EN VIII Mathematics cheat sheet

Order of operations: evaluate parentheses, then exponents, then multiplication and division left to right, then addition and subtraction left to right.

A number is divisible by 3 (or 9) exactly when the sum of its digits is divisible by 3 (or 9); by 4 when its last two digits form a multiple of 4.

For fractions: a/b + c/d = (ad + bc)/(bd); a/b ÷ c/d = a/b × d/c (multiply by the reciprocal).

Percent change = (new − old)/old × 100%; a percent increase of p% multiplies a quantity by (1 + p/100).

For any nonzero base: a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m−n), (a^m)^n = a^(mn), and a^(−n) = 1/a^n.

√(ab) = √a × √b and √(a/b) = √a/√b for nonnegative a, b; √(a²) = |a|.

Absolute value |x| equals x when x ≥ 0 and −x when x < 0; it represents distance from zero on the number line.

Scientific notation writes a number as a × 10^n with 1 ≤ |a| < 10; multiplying by 10^n shifts the decimal point n places.

Slope between points (x₁,y₁) and (x₂,y₂) is m = (y₂ − y₁)/(x₂ − x₁); parallel lines share m, perpendicular lines have slopes whose product is −1.

Slope-intercept form y = mx + b has slope m and y-intercept b; point-slope form is y − y₁ = m(x − x₁).

Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a) solves ax² + bx + c = 0; the discriminant b² − 4ac gives 2, 1, or 0 real roots when positive, zero, or negative.

Vertex of y = ax² + bx + c occurs at x = −b/(2a); the parabola opens upward when a > 0 and downward when a < 0.

A function assigns exactly one output to each input; f(g(x)) means substitute g(x) into f, and the domain excludes inputs making a denominator zero.

Exponential growth/decay: y = a·b^x grows when b > 1 and decays when 0 < b < 1; a is the initial value at x = 0.

Solving a system of two linear equations: substitution isolates one variable, while elimination adds or subtracts multiples of the equations to cancel a variable.

Laws of exponents and logs: log(xy) = log x + log y, log(x/y) = log x − log y, log(x^n) = n log x.

Area formulas: rectangle A = lw, triangle A = ½bh, trapezoid A = ½(b₁ + b₂)h, circle A = πr² with circumference C = 2πr.

Pythagorean theorem: in a right triangle a² + b² = c², where c is the hypotenuse; the converse tests whether a triangle is right.

Sum of interior angles of an n-gon is (n − 2) × 180°; each interior angle of a regular n-gon measures (n − 2)180°/n.

Volume: rectangular prism V = lwh, cylinder V = πr²h, cone V = ⅓πr²h, sphere V = (4/3)πr³.

Similar figures have equal corresponding angles and proportional sides; the ratio of areas equals the square of the side ratio, and volumes the cube.

Distance between points is √[(x₂ − x₁)² + (y₂ − y₁)²]; midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).

Central angle measure equals its intercepted arc; an inscribed angle equals half its intercepted arc.

Complementary angles sum to 90° and supplementary angles sum to 180°; vertical angles are equal.

Mean = sum of values ÷ number of values; median is the middle value of ordered data (average the two middle values if n is even); mode is the most frequent value.

Range = maximum − minimum; interquartile range IQR = Q₃ − Q₁, and outliers are commonly values below Q₁ − 1.5·IQR or above Q₃ + 1.5·IQR.

Standard deviation measures typical distance from the mean; larger spread gives larger standard deviation, and adding a constant to every value leaves it unchanged.

Probability of an event = favorable outcomes ÷ total equally likely outcomes; P(not A) = 1 − P(A).

For independent events P(A and B) = P(A) × P(B); for mutually exclusive events P(A or B) = P(A) + P(B).

Conditional probability P(A|B) = P(A and B)/P(B); events are independent when P(A|B) = P(A).

A normal distribution is symmetric about its mean; roughly 68% of data lie within 1 standard deviation and 95% within 2 standard deviations.

Correlation describes the strength and direction of a linear association (−1 ≤ r ≤ 1); correlation does not establish causation.

A random sample is representative when every member of the population has an equal chance of selection; larger samples generally reduce sampling variability.

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