages compute sine with angles in **radians**, not degrees. To convert: **radians = degrees x pi / 180**. In JavaScript:

```javascript
const radians = 30 * Math.PI / 180;      // 0.5236
console.log(Math.sin(radians));          // 0.5 (very close to it)
```

Forgetting this conversion is the most common bug in trade calculators: `Math.sin(30)` treats 30 as radians and returns -0.988.

## Worked example 3: checking a field measurement

An apprentice wants a 4 in offset with 22.5 degree bends and wonders whether 10 1/2 in between bends is right.

- sin(22.5 degrees) = 0.3827. Distance = 4 / 0.3827 = 10.45 in.
- To the nearest sixteenth: 0.45 x 16 = 7.2, rounds to 7, so **10 7/16 in**.

10 1/2 in (10 8/16) is 1/16 in long: a small but real difference on a tight offset.

## Common mistakes

- **Using cosine or tangent instead of sine** for the distance between bends. Remember: depth is opposite, distance is the hypotenuse.
- **Degrees versus radians** in calculators and code.
- **Unequal bends.** Both bends of an offset must be the same angle, or the conduit will not come back parallel.
- **Dog legs.** If the two bends are not in the same plane, the conduit twists. Sight down the conduit after bending.
- **Practising on the wrong things.** Bending practice belongs in a shop or apprenticeship lab with proper benders and supervision. At home, practise the math on paper.

## Try it

Open the **Do** tab. The offset calculator already converts degrees to radians, computes shrink, and rounds to the nearest sixteenth. You write the one line that computes the distance between bends.

## Quick recap

- An offset is a right triangle: depth is opposite, distance between bends is the hypotenuse.
- Distance between bends = depth / sin(angle); the multiplier is 1 / sin(angle).
- Shrink = depth x tan(angle / 2); trade tables round it per inch of offset.
- Convert degrees to radians before using `Math.sin`.


Compute the distance between bends and the shrink of a conduit offset with right-triangle trigonometry.

Offsets, angles, and bends

Pipe Trig

How To Make 30° Offset Bends Using Multipliers and Shrink

Fractions and tape measures, Ohm's law and area formulas, and conduit offsets with right-triangle trigonometry.