GRE · GRE · GRE 普通考试

Reading comprehension includes single-answer, select-one-or-more and sentence-selection tasks. Follow each instruction exactly.

Identify the conclusion separately from observations. An assumption is an unstated bridge the argument needs.

For select-all tasks, test each option independently. One correct option does not make the others false.

An author can describe a theory before challenging it. Track whose position each sentence reports.

Sales rose after advertising increased. “Advertising caused the rise” assumes no sufficient alternative explanation, such as a price reduction. A passage that only reports the timing supports an association, not that causal conclusion.

Graduate verbal tasks reward precise scope. Avoid an answer whose always or only overstates the passage.

Original passage: [1] A town library's loans rose after it began sending reminders. [2] During the same month, a nearby library closed for repairs. [3] The increase therefore does not establish that reminders alone caused more borrowing. [4] A comparison with an otherwise similar library unaffected by closure would help assess that claim.

What is the main purpose? A. Prove reminders never work. B. Limit a causal inference and suggest a comparison. C. Show repairs reduce reading. D. Recommend closing libraries.

Model and reasoning

B. Sentence 3 limits the inference from the observed timing; sentence 4 proposes evidence. A overstates uncertainty as proof of no effect. C changes the outcome from loans to reading and is not established. D turns a reported closure into a recommendation. The author accepts the rise in loans while questioning its explanation.

For the same passage, select every supported statement: A. Loans increased. B. Reminders caused none of the increase. C. Another change may help explain the increase. Then select the sentence that proposes a way to distinguish explanations.

Check after attempting

A and C; sentence 4. A is stated in sentence 1. C follows from the competing event in sentence 2 and caution in sentence 3. B is unsupported: failure to establish an exclusive cause does not establish zero contribution. Sentence 2 provides a competing observation; sentence 4 proposes a comparison. Judge each option independently and select the complete requested sentence.

For text completion, predict each blank from the passage. Support for a later blank may appear earlier.

A two- or three-blank item requires all blanks correct; do not treat choices as independent mini-scores.

Sentence equivalence requires two choices that both fit and produce equivalent sentence meanings.

A pair of synonyms can both fail the sentence. Check grammar, tone and logic as well as word meaning.

Although the design looked simple, building it was arduous. Although signals a contrast between appearance and effort. For equivalence, difficult and demanding could both fit this sentence; easy and effortless would reverse the needed contrast.

Do not search only for dictionary synonyms. The completed sentence controls equivalence.

Complete the original sentence: “Although the reviewer admired the report's detail, she found its conclusion ____ because its crucial experiment had not been repeated.” A. conclusive B. premature C. exhaustive D. irrelevant E. inevitable

Model and reasoning

B, premature. The unreplicated crucial experiment makes a settled conclusion too early. Although contrasts admiration for detail with criticism of the conclusion. Conclusive and inevitable oppose the criticism; exhaustive describes completeness, not unsupported timing; irrelevant would require evidence that the conclusion concerns the wrong issue. Predict “too early” before consulting the choices.

Sentence Equivalence: select exactly two of six choices. “Far from endorsing the proposal without reservation, the committee offered only ____ support.” A. qualified B. unconditional C. cautious D. unanimous E. enthusiastic F. immediate

Check after attempting

A and C. Both produce support limited by reservations. Unconditional contradicts the explicit contrast. Unanimous concerns how many members agree; immediate concerns timing. Enthusiastic concerns intensity and does not preserve the required limited-support meaning. Contextual fit and equivalent whole-sentence meaning are both necessary; two pleasant-sounding words are insufficient.

Quantitative Comparison has four fixed relationships: A greater, B greater, equal, or cannot determine.

Check all constraints. A variable can be negative, zero or fractional unless the stem restricts it.

Use strategic substitutions to disprove a fixed relation. They can show indeterminacy but a few examples do not prove universality.

Simplify both quantities without silently dividing by a value that could be zero or negative.

Given x²=9, x can be 3 or −3. Quantity A=x; B=0. A is greater for 3 and smaller for −3, so the relationship cannot be determined. If x is positive, only 3 is allowed and A is greater.

Figures are not necessarily drawn to scale. Given labels and constraints carry the evidence.

Quantitative Comparison uses these relationships: A. Quantity A is greater. B. Quantity B is greater. C. Equal. D. Cannot be determined. Given real x with $x^2=16$, compare Quantity A: x, Quantity B: 0. Repeat with the added condition $x>0$.

Model and reasoning

Without the added condition, D: x=4 makes A greater, while x=-4 makes B greater. With $x>0$, A, because only x=4 is permitted. The principal square root of 16 is four, but the equation has both roots. One chosen example cannot prove a relationship over all allowed values.

With the same four answer categories, let $a>0$ and $b<0$. Compare Quantity A: $a/b$, Quantity B: $b/a$. Give admissible examples if the relationship can vary, and identify when equality occurs.

Check after attempting

D. For a=2,b=-1, A=-2 and B=-1, so B is greater. For $a=1$,b=-2, A=-1/2 and B=-2, so A is greater. For $a=1$,b=-1 they are equal. More generally $a/b=b/a$ means $a^2=b^2$, and the stated opposite signs give a=-b. Multiplying an inequality by the negative product ab reverses direction; ignoring this sign would reverse conclusions.

