A ratio expressed per hundred. Different jobs use different bases: percent of a whole, what percent one number is of another, and percent change from old to new.
FORMULA
Percent of = (part ÷ whole) × 100EXAMPLE
45 is what percent of 180? (45 ÷ 180) × 100 = 25%. A price from 40 to 45 is +5 points but +12.5% relative change.
WHY IT MATTERS
Discounts, tax, tips, and growth rates all reduce to percent problems. Pick the right job and the right base.
COMMON MISTAKES
- Adding percents that do not share the same base
- Confusing percent increase with percent of
- Applying +10% twice and expecting +20% (compound is +21%)
A part of a whole written as numerator over denominator. Addition needs a common denominator; multiplication multiplies across; division multiplies by the reciprocal.
EXAMPLE
1/2 + 1/4 = 2/4 + 1/4 = 3/4. 1/2 × 1/4 = 1/8.
WHY IT MATTERS
Recipes, probability, slopes as rise/run, and algebra all lean on fraction fluency.
COMMON MISTAKES
- Adding numerators and denominators directly (1/2 + 1/4 is not 2/6)
- Canceling across addition as if it were multiplication
Compares two quantities as a : b or a/b. Equivalent ratios scale both parts by the same factor. Part:part and part:whole are different framings.
EXAMPLE
Mixing 2:3 concentrate to water means 2 parts concentrate for every 3 parts water. Scaling ×10 gives 20:30.
WHY IT MATTERS
Maps, scale models, dilution, and share splits are ratio problems in disguise.
COMMON MISTAKES
- Treating a part:part ratio as if the second number were the whole
- Scaling only one side of the ratio
States that two ratios are equal: a/b = c/d. When the relationship is linear, cross-multiply to solve for an unknown.
FORMULA
If a/b = c/d then a·d = b·cEXAMPLE
If 3 notebooks cost $12, then 5 cost x: 3/12 = 5/x → x = $20.
WHY IT MATTERS
Direct variation, unit conversions, and many word problems reduce to a proportion.
COMMON MISTAKES
- Setting up the ratio with mismatched units on the same side
- Using a direct proportion when the situation is inverse
A sentence-style proportion shortcut: if a corresponds to b, then c corresponds to x. Direct rule of three scales the same way; inverse rule of three flips when more of one means less of the other.
EXAMPLE
Direct: 4 workers finish in 6 days → 6 workers need fewer days (inverse), while double the paint for double the wall is direct.
WHY IT MATTERS
Quick workplace and homework framing for “if this, then that” problems. Still check whether the relationship is direct or inverse.
COMMON MISTAKES
- Applying direct rule of three to work-rate or shared-load problems that are inverse
ARITHMETIC
Scientific notation
Writes a number as a × 10^n with 1 ≤ |a| < 10 (normalized form). Moves the decimal by powers of ten so large and small magnitudes stay readable.
EXAMPLE
300,000,000 → 3 × 10^8. 0.00045 → 4.5 × 10^-4.
WHY IT MATTERS
Physics, chemistry, and calculator displays use scientific notation to avoid counting zeros.
COMMON MISTAKES
- Leaving a coefficient outside the 1-to-10 range and calling it normalized
- Mixing engineering notation (powers of 10³) with scientific without labeling
ALGEBRA
Quadratic equation
A polynomial equation ax² + bx + c = 0 with nonzero a. Solutions may be two real roots, one repeated root, or a complex pair, depending on the discriminant.
FORMULA
x = (-b ± √(b² - 4ac)) / (2a)EXAMPLE
x² - 5x + 6 = 0 → (x - 2)(x - 3) = 0 → x = 2 or x = 3.
WHY IT MATTERS
Projectile sketches, area optimization, and parabola intersections reduce to quadratics.
COMMON MISTAKES
- Skipping the discriminant b² - 4ac before claiming two real roots
- Sign errors on -b in the quadratic formula
Rise over run between two points: m = (y2 - y1) / (x2 - x1). Horizontal lines have slope 0. Vertical lines have undefined slope.
FORMULA
m = (y2 - y1) / (x2 - x1)EXAMPLE
Points (2, 3) and (6, 11) → m = (11 - 3) / (6 - 2) = 2. Line form y = 2x - 1 through (2, 3).
WHY IT MATTERS
Slope is rate of change: speed, marginal cost, and trend lines.
