Georgia Master's Exam: Integrated foundations: english, quant, formal, writing

Grammar and vocabulary through meaning and context

Grammar and vocabulary work together to express meaning. Start with the whole sentence, the speaker's intended time or relationship and the words around a gap. A word may be grammatically possible but unsuitable in meaning or collocation. Conversely, the correct idea may need a different word form. Read the completed sentence again and check that it expresses the intended message rather than selecting an answer only because a nearby word looks familiar.

Time, aspect and condition

Distinguish a completed past event from experience or a situation continuing into the present. Compare She visited last year with She has lived here since last year. The time expression is important evidence, but it does not replace understanding the full meaning. In conditionals, identify whether the speaker describes a real possibility, an imagined present situation or an unreal past. Had I known is an inverted form of If I had known; it is not a question merely because the auxiliary precedes the subject.

Sentence structure

Find the main subject and verb before deciding agreement. Identify whether a clause adds information, identifies a noun, expresses purpose or creates another relationship. Punctuation can distinguish defining and non-defining relative clauses. The museum, which opened recently, is busy adds information about an already identified museum. A passive construction changes information focus and should preserve the original tense and essential meaning. Avoid changing a possibility into a certainty when rewriting.

Words, phrases and register

Learn collocations and phrasal verbs in sentences: pay attention, make a decision, put off a meeting. A dictionary synonym is not automatically interchangeable in every context. Check complement patterns, prepositions, countability and tone. Word formation may require a noun, adjective, verb or adverb; resilience and resilient belong to different grammatical categories. Distinguish a familiar literal meaning from a contextual or idiomatic use.

Check and extend

For each error, record the intended meaning, the chosen form and why a competing form fails in this sentence. Then write a new sentence that tests the same distinction. At more advanced levels, attend to inversion, hedging, complex modification and precise lexical choices. For foundational practice, secure the core distinction before adding complexity. These shared exercises are not an independently calibrated measure of a particular examination level. Actual writing and speaking are needed to demonstrate that the language can be used productively, not only recognized among four options.

Read for information, inference and textual effect

Reading begins with the task and the text's purpose. Identify whether the text informs, argues, narrates, reviews, instructs or combines several functions. Skim for the overall direction, then return to the relevant evidence. A matching word can distract if the sentence negates it, attributes it to another person or describes an abandoned plan. Select the answer supported by the passage, even when a different answer matches your personal knowledge or opinion.

Detail and inference

A detail question asks what the text states. An inference asks for a conclusion reasonably supported by it. Neither permits inventing an unrelated explanation. Suppose a notice says the outdoor event will move indoors if rain continues. It establishes a condition, not an unconditional cancellation. Pay attention to may, only, unless, some and most. These small words limit the claim. A proposed answer saying every participant must attend goes beyond a passage describing an optional activity.

Follow structure and reference

Connectors reveal contrast, cause, concession, addition and sequence. Identify the noun or idea to which a pronoun refers, and check that a proposed linking sentence fits both sides of a gap. Coherence concerns the overall development of ideas; cohesion includes explicit language links. A text can contain many connectors and still lack a clear argument. In cross-text comparison, identify the actual view or evidence rather than treating a shared topic as proof that two authors agree.

Analyze language and literature

Distinguish author, speaker and narrator. Tone is an attitude conveyed through choices in wording and structure, not simply the subject being discussed. Explain how a particular image, contrast, repetition or narrative choice shapes meaning. Quoting a device name without describing its effect is incomplete analysis. Consider ambiguity and perspective when the evidence supports more than one reading; avoid assuming that a fictional narrator reports the author's personal position.

Produce an evidence-based answer

State the point, give a precise reference and explain the connection. A summary selects the relevant central content without inserting unsupported opinion. An analysis explains how meaning is created; an evaluation justifies a judgment with criteria and evidence. Short multiple-choice passages practise useful components but do not replace sustained reading or the required full works. Use original source passages and prescribed texts for the intended course, and check the current official reading and task requirements separately.

