TACHS cheat sheet

Main idea = the single point every paragraph supports; it is stated in the topic sentence or must be inferred from the whole passage, never from one detail.

Supporting details answer who, what, when, where, why, how; a detail that appears in the passage but does not back the main idea is a distractor.

Inference rule: the correct answer must be provable from the text plus common knowledge; if the passage never implies it, it is wrong even if it sounds true.

Vocabulary in context: substitute each answer choice into the sentence and reread; the correct choice preserves the sentence's logic, not just a dictionary meaning.

Tone and attitude words: positive (approving, admiring), negative (critical, skeptical), neutral (objective, informative); extreme words like 'outraged' rarely fit expository passages.

Sequence questions: locate time-order signal words (first, then, after, finally, meanwhile) and build a timeline before choosing.

Author's purpose options: to inform, to persuade, to entertain, to explain a process; purpose is signaled by verbs and by whether the passage argues or reports.

Paired-passage rule: answer questions about each passage separately first, then handle 'both authors' items by naming the exact point of agreement or disagreement.

Sentence fragment = missing subject or missing verb or a dependent clause standing alone; test by asking 'who or what does the action?'

Run-on/comma splice fix: join with a semicolon, a comma plus coordinating conjunction (for, and, nor, but, or, yet, so), or split into two sentences.

Subject-verb agreement: find the true subject, ignore prepositional phrases between subject and verb, and match singular/plural; indefinite pronouns (everyone, each, neither) take singular verbs.

Pronoun rule: a pronoun must agree with its antecedent in number and gender, and must refer to exactly one clear noun.

Parallel structure: items in a list or comparison must share grammatical form (all nouns, all -ing forms, all infinitives); 'like' compares nouns, 'as' compares clauses.

Modifier placement: a descriptive phrase must sit next to the noun it describes; an opening phrase implies the subject of the main clause.

Punctuation facts: commas set off nonessential clauses and items in a series; apostrophes show possession (singular 's, plural s') or contraction, never plural.

Verb tense consistency: keep one tense unless a time signal (before, after, by the time, since) requires a shift; 'had' marks the earlier of two past events.

Word choice: fewer for countable nouns, less for mass nouns; between for two, among for three or more; its = possessive, it's = it is.

Percent: part = percent x whole; percent change = (new - old) / old x 100; increase by p% means multiply by (1 + p/100).

Fractions: a/b + c/d needs a common denominator; to divide, multiply by the reciprocal (a/b ÷ c/d = a/b x d/c).

Ratios and proportions: set up a/b = c/d and cross-multiply; scale drawings use the same unit on both sides.

Area and perimeter: rectangle A = lw, P = 2(l + w); triangle A = ½bh; circle A = πr², C = 2πr; volume of a rectangular prism = lwh.

Mean = sum ÷ count; median = middle value of an ordered list; mode = most frequent value; range = highest minus lowest.

Average speed and rate: distance = rate x time; rearrange to r = d/t or t = d/r; combined work adds rates, not times.

Integer rules: negative x negative = positive; subtracting a negative adds; order of operations is PEMDAS (parentheses, exponents, multiply/divide left to right, add/subtract left to right).

Angles: straight line = 180°, triangle interior angles = 180°, vertical angles are equal; complementary = 90°, supplementary = 180°.

Probability of a single event = favorable outcomes ÷ total outcomes, always between 0 and 1; 'not' events use 1 - P(event).

Analogies: state the relationship in a sentence (e.g., 'a ___ is a type of ', ' is the tool used by ___') and apply the same sentence to each choice before picking.

Number series: check differences first, then ratios, then alternating patterns (two interleaved sequences), then squares/cubes or +/-1 offsets.

Letter series: convert letters to positions 1-26 (A=1, Z=26); constant or changing jumps usually mirror the number series rule.

Classification (odd one out): find the shared category of all but one item; the outlier may differ by function, category, or part-to-whole relationship.

Logic/deduction: draw a simple table or list of given conditions and eliminate; 'all A are B' does not imply 'all B are A'.

Ordering puzzles: translate 'before/after/taller/shorter' into a ranked line and place the most constrained item first.

Pattern/figural reasoning: track one attribute at a time - rotation, size, shading, count of sides - and confirm the rule holds across all given items.

Cause-and-effect and syllogism traps: reject choices that reverse the stated direction or that add information not given.