GRE General quantitative content includes arithmetic, algebra, geometry and data analysis; calculus is not required.

Read whether a problem asks for one answer, all answers or numerical entry. Check units and form before submitting.

For data interpretation, identify the denominator and whether a percentage is conditional on a subgroup.

The on-screen calculator is available for General quantitative sections. Estimate first to catch an entry error.

A class has 12 morning and 8 afternoon students. Of them, 3 morning and 4 afternoon students cycle. P(cycles)=7/20. P(afternoon given cycles)=4/7. The second denominator includes only cyclists.

Do not assume a geometric diagram is to scale or replace a conditional denominator with the whole population.

In an original survey table, morning students number 30 and evening students 20. Twelve morning and eight evening students cycle. Enter the probability that a randomly chosen student cycles as a decimal. Enter the probability that a cyclist is an evening student as a fraction.

Model and reasoning

Total students are 50 and cyclists 20. The unconditional probability is $20/50=0.4$. Restricting to cyclists gives evening probability $8/20=2/5$. The denominator changes from 50 to 20 because the second question supplies a condition. An evening student cycling has the reversed probability $8/20$ here only by numerical coincidence: the evening group and cyclist group both happen to have size 20.

A rectangular display has area $48\ \mathrm{cm^2}$, positive integer side lengths and perimeter at most $32\ \mathrm{cm}$. Select all possible lengths of its longer side: 6, 8, 12, 16, 24. Show why your list is complete.

Check after attempting

8 and 12. Let the longer side be L and shorter side $48/L$. Positive integer factor pairs are (1,48),(2,24),(3,16),(4,12),(6,8). Their perimeters are 98,52,38,32,28 cm. Only the last two meet the bound, with longer lengths 12 and 8. Six is the shorter side of the valid 6-by-8 rectangle, so it fails the requested role. Test both the geometric restriction and the word “longer”; numerical compatibility alone is insufficient.

The standard short GRE has one 30-minute Analyze an Issue task. Its specific instruction controls what you must discuss.

Choose a position with a reasoned scope. You may qualify a general claim rather than simply agree or disagree.

Explain examples. An anecdote alone does not show why a policy produces the claimed effect.

Consider a serious objection and answer it. Revise organisation and clarity; the emphasis is analysis, not ornate vocabulary.

A qualified position might support work experience in professional courses while allowing research alternatives in theoretical programmes. Explain how experience builds practical judgement, then consider unequal access to placements and a fair alternative.

Do not practise Analyze an Argument as a current standard GRE General writing component. Older materials can contain retired tasks.

Original Issue prompt: “Universities should require every degree student to complete a work placement.” Discuss how far you agree. Explain circumstances in which the recommendation would or would not be useful, using reasons and examples. Plan a qualified position before reading the model.

Model and reasoning

I support a requirement to connect university study with work, but a single compulsory placement is too narrow for every degree and every student. Universities should normally offer supervised practical experience while allowing a demanding alternative when a placement would add little learning or create an unfair barrier. The important test is whether the experience helps students apply and examine what they have learned.

In many professional subjects, a placement can reveal a gap between understanding a rule and using it. Consider a hypothetical engineering student who can calculate an ideal load correctly. In a supervised workshop, the student may also need to explain assumptions to a technician, notice a damaged component and record a change in the design. These activities require judgement that a written calculation alone may not show. A university can strengthen this learning by asking the student to connect a workplace decision with a course principle and explain the limits of the model used.

This benefit does not mean any period in an office is educational. If a placement consists mainly of unrelated routine work, attendance becomes evidence of time spent rather than of learning. A useful requirement needs supervision, appropriate duties and reflection. Where these conditions cannot be met, a carefully assessed project with a real client may achieve more than a nominal placement. The university should judge the demonstrated learning, rather than assume that a workplace address guarantees it.

Access also matters. A student with caring responsibilities may be unable to accept a distant or unpaid role. Requiring that role could reward financial flexibility more than ability. This is a reason to provide funded local opportunities and flexible schedules, rather than a reason to abandon practical learning entirely. If adequate support remains unavailable, an alternative project should ask for comparable independence, communication and evaluation, so that the adjustment preserves academic demand.

Some theoretical programmes raise a further difficulty. A student preparing for research in pure mathematics may gain more from an extended investigation than from an available commercial placement. Research still involves planning, criticism and communication, even when it produces no immediate product. Allowing a supervised research route recognises a different form of application without suggesting that theoretical students need no experience beyond lectures.

A universal work-placement rule therefore confuses a valuable educational aim with one particular method. Universities should require substantial application and reflection, and they should make placements a strong option where suitable work and support exist. They should also accept well-designed alternatives when these better meet the learning goal. This qualified policy preserves the benefits of experience while avoiding an empty or unfair attendance requirement.

Without reusing the model's examples, write a paragraph arguing the strongest case for a universal placement requirement. Then explain one condition that weakens that case and revise your thesis accordingly. This is original classroom practice, not an official scored response.