Writing, argument and audience-focused mediation

A writing task is a communication problem before it is a word-count exercise. Identify the audience, purpose, required content and appropriate genre. A personal email, public article, report and argumentative essay make different demands. Use the actual task instructions for length and format; numerical requirements from another examination do not become rules for this course. Plan the essential points before drafting so that fluent language does not hide an unanswered task.

Organize meaning

A paragraph should develop a clear point supported by explanation, evidence or a relevant example. Connect ideas according to their real relationship rather than inserting however or therefore merely to sound sophisticated. An argument should distinguish its claim, supporting reasons, a meaningful counterargument and a justified conclusion. A balanced discussion does not require equal praise for every position regardless of evidence. Avoid introducing a new unsupported major claim in the final sentence.

Genre and register

Choose vocabulary, tone and address suitable for the relationship and purpose. A request to a school principal usually needs clear courteous language; a message to a friend can be informal without becoming confusing. In traditional British letter conventions, Yours faithfully is often used when the recipient is unnamed and Yours sincerely with a named recipient, but follow the actual context and instructions. Formal writing benefits from precision, not unnecessary long words or mechanical passive constructions.

Mediate rather than copy

Mediation selects and explains source information for a particular audience and purpose, sometimes across languages. It is not automatically a word-for-word translation or permission to add personal views. Identify what the reader needs, preserve essential qualifications and explain unfamiliar institutions or references briefly when necessary. If a German source is being mediated into English, practise with an actual German source and check that the English accurately conveys its relevant content. English-only judgment questions here support the strategy but do not demonstrate complete cross-language performance.

Revise at several levels

First check task coverage and factual fidelity, then organization and register, and finally grammar, spelling and punctuation. Ensure that pronouns have clear references and examples actually support the claim. Read the result as the intended recipient: is the necessary action or position clear? Ask for feedback on an original response and use the course's real assessment criteria. Selecting a good model sentence among options is preparation for productive writing, not a substitute for producing and revising a complete text.

Algebra, functions, equations and data in explicit models

Algebraic Operations

What this domain tests

The Algebraic operations domain tests whether you can rewrite expressions accurately and recognize structure before calculating. The subject methods include polynomial arithmetic, factoring, rational expressions, exponent laws, radicals, and complex numbers. Most errors come from illegal cancellation or from ignoring restrictions on variables.

Polynomials and factoring

Add and subtract by combining like terms; multiply each term of one polynomial by each term of the other. For nonzero polynomials over the real numbers, the degree of their product is the sum of their degrees.
Memorize special products: $(a\pm b)^2=a^2\pm 2 ab+b^2$, $(a+b)(a-b)=a^2-b^2$, and $a^3\pm b^3=(a\pm b)(a^2\mp ab+b^2)$.
Factoring strategy: greatest common factor first, then grouping, difference of squares, sum or difference of cubes, or trinomial trial. State the coefficient domain: a factorization over integers need not be complete over the real or complex numbers.

Rational expressions and exponents

Cancel factors, never terms: $\frac{x+2}{2}$ cannot become $x+1$.
For $x^{m/n}$, take the $n$th root and raise to the $m$; negative exponents move the factor across the fraction bar.
State excluded values where a denominator is zero before simplifying. Simplifying does not restore them.

Radicals and complex numbers

Simplify radicals by extracting perfect-power factors; rationalize denominators using conjugates.
Since $i^2=-1$, add and subtract complex numbers componentwise, multiply as binomials and replace $i^2$, and divide by multiplying numerator and denominator by the conjugate.

Worked example

Simplify $\frac{x^2-4}{x^2-x-6}\cdot\frac{x^2-9}{x+2}$ and list restrictions.

Factor: $\frac{(x-2)(x+2)}{(x-3)(x+2)}\cdot\frac{(x-3)(x+3)}{x+2}$. Before canceling, note $x\ne 3$ and $x\ne-2$. Canceling $(x+2)$ and $(x-3)$ leaves $\frac{(x-2)(x+3)}{x+2}$, with $x\ne 3,-2$. Check $x=0$: the original equals $\left(\frac{-4}{-6}\right)\left(\frac{-9}{2}\right)=-3$, and the simplified expression equals $\frac{(-2)(3)}{2}=-3$.