COMMON MISTAKES
- Subtracting coordinates in inconsistent order
- Calling a vertical line slope zero
ALGEBRA
Scientific calculator modes
A scientific calculator evaluates expressions with trig, logs, exponents, and parentheses. Degree vs radian mode changes trig results for the same numeric input.
EXAMPLE
sin(30) in degree mode is 0.5. The same keystrokes in radian mode are sin(30 rad), a different value.
WHY IT MATTERS
Homework and engineering work fail quietly when the angle mode is wrong.
COMMON MISTAKES
- Leaving the calculator in radians during a degree-based homework set
- Ignoring order of operations and parentheses
GEOMETRY
Triangle properties
Interior angles sum to 180°. Right triangles obey Pythagoras: a² + b² = c² on the legs and hypotenuse. Area is often ½ × base × height. Ambiguous SSA cases can yield zero, one, or two triangles.
EXAMPLE
Legs 3 and 4 → hypotenuse 5; area = 6 square units.
WHY IT MATTERS
Surveying, graphics, and trigonometry start with triangle relationships.
COMMON MISTAKES
- Assuming SSA always determines a unique triangle
- Mixing degrees and radians in trig solutions
GEOMETRY
Pythagorean theorem
In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs: a² + b² = c².
FORMULA
a² + b² = c² (right triangle; c hypotenuse)EXAMPLE
a = 6, b = 8 → c = 10. Checking: 36 + 64 = 100.
WHY IT MATTERS
Distance in the plane, ladder problems, and many geometry checks reduce to Pythagoras.
COMMON MISTAKES
- Applying a² + b² = c² to a non-right triangle
- Using the longest side as a leg instead of the hypotenuse
STATISTICS
Mean, median, mode
Mean is the arithmetic average. Median is the middle value after sorting. Mode is the most frequent value. Each resists outliers differently.
EXAMPLE
Data 2, 3, 3, 100: mean is pulled up by 100; median stays near 3; mode is 3.
WHY IT MATTERS
Income and home-price summaries often report median because means are skewed by extremes.
COMMON MISTAKES
- Reporting only the mean on a skewed list
- Forgetting to sort before finding the median
STATISTICS
Standard deviation
Measures typical spread around the mean. Low σ means points cluster; high σ means more variability. Population and sample formulas differ in the denominator.
FORMULA
Population σ = √(Σ(x - μ)² / N); sample s uses (N - 1)EXAMPLE
Scores 80, 82, 78, 81, 79 have mean 80 and a modest sample standard deviation near 1.4.
WHY IT MATTERS
Compare consistency of grades, lab repeats, or returns, not just averages. Empirical-rule bands use σ as a yardstick.
COMMON MISTAKES
- Using population σ when you only have a sample
- Comparing standard deviations across lists with very different units without normalizing
PROBABILITY
Permutation & combination
Permutations count ordered arrangements (nPr). Combinations count unordered selections (nCr). Choose based on whether order matters.
FORMULA
nPr = n! / (n - r)! · nCr = n! / (r!(n - r)!)EXAMPLE
Choose 3 winners from 10 when order does not matter → 10C3 = 120. If order is 1st/2nd/3rd, use 10P3 = 720.
WHY IT MATTERS
Lottery odds, scheduling, and password-style counting use these rules.
COMMON MISTAKES
- Using combination when order matters (podiums, PINs)
- Using permutation when the problem only cares about the set
PROBABILITY
Random number (inclusive range)
A draw from a defined range. Inclusive integer ranges include both endpoints. Sampling with replacement can repeat; unique sampling cannot.
EXAMPLE
Integers from 1 to 6 inclusive are six tickets. Half-open [0, 1) continuous samples are a different model.
WHY IT MATTERS
Simulations, raffles, and classroom demos fail when the range or replacement rule is unclear.
COMMON MISTAKES
- Off-by-one errors on inclusive vs exclusive bounds
- Using RNG output as cryptographic secrets
A rectangular array of numbers with dimensions rows × columns. Addition needs matching shape. Multiplication needs inner dimensions to match and is not generally commutative.
EXAMPLE
[[1,2],[3,4]] × [[5,6],[7,8]] is defined. The reverse product is a different matrix when both exist.
WHY IT MATTERS
Graphics transforms, linear systems, and many data models rely on matrix operations.
COMMON MISTAKES
- Adding matrices of different shapes
- Assuming AB equals BA