Check after attempting

A defensible case is that a common minimum prevents departments from offering application only to students already confident enough to seek it. Develop how universal participation could broaden access and create a clear responsibility for the university. A weakening condition is lack of suitable supervised places: universality alone cannot create educational work. A revised thesis can require application for all while accepting equivalent projects where placements fail the stated educational standard. Other well-developed positions are valid. Review task fit, mechanisms, examples, counterargument and clarity; there is no uniquely correct stance or official score here.

Read the complete passage before starting a blank. Mark contrast cues such as although, support cues such as because, and a qualification that limits an apparent claim. Predict a plain-language role for each blank before looking for a difficult word. A later sentence may supply the clearest evidence for an earlier blank.

For two or three blanks, begin with the one whose context is most constrained. Use its proposed meaning to check the remaining passage, then return and verify the first choice. Answer columns do not restrict one another mechanically, but the resulting ideas must form one coherent account. Do not choose a superficially positive word merely because the subject sounds admirable.

A correct response needs one answer for every blank. The official task gives three choices per blank for multi-blank items, with no partial credit. Our additional checkbox drill labels each column explicitly: select exactly one entry from each labelled blank. It checks the full answer set locally; it is a practice adaptation, not the official column interface.

Read the completed passage using every chosen word. Check grammar, logical direction and strength. Reject a word that contradicts an explicit limitation even if it pairs well with one other blank. Keep a record of why each distractor fails: wrong polarity, wrong degree, wrong logical relation or wrong meaning in context.

Original two-blank case: “Although the proposal was initially (i) ____, the pilot’s consistent results made its critics (ii) ____.” Blank i: plausible / dubious / settled. Blank ii: reconsider / celebrate failure / lose interest. Choose dubious and reconsider: initial doubt contrasts with subsequent evidence prompting reassessment. Original three-blank case: “The curator’s account is (i) ____ about the object’s origin: it distinguishes verified records from (ii) ____, and its final conclusion is therefore (iii) ____ rather than absolute.” i: cautious / dogmatic / irrelevant; ii: certainty / conjecture / documentation; iii: qualified / categorical / unrelated. Cautious, conjecture and qualified preserve the passage’s evidential restraint.

Independent columns do not mean independent interpretations. Getting two of three blanks right is not a correct complete answer.

Original two-blank task: “The instrument's early readings were (i) ____, so the team treated the apparent trend as (ii) ____ rather than established.” Choose one per column. Blank i: erratic / consistent / definitive. Blank ii: provisional / inevitable / irrelevant.

Model and reasoning

erratic; provisional. Unstable readings justify a tentative interpretation. Consistent and definitive oppose the reason for caution. Inevitable contradicts “rather than established”; irrelevant discards the trend instead of limiting confidence in it. The “so” connects the evidence quality to the status of the inference. Both choices must work in the completed passage.

Original three-blank task: “Although the archive is (i) ____, its missing years do not make every conclusion (ii) ____: patterns repeated in the surviving independent records still justify a (iii) ____ claim.” Blank i: incomplete / comprehensive / fabricated. Blank ii: untenable / decisive / modest. Blank iii: qualified / universal / unrelated. Choose one per column and explain every rejection.

Check after attempting

incomplete; untenable; qualified. Missing years support incomplete, not comprehensive. Fabricated asserts falsification that the passage never supplies. The negation “do not make every conclusion” allows some justified inference: untenable means indefensible. Decisive would praise conclusions rather than name the criticism being rejected; modest is not the relevant failure. Repeated independent records support a limited claim, so qualified fits. Universal exceeds the incomplete evidence; unrelated breaks the stated support. The blanks are semantically dependent even though the answer columns are separate.

Predict what belongs in the blank from the whole sentence. Contrast and degree matter: “only a few errors” supports a different judgement from “errors throughout.” Avoid choosing pairs before reading the context. The pair must yield two coherent sentences that convey alike meanings, rather than merely share a dictionary association.

Group possible equivalents after establishing the needed sense. A word can have several meanings, and a candidate synonym may match the wrong one. “Reserved” can mean restrained in manner or set aside in advance. The complete sentence selects the sense, so test each candidate using a plain paraphrase.

Sentence Equivalence has six choices and requires two selections. There is no partial credit for one correct word, and selecting a third word invalidates the answer. The practice check requires exactly the correct pair; it does not award an official GRE score.

Compare the remaining plausible pair for scope and tone. Small versus negligible, uncertain versus impossible, or critical versus hostile may differ in strength. If the passage supports modest gains, do not infer that no gains occurred. Keep a vocabulary record with the contextual sense and one contrasting use rather than learning isolated translation pairs only.

Original case: “Despite the team’s ambitious targets, the pilot produced only ____ improvements.” Choices: dramatic, modest, erratic, slight, unprecedented, comprehensive. Modest and slight both fit the contrast between high ambitions and limited improvement. Dramatic and unprecedented may suggest a shared strong-result direction, but they do not fit only in this contrast. Original second case: “Far from being candid, the witness remained ____ about the missing receipt.” Evasive and noncommittal can preserve reluctance to give a direct account; eloquent concerns fluency and is not equivalent to either.

Do not choose a pair solely because the words are related. Degree, direction and local sense must all match.