Traps that cost marks

Writing $(a+b)^2=a^2+b^2$ and omitting the middle term $2 ab$.
Canceling a term shared by numerator and denominator rather than a common factor.
Forgetting that a simplified rational expression keeps the original domain restrictions.
Applying a negative exponent only to the coefficient, as in treating $(2 x)^{-2}$ as $2 x^{-2}$.

Equations and Inequalities

What this domain tests

The Equations and inequalities domain tests solving and interpreting solution sets: linear and quadratic equations, rational and radical equations, absolute value, systems, and inequalities. These exercises combine both algebraic solutions and correct interval notation, including excluded values.

Linear equations and systems

Solve linear equations by applying inverse operations to both sides. An identity is true for all values; a contradiction has no solution.
For systems, substitution or elimination gives an ordered pair. Parallel distinct lines produce no solution; the same line produces infinitely many solutions. Always check the ordered pair in both original equations.

Quadratic equations

Write in standard form $ax^2+bx+c=0$. Use factoring and the zero product property, the square root property, completing the square, or the quadratic formula.
The discriminant $b^2-4 ac$ determines the type of roots: positive gives two distinct real roots, zero gives one repeated real root, and negative gives two complex conjugate roots.
For $ax^2+bx+c=0$, the sum of roots is $-b/a$ and the product is $c/a$.

Inequalities and absolute value

Multiplying or dividing an inequality by a negative number reverses its direction.
For $k>0$, $|X|<k$ means write $-k<X<k$; and for $k>0$, $|X|>k$ means write $X<-k$ or $X>k$.
For rational inequalities, find critical values from numerator and denominator, test intervals on a sign chart, and exclude zeros of the denominator. Use a bracket at a numerator zero only if the inequality allows equality and the expression is defined there.

Radical and rational equations

Squaring both sides can introduce extraneous solutions, so every candidate must be checked in the original equation.

Worked example

Solve $\sqrt{2 x+3}=x$.

Square both sides: $2 x+3=x^2$, so $x^2-2 x-3=0$, which factors as $(x-3)(x+1)=0$. Candidates are $x=3$ and $x=-1$. Check $x=3$: $\sqrt{9}=3$, valid. Check $x=-1$: $\sqrt{1}=1$, but the right side is $-1$, so it is extraneous. The solution set is ${3}$. Squaring erased the sign information that the check restores.

Traps that cost marks

Forgetting to reverse an inequality after multiplying or dividing by a negative.
Using a closed interval at a value that makes a denominator zero.
Dividing both sides by a variable and losing the solution where that variable equals zero.
Skipping the check after squaring a radical equation.

Functions and Their Properties

What this domain tests

The Functions and their properties domain tests the function concept across algebraic and graphical representations: notation, domain and range, composition, inverses, transformations, and basic behavior of parent functions. Useful practice includes questions that combine two ideas, such as finding the domain of a composition.

Domain and range

The domain is the set of allowed inputs. Exclude values that make a denominator zero and, for even roots, values that make the radicand negative.
The range is the set of outputs. Use the graph, vertex, asymptotes, or end behavior to determine it.
Express domains and ranges in interval notation, using parentheses at excluded values and infinities.

Composition and inverses

$(f\circ g)(x)=f(g(x))$. Its domain requires $x$ in the domain of $g$ and $g(x)$ in the domain of $f$.
An inverse exists only for a one-to-one function; the horizontal line test detects whether each output has a unique input.
To find an inverse, swap $x$ and $y$ and solve for $y$. The domain and range of $f$ and $f^{-1}$ interchange, and $f(f^{-1}(x))=x$ on the appropriate domain.

Transformations and parent graphs

$y=f(x-h)+k$ shifts the graph $h$ units right and $k$ units up.
$y=af(x)$ stretches vertically when $|a|>1$ and compresses when $0<|a|<1$; negative $a$ reflects across the $x$-axis.
$y=f(-x)$ reflects across the $y$-axis. Apply transformations in a consistent order, handling horizontal shifts and scalings inside the argument carefully.

