ORT Main Test cheat sheet

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

Divisibility tests: a number is divisible by 3 if its digit sum is divisible by 3; by 4 if its last two digits form a multiple of 4; by 9 if its digit sum is divisible by 9.

Percent change = (new value - old value) / old value × 100%. Successive percent changes multiply factors: (1 + r1)(1 + r2).

For a linear equation ax + b = c with a ≠ 0, the solution is x = (c - b) / a.

Slope between two points: m = (y2 - y1) / (x2 - x1). Parallel lines have equal slopes; perpendicular nonvertical lines have slopes whose product is -1.

Exponent rules: a^m × a^n = a^(m+n); (a^m)^n = a^(mn); a^0 = 1 for a ≠ 0; a^(-n) = 1 / a^n.

Simple interest: I = Prt. Compound interest: A = P(1 + r/n)^(nt), where n is the number of compounding periods per year.

Quadratic formula: for ax^2 + bx + c = 0 with a ≠ 0, x = [-b ± sqrt(b^2 - 4ac)] / (2a). The discriminant b^2 - 4ac determines the number of real roots.

Ratio scaling: in a ratio a:b, the total is a + b parts; a quantity sharing total T in that ratio has parts T × a/(a + b) and T × b/(a + b).

Area formulas: rectangle = length × width; triangle = 1/2 × base × perpendicular height; circle = pi × r^2; trapezoid = 1/2 × (a + b) × height.

Circle measures: circumference = 2 × pi × r; arc length = (central angle / 360) × 2 × pi × r; sector area = (central angle / 360) × pi × r^2.

Pythagorean theorem: in a right triangle, a^2 + b^2 = c^2, where c is the hypotenuse. The converse identifies a right triangle from side lengths.

Volume formulas: prism = base area × height; cylinder = pi × r^2 × h; cone = (1/3) × pi × r^2 × h; sphere = (4/3) × pi × r^3.

Surface area: sphere = 4 × pi × r^2; closed cylinder = 2 × pi × r^2 + 2 × pi × r × h.

Similar figures with length scale factor k: corresponding lengths scale by k, areas by k^2, and volumes by k^3.

Angle facts: a straight angle is 180 degrees; triangle angles sum to 180 degrees; quadrilateral angles sum to 360 degrees; an exterior angle equals the sum of the two remote interior angles.

Coordinate geometry: distance = sqrt[(x2 - x1)^2 + (y2 - y1)^2]; midpoint = ((x1 + x2)/2, (y1 + y2)/2).

Unit conversions: 1 m = 100 cm; 1 km^2 = 1,000,000 m^2; 1 L = 1000 cm^3; 1 cm^3 = 1 mL.

Mean = sum of values / number of values. Median = middle ordered value, or average of the two middle values for an even count. Mode = most frequent value. Range = maximum - minimum.

Weighted mean = sum of (value × weight) / sum of weights. Use it when data points contribute unequally.

Probability of an event with equally likely outcomes = favorable outcomes / total outcomes. Complement rule: 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).

A linear function has the form f(x) = mx + b; m is the constant rate of change and b is the initial value or y-intercept.

Exponential growth or decay: y = a(1 ± r)^t, where a is the initial amount, r is the rate per period, and t is the number of periods.

Rate problems: distance = rate × time. For combined work, add work rates as reciprocals of individual completion times.

Mixture and total-value problems: total value = sum of (quantity × unit value) for each component; set this equal to the required combined value.

Correlation describes the direction and strength of association between two variables; correlation does not establish causation.

The main idea is the central claim or point of a passage; supporting details explain, illustrate, or prove that central point.

A valid inference must be supported by specific textual evidence combined with logical reasoning, not by personal opinion alone.

Tone is the author's attitude toward the subject, revealed through word choice, imagery, and sentence structure; mood is the emotional atmosphere created for the reader.

Common text structures: cause-effect, compare-contrast, problem-solution, chronological order, and description.

Signal words guide structure: 'however' signals contrast; 'therefore' signals conclusion; 'for example' signals illustration; 'because' signals cause.

A fact can be verified with evidence; an opinion expresses a judgment, belief, or evaluation that can be debated.

Author's purpose is the reason for writing, commonly to inform, persuade, explain, or entertain.

A pronoun's antecedent is the noun it refers to; the antecedent must agree with the pronoun in number and gender and be clear from context.

A paraphrase preserves the original meaning in new wording; a summary condenses the main points and omits minor details.

Subject-verb agreement: a singular subject takes a singular verb and a plural subject takes a plural verb; prepositional phrases between them do not change the subject's number.

Pronoun case: use subject pronouns (I, he, she, they) for subjects; use object pronouns (me, him, her, them) for objects and after prepositions.

Comma rules: use a comma before a coordinating conjunction joining two independent clauses; set off nonessential clauses with commas.

A semicolon joins two independent clauses without a conjunction; a colon introduces a list, explanation, or quotation after an independent clause.

Place modifiers next to the word they describe. A dangling modifier lacks a logical subject in the main clause and must be revised.

Apostrophes: singular possessive = apostrophe + s; most plural possessives = apostrophe after the plural s; 'it's' means 'it is,' while 'its' is possessive.

Parallel structure requires items in a list or comparison to share the same grammatical form, such as all nouns, all gerunds, or all infinitives.

Context clues include nearby definitions, contrasts, examples, and restatements that help a reader infer an unfamiliar word's meaning.

Active voice: the subject performs the action. Passive voice: the subject receives the action, often forming 'be' + past participle.

Theme is a universal idea or message about life explored through plot, character, and imagery; plot is the sequence of events.

Conflict types include person versus person, person versus self, person versus nature, and person versus society.

Direct characterization states a character's traits; indirect characterization reveals traits through speech, actions, thoughts, appearance, and others' reactions.

Point of view: first person uses 'I'; third-person limited follows one character's perspective; third-person omniscient knows all characters' thoughts.

Irony types: verbal irony says the opposite of what is meant; situational irony produces an outcome opposite to expectation; dramatic irony gives the audience knowledge a character lacks.

A symbol is an object or concept representing an abstract idea; an allegory is a sustained narrative in which symbols convey a second meaning.

An allusion is a brief reference to a well-known person, event, text, or cultural work; understanding it depends on shared cultural knowledge.

Dramatic structure: exposition, rising action, climax, falling action, and resolution.

Poetic forms: a sonnet has 14 lines; iambic pentameter has five unstressed-stressed pairs per line; a haiku has three lines with 5, 7, and 5 syllables.

Historical context shapes a text's assumptions and values; presentism is judging past works solely by modern norms.