Select exactly two of six options: “The curator openly acknowledged gaps in the records; her account was therefore unusually ____ about what remained unknown.” A. candid B. evasive C. ornate D. forthright E. repetitive F. hostile

Model and reasoning

A and D. Candid and forthright both express openness about uncertainty. Evasive reverses the meaning. Ornate describes style; repetitive describes recurrence; hostile adds an unsupported attitude. The sentence praises direct acknowledgement, not certainty about every detail.

Select exactly two of six options: “Despite its imposing appearance, the device proved remarkably ____ to operate; first-time users completed the procedure without assistance.” A. cumbersome B. intuitive C. unwieldy D. straightforward E. delicate F. novel

Check after attempting

B and D. Both make operation readily understandable in this context, supported by unaided new users. Cumbersome and unwieldy form a tempting negative pair but conflict with the evidence and contrast. Delicate concerns fragility or care, and novel concerns newness; neither matches ease of operation. Equivalent completed meanings matter more than choosing a dictionary synonym pair in isolation.

Read a passage for its claim and structure as well as local details. Identify what the author accepts, reports or questions. For select-one-or-more items, classify each option as supported or unsupported using the passage. An unsupported statement need not be disproved; absence of evidence is enough to reject it as a passage-based conclusion.

For inference, retain the text’s conditions and degree of confidence. If a study examined volunteers at one site, do not infer the result holds for every person or site. A statement consistent with the passage can still be unsupported. Select-all reading tasks use three choices, and require all correct choices with no extra selections.

Select-in-Passage asks for a sentence performing a specified role within the allowed text. Name the requested function: evidence, concession, explanation, challenge or conclusion. Read complete sentences rather than choosing the one containing a keyword. A sentence repeating the topic may do a different job. The local lesson uses numbered sentences and option labels; official delivery selects a sentence in the passage itself.

Create a short evidence note for every selected answer. For a rejected option, identify the extra assumption or scope change. This makes review productive and separates vocabulary difficulty from reasoning errors. These original short passages practise question types; they do not replicate adaptive routing or create an official verbal score.

Original passage: [1] A workshop extended its opening by one hour during a four-week trial. [2] Evening attendance increased, but the same month a nearby workshop closed. [3] The director therefore cannot attribute the entire increase to longer hours. [4] A second trial, at a site without a nearby closure, would help separate the explanations. Supported statements: attendance increased; another event provides an alternative explanation. Unsupported: longer hours had no effect. For “select the sentence stating a limitation on the causal conclusion,” choose sentence 3, not sentence 2: sentence 2 reports the confounding observation, while sentence 3 explicitly states the limit.

“Not proved to cause the increase” does not mean “proved to have no effect.” Match the requested sentence function exactly.

Original passage: [1] A laboratory replaced handwritten booking with an online system. [2] Booking errors fell during a two-month trial. [3] The laboratory also reduced the number of users during that period. [4] Although the new system may have helped, the trial cannot establish its effect under normal demand. [5] Testing it with the usual number of users would address this limitation.

Select every supported statement: A. Errors fell during the trial. B. Lower demand may affect the interpretation. C. Online booking cannot reduce errors. Select the sentence that explicitly limits the generalisation.

Model and reasoning

A and B; sentence 4. A restates sentence 2. B follows from the changed user count and stated limitation. C turns lack of decisive evidence into evidence of impossibility. Sentence 3 reports the confounding change; sentence 4 states why the normal-demand conclusion remains limited. Select the whole sentence, not just its caution phrase.

For that passage, identify the role of sentence 5 and explain whether a successful normal-demand trial would prove that every laboratory should adopt the system. Name one additional fact needed for that wider recommendation.

Check after attempting

Sentence 5 proposes a test addressing the specified limitation. A successful trial could support an effect in this laboratory under usual demand, but it would not establish a universal policy. Other laboratories may have different booking tasks, user access or costs. For example, evidence that the system remains usable for laboratories with limited internet access would inform transfer. The answer must connect the new fact to the proposed scope, not merely ask for unspecified “more research”.

Translate a short argument into evidence, conclusion and missing bridge. An assumption is what the reasoning needs but has not established. Avoid searching for a sentence that merely sounds reasonable or morally desirable. The target is the inference from these premises to this conclusion.

For a causal claim, test alternative explanations, reverse direction and relevant baseline differences. Evidence that an outcome followed a change establishes sequence; it does not isolate the change’s effect. A comparison group helps only if important conditions are sufficiently comparable or accounted for.

Strengthening and weakening are directional rather than absolute. A relevant new fact can make the conclusion more or less plausible without proving or refuting it. Ask how the fact changes the link: supports a missing mechanism, removes a competing cause, exposes a selection bias or shows the result under opposite conditions.

For a necessary-assumption task, negate a candidate carefully and ask whether the stated argument loses its support. This technique identifies dependence, not universal truth. Some options are sufficient for a conclusion but stronger than necessary. Keep claim scope precise: average improvement differs from every participant improving.

Original argument: “Students who chose the optional revision workshop scored higher than those who did not. Therefore the workshop caused the higher scores.” Missing bridge: self-selection did not adequately explain the difference. If workshop attendees were already scoring higher before it began, the conclusion weakens. If comparable students were randomly allocated and only attendance differed, the causal inference strengthens. Neither the workshop’s popularity nor its colourful materials isolates an effect. For “every attendee improved,” an average group increase alone is insufficient because individual outcomes can differ.