Worked example

Find the inverse of $f(x)=\frac{2 x+1}{x-3}$.

Write $y=\frac{2 x+1}{x-3}$ and swap: $x=\frac{2 y+1}{y-3}$. Multiply: $x(y-3)=2 y+1$, so $xy-3 x=2 y+1$. Collect $y$ terms: $xy-2 y=3 x+1$, hence $y(x-2)=3 x+1$ and $y=\frac{3 x+1}{x-2}$. Thus $f^{-1}(x)=\frac{3 x+1}{x-2}$, with $x\ne 2$. The original function excludes $x=3$ from its domain, and its inverse excludes $x=2$ from its domain, matching the range restriction.

Traps that cost marks

Assuming $f(g(x))=g(f(x))$; composition is generally not commutative.
Ignoring the inner function's domain when finding the domain of a composition.
Reading $f^{-1}(x)$ as $1/f(x)$ rather than as inverse notation.
Confusing horizontal and vertical line tests, or shifting a graph in the wrong direction.

Algebra, functions and data in stated models

Translate a numerical story into a model before manipulating symbols. Identify the variable and its unit, the fixed quantities, and any restrictions such as whole-number counts or nonnegative time. A fixed joining fee plus a monthly charge has the formC=b+mx. The intercept represents the initial charge and the slope the change per additional month in that stated model. A line may fit a classroom example without describing every real membership contract or every future observation.

For an inequality, interpret the boundary phrase carefully. At least includes the threshold and gives a greater-than-or-equal relation; at most includes it with a less-than-or-equal relation. Check a candidate in the original story after solving. A negative number of tickets may solve an algebraic equation but fail the count restriction. For two equations, a common solution must satisfy both; parallel distinct lines do not share a point, while two forms of the same line describe one common set.

Functions assign each permitted input exactly one output. Their tables, graphs, formulas and verbal descriptions emphasize different properties. Distinguish an input restriction from an output range, an intercept from a rate of change, and a zero from a point where the function is undefined. In a quadratic model, identify what the vertex represents and whether it lies in the allowed domain. A mathematical vertex outside a supplied time interval need not be an attainable physical maximum.

Data questions add another layer of interpretation. Compute the mean from sum divided by count; use the ordered middle observation or pair for the median. An extreme observation can affect the mean differently from the median. Compare a spread measure when two groups have the same center, and check whether a graph changes its scale or begins above zero. A scatterplot association can suggest a pattern without proving that one variable causes the other. The sample, method and possible confounding variables matter.

For example, a fee modelC=40+25 m with budget 215 gives 40+25 m<=215, hence m<=7. If months must be nonnegative whole numbers, the permitted counts are 0 through 7. The original inequality and 25 m<=175 describe the same feasible set; a question claiming they are different answers would be defective. This check illustrates why equivalent algebraic forms and units must be evaluated before accepting a proposed multiple-choice key.

Data interpretation, information sufficiency and source reasoning

Data Insights

Independent data-reasoning practice develops table and graph reading, information sufficiency, multi-source reconciliation and linked constraints. Current official question counts, timing, calculator rules and interaction formats are not established by these notes.

The five question types

Data Sufficiency: a question plus two statements; decide whether the data are sufficient, not what the value is.
Multi-Source Reasoning: two or three tabs of text, tables, or charts; questions ask you to judge statements as true/false or to select an option.
Table Analysis: a sortable table; each statement is judged true or false against it.
Graphics Interpretation: a graph with drop-down, fill-in-the-blank statements.
Two-Part Analysis: a table where you pick one value in each of two columns; both must be correct.

A five-case information-classification model

A: (1) alone sufficient, (2) alone not. B: (2) alone sufficient, (1) alone not. C: together sufficient, neither alone. D: each alone sufficient. E: together still not sufficient.

Worked example

Is x > 0? (1) x² > 0. This only establishes x ≠ 0; x could be 3 or −3. Not sufficient. (2) x³ > 0. Cubing preserves sign, so x must be positive. Sufficient.

