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Trades Math Review Flashcards: Formulas, Geometry, Percent, Ratios, Notation, and Error Analysis

  • Formulas (Unit 2, Module 7)

    • Key idea: formulas use variables; to evaluate, you know all variables except one (the unknown you solve for).
    • Notation for operations can vary: multiplication as 2A, 2\cdot A, 2(A); division as \frac{3}{B}, 3 \div B, 3/B.
    • Example a (nails): total cost C = 2.42\,w + 15 where w = pounds; calculate for w=7: C = 2.42(7) + 15 = 33.94 + 15 = 48.94\$.
    • Example b (Uber fare): Fare = Base + (cost per minute) × (time) + (cost per mile) × (distance) + Booking Fee
    • Given: Base = 6.00, per minute = 0.15, time = 27\text{ min}, per mile = 1.57, distance = 19.8\text{ mi}, Booking Fee = 0.55
    • Formula: \text{Fare} = 6 + 0.15\times 27 + 1.57\times 19.8 + 0.55
    • Calculation: 6 + 4.05 + 31.086 + 0.55 = 41.686 \$ \approx 41.69\$.
    • Example c (Germany forecast): high today 29^\circ\text{C}; likely wear shorts (warm weather). No jacket needed.

  • Perimeter & Circumference (Module 8)

    • Polygon basics: a polygon is a closed plane figure with 3 or more straight sides. Perimeter = sum of side lengths.

    • Common quadrilaterals:

    • Parallelogram: opposite sides are parallel and equal in length.

    • Rectangle: parallelogram with right angles (all corners 90°).

    • Square: rectangle with all sides equal.

    • Trapezoid: two parallel sides; two non-parallel sides.

    • Quadrilateral: four sides, not necessarily with parallel sides.

    • Area and perimeter (selected formulas):

    • Rectangles: A = L\cdot W, \quad P = 2L + 2W

    • Square: A = s^2, \quad P = 4s, \quad s = \sqrt{A}

    • Parallelograms: A = b h, \quad P = 2a + 2b

    • Trapezoid: A = \frac{(b1 + b2)}{2}h (equivalently A = \frac{h}{2}(b1+b2))

    • Circles:

    • Radius and diameter: d = 2r, \quad r = \tfrac{d}{2}

    • Circumference: C = \pi d = 2\pi r

    • Example: if diameter = 28, then C = \pi \cdot 28 = 28\pi.

  • Circle basics and parts (Page 7–8)

    • Circumference ratio: for all circles, circumference to diameter is constant, equals (\pi).
    • Basic conversions and arcs: primary focus is using C = \pi d = 2\pi r; d=2r.

  • Percent Part 1 (Module 9)

    • Percent = part of a whole per 100; base is the whole quantity.
    • Conversions:
    • Decimal to percent: multiply by 100%.
    • Fraction to percent: convert to decimal, then multiply by 100%.
    • Percent to decimal: divide by 100%.
    • Basic relation: Amount = Rate × Base; A = R\cdot B (where R is decimal form of percent).
    • Examples of conversions:
    • Decimals to percent: 0.5\to 50\% , 0.42\to 42\% , 0.123\to 12.3\% , 0.07\to 7\%
    • Percent to decimal: 23\%\to 0.23, 17\%\to 0.17, 50\%\to 0.50, 62\%\to 0.62
    • Percent to fraction: convert to fraction with denominator 100 (then reduce).

  • Ratios, Rates, Proportions (Module 10)

    • Ratio: comparison of two quantities with same units (as a fraction). Examples: find ratio of differing units (e.g., time, distance).
    • Rate: ratio with different units; often expressed as unit rate (denominator = 1). Examples: /$/unit (e.g., $/lb, $/gal).
    • Unit price: price per single unit of measure. Example: 20 ft pipe costs 12.82 → unit price = 12.82/20 = 0.641\$/ft.
    • Proportions: two ratios are equal. Cross-multiplication: if \frac{a}{b} = \frac{c}{d} then ad = bc.
    • Solving for an unknown: use cross-multiplication to find missing value in a proportion (e.g., problems like \frac{n}{8} = \frac{18}{72}, etc.).
    • Example unit costs: carpet 12 ft × 18 ft for 686.88 → unit price per ft² = (686.88 / (12\cdot 18)). (Use as needed for quick checks.)

  • Scientific Notation (Module 11)

    • Negative exponents: x^{-n} = 1/x^{n}.
    • Scientific notation form: P \times 10^{k} where 1 \le P < 10 and k\in\mathbb{Z}.
    • Multiplication: ((A\times 10^{B})(C\times 10^{D}) = (AC)\times 10^{B+D}).
    • Division: ((A\times 10^{B})/(C\times 10^{D}) = (A/C)\times 10^{B-D}).
    • Engineering notation: exponent is a multiple of 3; uses metric prefixes (kilo, mega, giga, etc.).
    • Conversions: write numbers like 1000, 48000, 0.00063, 0.0021 in scientific notation; convert back to decimal as needed.

  • Percent Part 2 & Error Analysis (Module 12)

    • Base equation for percent problems: A = R\cdot B; real-world phrasing translates via: "is" = equals, "of" = multiply, "what" = unknown variable.

    • Solving percent problems via proportions: three quantities involved – Amount, Base, Percent.

    • Examples of proportional setups: e.g., determining what percent is part of a whole, or what amount corresponds to a given percent.

    • Percent Increase / Decrease: changes expressed as percent of the original value.

    • Percent Increase: compute difference, divide by original, multiply by 100\%.

    • Percent Decrease: compute difference, divide by original, multiply by 100\%.

    • Error terminology:

    • Absolute Error = |measured − expected|

    • Relative Error = (Absolute Error) / (Expected)

    • Tolerance: acceptable range around the target value; e.g., a measurement should be within ± tolerance of the target.

    • Examples of error/applications:

    • Relative error examples: shaft length measured 28.89 in vs expected 28.50 in; weight measurement example; tolerance values for diameters and weights.

  • Quick reference formulas (summary)

    • Perimeter of a polygon: P = \sum{i} si
    • Rectangle area/perimeter: A = L W, \quad P = 2L + 2W
    • Square: A = s^2, \quad P = 4s
    • Parallelogram: A = bh, \quad P = 2a + 2b
    • Trapezoid: A = \frac{(b1 + b2)}{2}h
    • Circle: d = 2r, \quad C = \pi d = 2\pi r
    • Percent: A = R\cdot B or A = \frac{p}{100} \cdot B
    • Scientific notation: P \times 10^{k} with rules for multiplication/division as above
    • Error measures: \text{Absolute Error} = |\text{measured} - \text{expected}|, \quad \text{Relative Error} = \frac{\text{Absolute Error}}{\text{Expected}}

Note: The above captures the essential formulas, definitions, and typical problem-setup patterns from the transcript for quick last-minute review.