A fact can be relevant to the topic without affecting the inference. Identify the mechanism by which it changes support.

Original argument: “Participants who chose an optional planning session finished their projects earlier. Therefore the session caused faster completion.” Which most weakens the inference? A. Participants already completed previous projects earlier than nonparticipants. B. The room was recently painted. C. The session used printed notes. D. Many participants liked the session.

Model and reasoning

A. A pre-existing completion difference supplies a self-selection explanation. B and C have no stated connection to the outcome; D measures satisfaction rather than causal effect. A does not prove the session had no effect. It weakens the inference that the observed group difference establishes the claimed cause.

An organiser argues: “A guide will prevent all registration errors because every applicant can read it.” State a necessary assumption about accessibility, test its negation and distinguish it from the stronger claim that everyone actually reads the guide. Propose evidence that would strengthen the recommendation without proving “all”.

Check after attempting

A necessary accessibility bridge is that no applicant is unable to understand the guide's relevant instructions. Negating it gives at least one applicant unable to understand them; the universal prevention argument loses its stated support. Accessibility is weaker than actual reading, and neither alone guarantees that every action is performed correctly. A comparison showing fewer errors among comparable applicants using the guide would strengthen an effectiveness claim, while still not proving zero errors for every applicant. Other necessary assumptions, such as instructions covering the possible errors, are valid when their negation is tied to the exact universal claim.

Read the response instruction before calculation. Select-one asks for one option; select-one-or-more can require several; Numerical Entry has no options to cue an approximate answer. Identify restrictions such as integer, positive or rounded to a stated precision. In select-all tasks, test every option against those restrictions.

Solving a transformed equation yields candidates. A denominator restriction or a sign requirement in an unsquared radical can remove one. Substituting into the original is more reliable than recognising a familiar-looking list of roots. For inequality options, test the strict or inclusive boundary as written.

A requested fraction and a requested decimal require different entry forms. Equivalent fractions are accepted in official Numeric Entry; simplify if needed to fit the boxes. Apply rounding only at the end when instructed. Check units: a percentage answer 25 differs from a probability answer 0.25. The free local numerical drill below requests an exact terminating decimal; it is a teaching check, not a replacement for the official two-box fraction interface.

Estimate before using the on-screen General-test calculator and verify after entry. An implausible order of magnitude often reveals misplaced parentheses or a percent converted incorrectly. The GRE General mathematical level does not require calculus; sophisticated reasoning can still arise from elementary constraints and careful instructions.

Original select-all: x is real and (x−2)(x+3)=0. Choices −3, −2, 0, 2, 3. Select −3 and 2, and verify each factorisation root in the original. If the stem additionally requires $x>0$, only 2 remains. Original numerical entry: three of eight equally likely counters are marked; enter the probability as a decimal. P(marked)=3/8=0.375. Enter 0.375, not 37.5 (a percent value) or 3 (the count).

The number of correct select-all choices comes from the constraints, not from a guessed pattern. Numeric form is part of the task.

Select all real solutions of $\sqrt{x+6}=x$ from -3, -2, 0, 2, 3. Then enter the probability of drawing one marked counter from a bag with seven marked and thirteen unmarked counters, as a decimal.

Model and reasoning

3 only; 0.35. The original radical requires x nonnegative. Squaring gives $x^2-x-6=(x-3)(x+2)=0$. The candidate -2 fails both the sign condition and substitution. At 3 both sides equal three. For the probability, the denominator is twenty, so $P=7/20=0.35$. Enter a probability, not the percent value 35.

Select every listed integer satisfying $-2\le x<4$ and $x^2>1$: -3, -2, -1, 0, 1, 2, 3, 4. Then enter $7/12$ as a decimal rounded to the nearest hundredth.

Check after attempting

-2, 2 and 3; 0.58. The first restriction keeps integers -2 through 3. The strict squared inequality excludes -1,0,1. Four fails the strict upper endpoint; -3 fails the lower bound. The decimal $7/12=0.58333\ldots$ rounds to 0.58. The rounding instruction here is explicit; an exact fraction response would require the fraction representation instead. Check each listed value and retain every valid one.

An integer divisible by d can be written dk; a remainder r gives dk+r with 0≤r<d for a positive divisor. Prime factorisation identifies divisibility and common multiples. The greatest common factor takes shared prime powers at their smaller exponents; the least common multiple takes every needed prime power at its larger exponent.

Keep odd/even and positive/negative properties separate. A product of two odd integers is odd; a product with an even integer is even. A statement about integers need not hold for real numbers. For quantitative comparison, exploit a remainder form to expose possible cases rather than assume the smallest positive example is the only value.

A ratio a:b allocates a whole into a+b equal parts when the groups exhaust it. A percentage change multiplies the original value by 1+r or 1−r with r expressed as a decimal. Successive changes multiply factors. A part-of-a-part calculation changes the reference group at each stage; label the denominator before multiplying.

Estimate magnitude and check integer restrictions after solving. Fractions, roots and negative exponents follow their ordinary domain restrictions; even roots represent the nonnegative principal value. Do not cancel a term in a sum as if it were a factor. Exact arithmetic can be faster and safer than decimal calculator entries.