Within that five-case model, statement 2 alone is sufficient. Classify against the options actually given in a practice item.

The discipline is to test awkward values - negatives, zero, proper fractions, non-integers - before declaring sufficiency. Statement (1) looks sufficient only if you try positive numbers exclusively.

Traps that cost marks

Calling a statement sufficient after finding a single satisfying value.
Forgetting that a variable may be negative or fractional when the question imposes no restriction.
In Two-Part Analysis, choosing two values that satisfy one constraint but violate the other.
In Table Analysis, misreading a column or failing to re-sort before judging a statement.
Skipping a tab in Multi-Source Reasoning and missing the data that decides the question.
Reaching for the calculator before deciding what actually needs computing; it verifies arithmetic, not reasoning.

Evidence, sufficiency and quantitative comparison

Information is sufficient when it determines the answer to the question being asked, within the stated domain. A statement need not identify an exact value to answer a yes-or-no question consistently. Conversely, finding one example that gives a yes does not establish that every permitted example does. First rewrite the target question precisely; then test each supplied statement independently before combining them.

For example, if x is a real number and x²=49, its value could be 7 or-7. This is insufficient to determine x uniquely, but it is sufficient to answer whether x is nonzero. Adding x>0 determines x=7. The question matters as much as the numerical equation. Do not import a second statement when checking the first alone, and do not assume positivity, integrality or distinctness unless it is actually given.

Tables and charts introduce source discipline. Read the label, unit, date or period, denominator and any note before comparing values. A total in thousands is not directly comparable to a single-unit count without conversion. A percentage based on all responses can differ from the percentage based on respondents to one item. A share of a total, a percent increase from a baseline and a percentage-point difference answer different questions. State which one the problem requests.

In a multi-source model, assign each fact to its source and time. An original estimate and a revised estimate are not necessarily contradictory: they may answer different versions of the same planning question. If three people cost 120 per hour and two cost 90 per hour, the combined hourly rate is 540. At 200 hours the modeled labor cost is 108000; at 250 hours it is 135000. A budget note cannot silently replace the requested original duration. These are fictional calculations, not labor-market facts.

Finally, distinguish consistency from proof. A rule may fit every supplied term of a finite sequence without being the only mathematical continuation possible. Such exercises ordinarily ask for the intended simple pattern represented by the available choices. Explain that rule and test it on all supplied pairs. In an argument question, similarly identify the premise, proposed conclusion and any missing assumption. A plausible conclusion is not automatically entailed. A counterexample can disprove a universal claim, while several supporting examples alone do not establish one.

Formal reasoning from explicit premises

A reasoning question supplies a model, not necessarily a report of how the real world works. Identify its stated premises and the exact conclusion before applying background knowledge. If a passage says every member of group A belongs to B, it establishes inclusion in that direction. It does not establish that every member of B belongs to A. A diagram or small counterexample can help keep the direction clear.

Quantifiers and existence

All, some and none make different claims. All engineers use a specified tool and no user of that tool lacks a stated skill together imply that no engineer lacks that skill, within the supplied model. However, all A are B and some B are C do not by themselves imply that some A are C. The B members in C may lie outside A. A universal statement does not always establish that its subject class has a member, so do not silently turn all into some.

Conditions and alternatives

If P then Q makes P sufficient for Q and Q necessary for P in the stated implication. It does not prove the converse. If the model says an event moves indoors if rain continues, an indoor event is not proof of rain: there may be another reason. A valid contrapositive is if not Q then not P. Keep and, or, only if and unless distinct, and state any interpretation needed for an ambiguous everyday use of or. Do not invent a condition that the text has not supplied.

Cases and counterexamples

For a must-follow question, try to construct a permitted case where the proposed conclusion is false. One such case refutes necessity. For a possibility question, one coherent case can establish consistency, but it does not establish that the outcome must happen. In a number pattern, explain the intended simple rule and verify every supplied pair; do not claim a finite list has only one unrestricted mathematical continuation.

