OSAT Middle Level/Intermediate Mathematics (225) cheat sheet

Rational numbers are exactly the fractions a/b with integers a, b and b ≠ 0; their decimal expansions either terminate or eventually repeat.

A decimal terminates iff its reduced denominator has only 2s and 5s as prime factors; any other prime factor forces eventual repetition.

Order of operations: grouping symbols, exponents and roots, multiplication and division left to right, addition and subtraction left to right.

Integer exponent rules: a^m · a^n = a^(m+n), a^m / a^n = a^(m−n), (a^m)^n = a^(mn), a^(−n) = 1/a^n, a^0 = 1 for a ≠ 0.

Radical rules: √(ab) = √a·√b and √(a/b) = √a/√b for a, b ≥ 0; rationalize by multiplying by the conjugate, e.g. 1/(√3+1) · (√3−1)/(√3−1).

Percent change = (new − old)/old × 100%; successive percent changes multiply, so +20% then −20% yields 1.2 × 0.8 = 0.96, a net 4% decrease.

Scientific notation: a × 10^n with 1 ≤ |a| < 10; to multiply, multiply coefficients and add exponents; to divide, divide coefficients and subtract exponents.

Divisibility tests: by 3 or 9 when digit sum is divisible by 3 or 9; by 4 when the last two digits form a multiple of 4; by 11 when the alternating digit sum is a multiple of 11.

Absolute value |x| gives distance from 0 on the number line, so |a − b| is the distance between a and b; |x| = k has solutions x = k and x = −k when k > 0.

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

Line forms: slope-intercept y = mx + b, point-slope y − y₁ = m(x − x₁), standard Ax + By = C with slope −A/B.

Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a); the discriminant b² − 4ac gives two, one, or zero real roots as it is positive, zero, or negative.

Vertex of y = ax² + bx + c occurs at x = −b/(2a); the vertex form y = a(x − h)² + k has vertex (h, k) and axis x = h.

Factoring patterns: difference of squares a² − b² = (a − b)(a + b); perfect square trinomials a² ± 2ab + b² = (a ± b)².

Function notation: f(g(x)) means substitute g(x) into f; f⁻¹ undoes f, so f(f⁻¹(x)) = x, and the graph of f⁻¹ reflects f over y = x.

Exponential growth/decay y = a·b^x: growth when b > 1, decay when 0 < b < 1; compound interest A = P(1 + r/n)^(nt).

Systems of two linear equations: solve by substitution or elimination; no solution when slopes are equal but intercepts differ, infinitely many when the equations are equivalent.

Laws of logarithms: log(xy) = log x + log y, log(x/y) = log x − log y, log(x^k) = k log x; log_b b = 1 and log_b 1 = 0.

Angle relationships: vertical angles are equal; angles on a straight line sum to 180°; angles around a point sum to 360°.

Parallel lines cut by a transversal make alternate interior, alternate exterior, and corresponding angles congruent; same-side interior angles are supplementary.

Triangle angle sum is 180°; the exterior angle equals the sum of the two remote interior angles; the longest side faces the largest angle.

Pythagorean theorem a² + b² = c² for right triangles; common triples are 3-4-5, 5-12-13, 8-15-17, and 7-24-25.

Area formulas: triangle ½bh, parallelogram bh, trapezoid ½(b₁ + b₂)h, circle πr²; circumference C = 2πr = πd.

Volume: prism/cylinder V = Bh; pyramid/cone V = ⅓Bh; sphere V = (4/3)πr³ with surface area 4πr².

Similar figures have proportional sides and equal angles; the ratio of areas equals the square of the scale factor and the ratio of volumes equals its cube.

Distance and midpoint: d = √[(x₂ − x₁)² + (y₂ − y₁)²]; midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).

Conversions within the metric system shift the decimal by powers of 10; 1 in = 2.54 cm, 1 L ≈ 1.057 qt, and 1 kg ≈ 2.2 lb are common customary-metric bridges.

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

Range = max − min; interquartile range IQR = Q₃ − Q₁; outliers are commonly values below Q₁ − 1.5·IQR or above Q₃ + 1.5·IQR.

For a roughly symmetric distribution the mean and median are close; a long tail pulls the mean toward it while the median resists outliers.

Theoretical probability P(E) = favorable outcomes / total equally likely outcomes; experimental probability uses observed frequencies.

Complement rule P(not E) = 1 − P(E); addition rule P(A or B) = P(A) + P(B) − P(A and B).

Independent events: P(A and B) = P(A)·P(B); conditional probability P(A|B) = P(A and B)/P(B).

Expected value = Σ (value × probability); it need not be an attainable outcome and equals the long-run average per trial.

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

Correlation r ranges from −1 to 1 and measures linear association only; correlation does not establish causation, and a strong nonlinear pattern can still have r near 0.