Known: $18=2\cdot3^2$ and $24=2^3\cdot3$. Shared smaller powers give GCF six; needed larger powers give LCM seventy-two. If $n=5k+2$, then $2n=5(2k)+4$, so the remainder on division by five is four.

For a model price of 100 CNY decreased then increased by 20%, multiply the two changes:

The price is not restored because the second percentage uses a different base.

Dividing 250 in the ratio 2:3 gives 100 and 150: each of five parts is fifty.

Equal percentage increases and decreases use different bases. A remainder must be reported relative to the named divisor.

Find the greatest common factor and least common multiple of 36 and 60. A price rises 25% then falls 20%. Find its overall percentage change.

Model and reasoning

$36=2^2\cdot3^2$, $60=2^2\cdot3\cdot5$. Shared smaller powers give GCF=12; larger required powers give LCM=180. Their product $12\cdot180=36\cdot60$ checks the result. The price multiplier is $1.25\cdot0.80=1$, so the overall change is zero percent. The two changes use different reference prices; adding +25 and -20 would give a wrong five-percent increase.

An integer n has remainder 5 on division by 8. Find the remainder of $3n+7$ on division by 8, including for negative n. A value rises 10% in one year and 20% the next. What single percentage reduction restores its original value?

Check after attempting

Write $n=8k+5$ for integer k. Then $3n+7=24k+22=8(3k+2)+6$, so the remainder is six for all integer k. The final value is 1.32 times the original. Restoring it multiplies by $1/1.32=25/33$, a reduction of $1-25/33=8/33$, or about 24.24%. A 30% reduction does not reverse compounded growth. The remainder convention uses $0\le r<8$ even when n is negative.

Write restrictions before manipulating an equation. A denominator must be nonzero; a real even root needs a nonnegative argument. Cancelling factors preserves the expression only on its original domain. Squaring can introduce candidates because a square loses sign information; verify every candidate in the unsquared relation.

Solving a system requires the same ordered pair to satisfy all relations. Substitution or elimination can expose one, no or infinitely many linear solutions. For an inequality, multiplication or division by a negative reverses direction. For a factored quadratic inequality, order roots and test interval signs, including endpoints only when equality is allowed.

A function assigns one output to each permitted input. Evaluate the inner function first in a composition, preserving domain restrictions. Read a slope as change in output per input unit and an intercept as the output at input zero only when that input is meaningful. A verbal model should name variables and keep quantities in compatible units.

For Quantitative Comparison, preserve all allowed values. A variable with x²=4 could be −2 or 2 unless constrained. One counterexample can reject a universal relationship; several favourable substitutions cannot prove it for every real value. Algebraic structure can establish a relationship without exhaustive numerical sampling.

For √(x+2)=x, x≥0. Squaring gives x²−x−2=0, candidates 2 and −1. Only 2 satisfies the original. For (x−1)(x−4)≤0, an upward quadratic is nonpositive on 1≤x≤4, including roots. In the system x+y=10, x−y=2, adding gives 2x=12; x=6 and y=4. For f(t)=2t+1 and g(x)=x², f(g(3))=2·9+1=19, applying g first.

A transformed equation supplies candidates, not automatic original solutions. Negative inequality multipliers and strict endpoints need explicit checks.

Solve $(x^2-4)/(x-2)=5$ on its original real domain. Solve $(x+1)(x-3)\le0$ and find $f(g(2))$ for $f(t)=t^2$, $g(x)=x-3$.

Model and reasoning

The rational equation excludes x=2. For allowed x it reduces to x+2=5, giving x=3, which substitutes correctly. The quadratic is nonpositive between its roots, including them: $-1\le x\le3$. The composition applies g first: $g(2)=-1$ and $f(g(2))=1$. Cancelling a factor does not restore excluded inputs.

For real a, classify the system $x+y=4$, $2x+ay=8$ as having one, none or infinitely many solutions. Compare with the second equation $2x+ay=9$. Give solutions in every consistent case.

Check after attempting

Subtract twice the first equation. In the first system $(a-2)y=0$: for a not equal to two, y=0,x=4, uniquely. For a=2, both equations are the same relation, giving every pair $(4-t,t)$. There is no inconsistent case in this first system. With right side nine, $(a-2)y=1$. For a not equal to two, $y=1/(a-2)$ and $x=4-1/(a-2)$, uniquely. For a=2 it demands 0=1, so no solution. A zero coefficient row means inconsistency only when the right side is nonzero.

Extract the facts a diagram supplies: lengths, right-angle marks, parallel lines, circle radii and stated shape properties. Avoid measuring the screen. Angle sums and similarity give relationships when their conditions hold. Pythagorean reasoning needs a right triangle; it does not apply to every drawn triangle.

For two coordinates, use midpoint ((x₁+x₂)/2,(y₁+y₂)/2) and distance √((x₂−x₁)²+(y₂−y₁)²). Slope is Δy/Δx when Δx≠0. A vertical line has no finite slope. Combine coordinate and shape facts where useful; perpendicular nonvertical lines have slopes whose product is −1.