Explain the inferential step

Write the premises, the relation used and the conclusion in a short chain. A persuasive story or familiar technical setting cannot replace that chain. When a passage quotes a fictional policy, use it only for the reasoning exercise and never transfer it into a real legal, clinical, laboratory or professional procedure. If essential information is absent, identify the missing link. Clear reasoning includes knowing when the evidence supports a limited conclusion and when it does not determine one.

Quantitative, verbal and issue-essay reasoning foundations

Arithmetic: Number Properties, Percents, Ratios, and Exponents

What you must know

This arithmetic practice develops number-property reasoning more than long computation. Be fluent with divisibility rules (3 and 9 by digit sum, 4 by the last two digits, 8 by the last three), prime factorization, greatest common factor and least common multiple, remainders, even/odd and sign rules, fraction–decimal–percent conversions, ratios and proportions, exponent laws, roots, absolute value, and order of operations. Memorize common fraction–percent equivalents (1/6 ≈ 16.7%, 1/8 = 12.5%, 1/7 ≈ 14.3%) to save time under the clock.

Traps that cost points

Percent change is always measured against the base. A 25% increase followed by a 20% decrease returns to the original value; successive percents multiply, they never add.
Distinguish percent of from percent more than, and percent from percentage points.
Never divide both sides by a quantity that could be zero - it silently destroys a case.
Convert units before comparing, and honor any integer constraint the question states.
If a question asks only for the units digit of an integer power, study the base-10 final-digit cycle; do not discard other digits when the question asks for a different quantity.

Worked example

A jacket costs $80. Its price is raised 25%, then the new price is discounted 20%. Final price = 80 × 1.25 × 0.80 = 80 × 1.00 = $80 - unchanged. Rule of thumb: chain percent changes by multiplying factors of the form (1 + r), never by adding percentages.

Algebra: Equations, Inequalities, Quadratics, and Translation

Core skills

Algebra questions reward translation and structure over brute force. You need: solving linear equations and inequalities; systems of two equations (substitution and elimination); quadratic techniques including factoring, the quadratic formula, the discriminant b² − 4 ac, and vertex form; function notation and composition such as f(g(x)); exponent and radical laws; rational expressions; absolute value equations; and modeling word problems - distance/rate/time, combined work rates, mixture and weighted-average setups, and simple sequences.

Traps

Dividing both sides by a variable expression: from x² = 5 x, dividing by x leaves x = 5 and silently drops x = 0. Factor instead.
Multiplying or dividing an inequality by a negative number reverses the inequality sign.
Squaring both sides can introduce extraneous roots; always test candidates in the original equation.
In a quantitative comparison, test permitted cases, including boundary or sign cases where applicable. Values such as 0 or 1 are not valid test substitutions if the stated domain excludes them.
Translated phrases: 5 less than x is x − 5, not 5 − x.
If the question fixes a range such as x > 2, report only the root that satisfies it.

Worked example

Solve x² − 5 x + 6 = 0. Factor: (x − 2)(x − 3) = 0, so x = 2 or x = 3. If the problem also states x > 2.5, the only valid answer is 3. Rule of thumb: move everything to one side, factor, then apply every stated constraint before answering.

Geometry: Figures, Special Triangles, Circles, and Scaling

Core skills

Geometric reasoning compares relationships rather than measurement. Master: angle rules for parallel lines cut by a transversal; triangle properties (angle sum, isosceles base angles, exterior angle, triangle inequality); the Pythagorean theorem and special right triangles 45–45–90 (sides x, x, x√2) and 30–60–90 (x, x√3, 2 x); area and perimeter of standard figures; circles (circumference, area, arcs, sectors, with an inscribed angle equal to half the central angle subtending the same arc); coordinate geometry (slope, distance, midpoint, parallel and perpendicular slopes); and volumes of rectangular solids, cylinders, and spheres.

Traps

Figures are explicitly not drawn to scale: never infer a right angle, equal length, or parallel line from appearance alone.
Similar figures scale lengths by k, areas by k², and volumes by k³.
Area and perimeter are not linked proportionally - doubling a square's perimeter quadruples its area.
The diagonal of a rectangle is not one of its sides; it comes from the Pythagorean theorem.
For sectors, the degree measure and the arc length must use the same fraction of the whole circle.

