4.1 Craft-Related Mathematics: Fractions, Units, Area, and Volume

Why this domain is worth 15 items

Domain 3, Math and Measurements, contributes 15 of 115 items — tied for the second-largest block. It draws on four modules: 32106 Craft-Related Mathematics, 32201 Basic Layout, 32301 Advanced Trade Math, and 32302 Precision Measuring Tools. The assessment specification explicitly lists "area, volume, sine, cosine, hypotenuse, and Pythagorean concepts" in the focus statement, so these are not incidental items.

Remember the constraint from Chapter 1: you may use a basic function, non-printing calculator and nothing else. No formula sheet, no conversion table, no trig table.

Shop fractions and decimals

Steel rules, drill sets, wrench sizes, and shim stock are all fractional. Convert by division: the fraction bar is a division sign.

Two habits prevent most arithmetic errors:

1. Convert to decimals immediately and stay in decimals until the final answer, then convert back if the question asks for a fraction.
2. Add a common denominator only when you must stay fractional — for example, when totaling shim thicknesses that are supplied in fractional sizes.

Unit conversions worth memorizing

The psi-to-feet-of-head pair earns its place because it reappears in the pump chapter: a centrifugal pump develops head, and converting a discharge gauge reading into feet of head is routine.

Area

$A_{\text{rectangle}} = L \times W \qquad A_{\text{triangle}} = \frac{1}{2} b h \qquad A_{\text{circle}} = \pi r^2$

The circle formula uses the radius, not the diameter. Squaring the diameter by mistake gives an answer four times too large — and a four-times-too-large distractor is almost always among the options.

Circumference:

$C = \pi d = 2 \pi r$

Worked example. A round inspection cover is 18 inches in diameter. Its area is pi times 9 squared, or 254.5 square inches, and its circumference is pi times 18, or 56.5 inches of gasket.

Volume and capacity

$V_{\text{rectangular}} = L \times W \times H \qquad V_{\text{cylinder}} = \pi r^2 h$

Worked example — reservoir capacity. A cylindrical hydraulic reservoir is 30 inches in diameter and 48 inches tall. Working in inches:

$V = \pi \times 15^2 \times 48 = 33{,}929 \text{ in}^3$

Convert to cubic feet by dividing by 1,728, since 12 cubed is 1,728: 19.6 ft³. Convert to gallons by multiplying by 7.48: about 147 gallons.

Worked example — plate weight. A steel plate 4 ft by 8 ft by 1/2 in thick. Volume is 4 times 8 times 0.0417, or 1.333 ft³. At 490 lb/ft³ that is 653 pounds — a two-person handling problem, not a one-person lift.

Working an area or volume problem under exam conditions

With a basic calculator and no reference sheet, a repeatable routine beats cleverness:

1. Sketch the shape and write the given dimensions on the sketch. Half of all errors are transcription errors.
2. Convert every dimension to one unit before doing any arithmetic. Mixing feet and inches inside a single multiplication is the most common failure.
3. Identify the shape and write the formula before entering anything into the calculator.
4. Compute, then convert the result to the unit the question asked for.
5. Sanity check the magnitude. If a 55-gallon-drum-sized vessel comes out at 4,000 gallons, a decimal or a radius-versus-diameter error is in the work.

Worked example. A rectangular sump measures 6 ft long by 4 ft wide, and the operator reports 18 inches of water in it. How many gallons must be pumped out, and what does that water weigh?

• Convert: 18 inches is 1.5 ft.
• Volume: 6 x 4 x 1.5 = 36 ft^3.
• Gallons: 36 x 7.48 = 269 gallons.
• Weight: 36 ft^3 x 62.4 lb/ft^3 = 2,246 lb, or equivalently 269 gal x 8.34 lb/gal = 2,243 lb. The small difference is rounding.

That last cross-check — computing the same answer two ways from two different conversion factors — is worth the fifteen seconds it takes, because it catches a wrong conversion factor immediately.

Percentages and ratios in maintenance work

• Percent efficiency: output divided by input, times 100.
• Percent of capacity: load divided by rating, times 100. A crane lifting 6,800 lb on a chart capacity of 8,000 lb is at 85% of capacity — the threshold at which most plants require a critical lift plan.
• Ratio: expressed as driver to driven. A 1,750 rpm motor driving a shaft at 350 rpm is a 5:1 reduction.

Ratios return in the mechanical drive and gearbox chapters, so learn to set them up as a fraction and reduce.