A circle has circumference 2πr and area πr². An arc or sector uses its fraction of a full turn. Rectangular solids use volume lwh; cylinders use πr²h. Surface area counts exposed faces or surfaces, not the enclosed volume. Attach units so an area answer is not confused with a length or volume.

For similar figures with linear factor k, corresponding areas multiply by k² and volumes by k³. If a question provides only an area ratio, take a square root to recover the length ratio; do not copy the area ratio directly. General-test geometry uses elementary relationships, with no trigonometry or calculus required for this lesson’s tasks.

Points A(−1,2) and B(5,10) have Δx=6, Δy=8. Distance d=√(6²+8²)=10 and midpoint M=((−1+5)/2,(2+10)/2)=(2,6). For a circle of radius 6, a 60° sector is 1/6 of a full circle: area=(60/360)π6²=6π; arc length=(60/360)2π6=2π. Enlarging every length by 3 multiplies area by 9 and volume by 27, not 3.

Use labelled facts, include required π or units, and distinguish a perimeter from an area or a sector from its arc.

Points A=(-1,2) and B=(5,10) are given. Find their distance and midpoint. A right circular cylinder has radius 3 cm and height 4 cm. Find its volume and total surface area, including both ends.

Model and reasoning

Coordinate differences are six and eight. Distance is $\sqrt{6^2+8^2}=10$ and midpoint $(({-1}+5)/2,(2+10)/2)=(2,6)$. Cylinder volume is $V=\pi r^2h=\pi(3\ \mathrm{cm})^2(4\ \mathrm{cm})=36\pi\ \mathrm{cm^3}$. Total area is $S=2\pi r^2+2\pi rh=2\pi(3\ \mathrm{cm})^2+2\pi(3\ \mathrm{cm})(4\ \mathrm{cm})=42\pi\ \mathrm{cm^2}$. Volume and surface area measure different quantities.

A rectangle has diagonal 10 cm and one side 6 cm. A similar rectangle has area nine times as large. Find its perimeter. Explain which labelled facts permit Pythagoras, and whether the same calculation would hold for an arbitrary quadrilateral.

Check after attempting

A rectangle's adjacent sides are perpendicular. Its diagonal forms a right triangle, so the other side is $b=\sqrt{d^2-a^2}=\sqrt{(10\ \mathrm{cm})^2-(6\ \mathrm{cm})^2}=8\ \mathrm{cm}$. Area scale nine means length scale three. The new sides are 18 and 24 cm, giving perimeter $P=2(a+b)=2(18+24)\ \mathrm{cm}=84\ \mathrm{cm}$. An arbitrary quadrilateral need not supply a right angle, so its drawing alone would not justify the equation.

Mean is sum divided by count; median is the central ordered value or the mean of the middle two. Range is maximum minus minimum. An extreme value can alter mean and range without changing the median. Standard deviation measures spread about the mean; adding a constant changes the centre but not spread, while multiplying all values by c multiplies standard deviation by |c|.

Combine groups with their counts, not an unweighted mean of group means unless sizes are equal. A group of 10 with mean 6 contributes total 60; a group of 30 with mean 10 contributes total 300. The combined mean is (60+300)/40=9. A percentage of a subgroup uses that subgroup’s denominator.

For equally likely discrete outcomes, probability is favourable count over total count. Conditional probability restricts the denominator to the given condition. Without replacement changes later counts. Overlapping events need subtraction of the overlap when counting a union. Independence must be given or justified; it is not the same as mutually exclusive.

Read displays for their axes, units and sample description before making inferences. A correlation does not isolate a cause. A convenience sample may not represent the target population. Basic counting can use ordered multiplication when roles differ, or divide duplicated arrangements when order does not matter. Keep the events and outcomes explicitly defined.

A bag holds four red counters, three marked, and six blue counters, one marked. There are four marked counters in total. P(red)=4/10, P(marked)=4/10, P(red given marked)=3/4. P(two red without replacement)=(4/10)(3/9)=2/15, not (4/10)². For data 1,2,3,4,20, mean=30/5=6 and median=3. Adding 5 to each shifts mean and median by 5 but leaves their pairwise deviations and standard deviation unchanged.

A condition changes the reference population. Group means need weights, and no-replacement draws need changing denominators.

Group A has 12 observations with mean 5; group B has eight with mean 10. Find the combined mean. For the data 1,2,3,4,20 find mean, median and range. Describe how adding 7 to every value changes the standard deviation.

Model and reasoning

The combined total is $12(5)+8(10)=140$ across twenty observations, so mean is seven. An unweighted average of five and ten would incorrectly ignore group sizes. In the second data set, sum is thirty, mean six, median three and range nineteen. Adding seven shifts all values and their mean together; every deviation from the mean is unchanged, so standard deviation is unchanged.

Of 40 students, 24 study art, 18 study music and 10 study both. Find the probability of music given art and the probability of at least one subject. Are the events independent? Then explain what happens to standard deviation if all measurements are multiplied by -3.

Check after attempting

Conditioning on art gives $P(M\mid A)=10/24=5/12$. The union count is $24+18-10=32$, so union probability is 4/5. Independence would require $P(A\cap M)=P(A)P(M)$, but $10/40=1/4$ differs from $(24/40)(18/40)=27/100$. The events are not independent. Scaling every deviation by -3 scales its square by nine and the square root by three, so standard deviation triples. It cannot become negative.