Worked example

Two similar triangles have corresponding sides in the ratio 3:5. If the smaller triangle's area is 18, the larger area is 18 × (5/3)² = 18 × 25/9 = 50. Rule of thumb: write the similarity ratio first, then apply the correct power - square for area, cube for volume - before computing.

Data Analysis: Statistics, Counting, Probability, and Graphs

Core skills

Data analysis covers descriptive statistics (mean, median, mode, range, standard deviation, quartiles, interquartile range); interpreting bar, line, and circle graphs, scatterplots, and tables - often two or three linked questions per set; the normal distribution's 68–95–99.7 rule; counting via the fundamental counting principle, permutations, and combinations; probability including complements and independent events; and overlapping sets through Total = A + B − Both + Neither.

Traps

With an even number of data values, the median is the average of the two middle values, not one of them.
Adding a constant to every value shifts the mean and median but leaves the standard deviation and range unchanged; multiplying every value by k multiplies the standard deviation by |k|.
Standard deviation measures spread only - two sets can share a mean and have very different spreads.
At least one is almost always fastest as 1 − P(none).
Ask whether order matters: if it does, it is a permutation; if only selection matters, it is a combination.
On graph questions, check axis labels, units, and whether the scale starts at zero before estimating.

Worked example

A set of five numbers has mean 20, so its sum is 100. Adding a sixth value of 32 gives a new mean of (100 + 32)/6 = 132/6 = 22. Rule of thumb: convert every mean back into a sum (sum = mean × count) before combining groups.

Reading Comprehension: Structure, Function, and Inference

Read for structure before detail

A passage may have a structure of claim, evidence, concession or rebuttal; identify what is actually present rather than forcing all four functions onto every text. Before answering anything, summarize in one sentence what the author wants you to believe, and label how the parts move (e.g., phenomenon → rejected explanation → preferred explanation). Most questions, including detail questions, get easier once that skeleton is explicit.

Question types you must recognize

Main idea; function (“in order to”); inference; author's attitude; detail and EXCEPT; strengthen/weaken; vocabulary in context; analogy. For function questions, insert “in order to” before each choice - the correct one completes a statement of purpose, not a paraphrase of the sentence's content.

Traps

True but irrelevant: supported by the passage, yet it does not answer the question asked.
Extreme language: all, never, proves, impossible usually outruns hedged academic prose.
Scope creep: an answer about the wrong paragraph, or one that generalizes a carefully limited claim.
Outside knowledge: your own expertise is never the evidence; only the text is.
Reversed direction: in strengthen/weaken items, choices that do the opposite are common.

Rule of thumb

Phrase your own answer in the author's cautious register before looking at the choices, then eliminate anything stronger or broader than the text. For an inference, require support for the proposed conclusion - a statement that must be true if the passage is.

Worked example

Passage: “Historians once treated guild records as mere administrative residue. Recent work reads them as negotiated documents in which masters and journeymen contested wage norms.” Question: the author mentions “administrative residue” primarily to - the answer names a view the passage then revises. A choice like “to describe medieval bookkeeping practices” is true to the phrase but misses its function, and is wrong.

Text Completion: Logic First, Vocabulary Second

The sentence is a logic puzzle

Each blank is constrained by a signal: contrast (although, yet, paradoxically, despite), support (because, indeed, moreover, so), or definition (a phrase that restates the blank). Locate the signal before weighing vocabulary; the words are the last step, not the first.

Method

Read the whole sentence ignoring the blanks and predict a plain word of your own for each. Then match your prediction to the choices. For two- and three-blank items, no choice is correct until every blank produces a coherent sentence - a word can be perfect in blank (i) and impossible in blank (iii).

Traps

Neighbor-word attraction: a choice that echoes a nearby word often lacks any logical fit.
One-blank-at-a-time thinking: the tempting pair frequently collapses on the remaining blank.
Tone mismatch: grammatically fine but wrong register - praise where the sentence describes criticism.
Ignoring the second sentence: it often supplies the definition the first blank needs.

