SAAT cheat sheet

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

Divisibility: by 3 if digit sum is divisible by 3; by 4 if the last two digits form a multiple of 4; by 9 if digit sum is divisible by 9; by 6 if divisible by both 2 and 3.

Percent change = (new − old) / old × 100; a 20% increase followed by a 20% decrease yields a net 4% decrease.

Repeated percent changes multiply: 1.15 × 0.90 = 1.035, a 3.5% net increase.

Absolute value |x| is the distance from 0; |a − b| is the distance between a and b on the number line.

Scientific notation a × 10^n requires 1 ≤ |a| < 10; multiply coefficients and add exponents when multiplying powers of ten.

Rules of exponents: x^a · x^b = x^(a+b); x^a / x^b = x^(a−b); (x^a)^b = x^(ab); x^0 = 1 for x ≠ 0; x^(−n) = 1/x^n.

For fractions a/b and c/d, cross-multiply to compare: a/b > c/d exactly when ad > bc (with b, d > 0).

LCM × GCF of two positive integers equals their product; use prime factorization to find either.

Slope m = (y₂ − y₁)/(x₂ − x₁); point-slope form y − y₁ = m(x − x₁); slope-intercept y = mx + b.

Parallel lines have equal slopes; perpendicular lines have slopes whose product is −1 (negative reciprocals).

Quadratic formula: x = [−b ± √(b² − 4ac)] / (2a); 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 axis of symmetry is the vertical line x = −b/(2a).

Sum and product of quadratic roots: r₁ + r₂ = −b/a and r₁r₂ = c/a.

For f(x) = a(x − h)² + k, the graph shifts h units horizontally and k units vertically from y = ax².

Exponential growth/decay: y = a·b^x, where b > 1 is growth and 0 < b < 1 is decay; percent rate = (b − 1) × 100%.

Solve systems by substitution or elimination; a linear system has no solution when slopes are equal and intercepts differ.

Direct variation y = kx; inverse variation y = k/x; joint variation y = kxz.

Circle: circumference C = 2πr, area A = πr²; arc length = (θ/360)·2πr and sector area = (θ/360)·πr² for central angle θ in degrees.

Triangle area = ½bh; trapezoid area = ½(b₁ + b₂)h; parallelogram area = bh; circle sector formulas use fraction of full circle.

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

Special right triangles: 45-45-90 has side ratio 1:1:√2; 30-60-90 has side ratio 1:√3:2.

Volume: rectangular prism lwh; cylinder πr²h; cone ⅓πr²h; sphere (4/3)πr³; pyramid ⅓Bh.

Surface area of a sphere = 4πr²; lateral area of a cylinder = 2πrh; total cylinder area = 2πr² + 2πrh.

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

Similar figures: corresponding sides scale by k, areas by k², and volumes by k³.

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

Mean = sum / count; median is the middle value of ordered data; mode is the most frequent value; range = max − min.

Standard deviation measures spread about the mean; adding a constant to every value leaves standard deviation unchanged, while multiplying by k multiplies it by |k|.

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).

General addition rule: P(A or B) = P(A) + P(B) − P(A and B).

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

Combinations count unordered selections: nCr = n!/[r!(n − r)!]; permutations count ordered arrangements: nPr = n!/(n − r)!.

Outliers pull the mean toward them but leave the median largely unchanged; use median for skewed data.

In a normal distribution, about 68% of values lie within one standard deviation of the mean and about 95% within two.

Main idea = the central claim the whole passage supports; it must cover the entire passage, not just one paragraph.

Primary purpose questions ask why the author wrote the passage (to persuade, explain, compare, or criticize), not what it says.

Inference answers must be logically forced by the text; an answer that is merely plausible but unsupported is wrong.

Tone and attitude questions are answered by connotation words and qualifiers (e.g., "arguably," "unfortunately"), not by topic alone.

"According to the passage" questions require explicit textual support; avoid answers that rely on outside knowledge.

Except/least/not questions invert the task: three choices are supported by the text and one is not.

When comparing two passages, identify where they agree, where they differ, and whether one responds directly to the other.

Vocabulary-in-context: substitute each answer choice into the sentence and check that the passage's logic still holds.

Scope traps include answers that are too broad, too narrow, or that reverse the author's stated cause and effect.

A complete sentence needs a subject and a finite verb and cannot join two independent clauses with only a comma (comma splice).

Fix run-ons with a period, a semicolon, or a comma plus a coordinating conjunction (for, and, nor, but, or, yet, so).

Nonessential information is set off by a matched pair of commas, dashes, or parentheses; essential information takes no commas.

Subject-verb agreement: the verb agrees with the grammatical subject, ignoring prepositional phrases between them; indefinite pronouns like everyone and each are singular.

Pronouns must have a clear, singular or plural antecedent that matches in number; "they" cannot refer ambiguously to two different nouns.

Modifiers must sit next to what they modify; an introductory participial phrase modifies the subject of the main clause.

Parallel structure: items in a list or comparison must share the same grammatical form (all nouns, all gerunds, or all infinitives).

Punctuation: a semicolon joins two independent clauses; a colon follows a complete clause and introduces a list or explanation.

Diction and idiom questions test word choice and fixed prepositions (e.g., "differ from," "capable of," "comply with"); shorter is not automatically correct.