Rule of thumb

Start with the blank carrying the most constraints (the one with a definition or a strong signal). Eliminate there, then verify the surviving combination by rereading the full sentence.

Worked example

“Although the committee's report was ______, its recommendations were adopted almost unchanged.” The although demands a negative word in contrast with acceptance: reviled or derided fits. A choice such as conciliatory or lauded reverses the logic, and a word that merely echoes “report” (e.g., documentary) has no evaluative force at all. Predict first, match second.

Sentence Equivalence: Finding the Coherent Pair

Two answers, one meaning

The paired-completion method compares two words that produce coherent sentences of equivalent meaning; this practice page uses its own four-option interface and does not duplicate an official interaction. They need not be synonyms; they must produce two sentences that mean the same thing, and both must be independently defensible. Treat the item as a single-blank Text Completion that happens to have six options.

Method

Read the sentence and name its logic signal (contrast, cause, restatement).
Predict a word in your own vocabulary.
Scan for the pair nearest your prediction, then reread the sentence with each word substituted in full.

If two choices are near-synonyms but only one fits the logic, neither is correct - both members of the pair must fit.

Traps

The synonym decoy pair: two attractive synonyms that together violate the sentence's contrast.
The lone correct word: one excellent choice whose partner is wrong; both members must support the intended paired reading.
Intensity mismatch: words close in meaning but far apart in strength (unhelpful vs. catastrophic).
Skipping the second clause: it frequently defines or reverses the blank.

Rule of thumb

Substitute each candidate into the whole sentence and ask whether the two resulting sentences say the same thing. If you cannot state the sentence's logic in one clause - “the writer concedes X but argues Y” - you are guessing, not solving.

Worked example

“Far from being ______, the proposal drew immediate praise from every faction.” The far from signals the blank is the opposite of what praise implies. Contentious and divisive produce the same sentence and both fit. Novel and innovative are a tidy synonym pair, but they are coherent with nothing in the sentence - the classic decoy.

Analyze an Issue: Building a Nuanced, Well-Supported Position

An issue essay develops a reasoned position on a stated claim, explains evidence and examines relevant complexity. This independent chapter teaches argument construction rather than predicting an official score or supplying current timing and scoring policy.

Read the instruction, not just the prompt

Each Issue prompt is followed by a specific instruction, such as discussing the extent to which you agree or disagree, the circumstances in which a recommendation would be advantageous, or the consequences of adopting a claim. These instructions vary across the pool; answering a different question fails to address the assigned task. Underline the verb and the scope words-'extent', 'specific circumstances', 'consequences'-and make your thesis answer that exact question.

Build a position with visible complexity

High-scoring responses do not simply agree or disagree; they define the claim's scope. Identify the key qualifier-'only', 'primary', 'must', 'most important'-and test it. Concede the strongest opposing point, then explain why your position still holds. Organize reasons as paragraphs, each with a claim, explanation, and a concrete example from history, science, current events, or lived experience. Avoid invented statistics.

Worked example

Prompt: 'Governments should fund the arts only when all other public needs have been met.' Instruction: discuss the extent to which you agree or disagree, considering ways the statement might or might not hold true.

Reasoning: the word 'only' makes arts funding residual. Taken literally, it would almost never apply, because public needs are never fully satisfied. Concede that emergency needs-pandemic hospital capacity, fire response-should outrank museums. Then argue that a blanket residual rule is poor policy: arts funding supports tourism, education, and community cohesion, and the resource trade-off should be examined explicitly rather than ignored. Address the counterargument that arts subsidies can favor elite institutions; the remedy is accountability, not elimination. Thesis: urgent needs should take precedence, but 'only when' is too absolute; governments should maintain a modest, accountable baseline.

Traps that cost marks

Ignoring the instruction line and writing a generic agree/disagree essay.
A false dichotomy: 'I completely agree' followed by no acknowledgment of exceptions.
Vague examples ('studies show') or invented facts instead of explainable cases.
Listing examples without explaining how they support the claim.
No counterargument, so the position looks unexamined.