The current standard GRE writing measure has one 30-minute Issue task. The issue and the instruction together define the response. Some instructions ask when a statement holds; some ask when a recommendation helps; others ask for the strongest challenge to your position. Identify the requested reasoning action before generating examples. A generic agree/disagree template can omit the central task.

A two-view instruction requires engagement with both views, not just a chosen side. A claim-and-reason instruction requires evaluation of the claim and the stated justification separately. The conclusion may be reasonable while its reason is weak, or vice versa. Make the relationship explicit in the thesis and development rather than leaving the reader to infer it.

For a policy instruction, explain consequences and how they shape your view. Separate intended benefits from plausible unintended costs. Use concrete mechanisms and circumstances: who benefits, what resources are needed, where the recommendation may fail and what limit improves it. Hypothetical examples can illuminate reasoning when labelled as such; unsupported invented statistics should not be presented as measured evidence.

Use a flexible plan, not a required paragraph count. A suggested practice budget is four minutes to interpret and outline, twenty-three to write and three to revise. Check task fit, development, logical organisation and clear language. An original classroom response is reviewed by a teacher; the local formative checks do not produce an official GRE writing score. Current Issue practice must not be relabelled as the retired Argument task.

Original claim-and-reason case: Claim: universities should let every student replace a laboratory requirement with independent reading. Reason: all meaningful learning comes from books. A qualified thesis might reject universal replacement while accepting reading as preparation. The reason overlooks learning from direct measurement, procedural errors and collaboration. Model development: “In a hypothetical introductory measurement course, a student may correctly describe uncertainty after reading but still misalign an instrument during use. A supervised trial exposes the difference between knowing a rule and applying it. Reading therefore supports the laboratory rather than always replacing it. A student with verified equivalent practical experience could reasonably receive an alternative task, so preserving a laboratory requirement need not mean ignoring individual circumstances.” This evaluates both the recommendation and its reason, gives a mechanism and introduces an explained exception.

Do not merely label an example impressive. Explain its connection to the claim, the required instruction and the limits of your conclusion.

Original Issue prompt. Claim: “Public learning centres should provide help only through written guides.” Reason: “Giving every visitor identical information is the fairest policy.” Discuss how far you agree with both the claim and its reason. Explain circumstances and consequences that affect your position.

Model and reasoning

Written guides can make public services clearer and more consistent, but they should not be the only source of help in a learning centre. I also question the reason offered for that policy. Fairness requires dependable information, yet giving everyone identical words does not ensure that everyone can use them. A better policy combines shared written guidance with appropriate personal support.

The argument has a real strength. Visitors should not receive contradictory instructions simply because different staff members answer their questions. A clear guide can state opening times, booking rules and available services in a form that visitors can consult again. In a hypothetical centre where staff previously gave different answers about course registration, a common written procedure could reduce confusion. It could also free staff from repeating routine explanations and allow them to spend more time on difficult questions. These are practical reasons to invest in guides.

However, consistency of information is different from equality of access. A visitor with limited reading ability may be unable to follow the guide. Another may understand each sentence but not know which rule applies to an unusual case. If both receive only the same document as a confident regular visitor, the centre has treated them identically while leaving them with different opportunities to learn. Personal explanation can remove that barrier without changing the underlying rule. Fairness therefore supports consistent decisions and suitable assistance, rather than one communication method for all.

The proposed restriction may also produce costs that its reason ignores. Suppose a visitor misunderstands a booking instruction and repeatedly submits an incomplete form. A short conversation might identify the difficulty immediately. Refusing that conversation can increase delay for the visitor and create extra correction work for staff. The centre should compare the full process, not just the time saved at the first enquiry. This example is hypothetical, but it explains a mechanism through which a seemingly efficient policy could waste resources.

There are circumstances in which written-only support may be reasonable for a limited service. A simple self-service room used by experienced members might need only clear notices and an accessible guide. Even then, the centre should provide a way to report an exceptional problem and check whether users actually understand the instructions. A limited case does not justify extending the restriction to every service and every visitor.

I would therefore retain written guides as the common reference and allow staff to explain and apply them. Staff should record recurring difficulties so the guide improves, and unusual decisions should remain accountable to shared rules. This approach accepts the value of identical core information while rejecting the claim that identical communication is always fair. It preserves consistency without making a visitor's ability to interpret a document the hidden condition for receiving help.

Compare three instruction families using this issue: (a) assess when a recommendation works, (b) discuss both a claim and its stated reason, (c) address the strongest challenge to your own position. Write a different planning sentence for each. Explain why a fixed agree/disagree template could miss the task.

Check after attempting

(a) “Written-only help may work for simple routine services used by experienced visitors, but fails when accessibility or unusual cases demand explanation.” (b) “I reject exclusive written help and distinguish the reason's valid concern for consistency from its unsupported equation of identical words with fairness.” (c) “The strongest challenge is that personal support may introduce inconsistent advice; shared rules, records and staff training can reduce that risk.” Other plans are valid if they perform the requested action. A generic position without conditions, a judgement on the reason or an answered challenge can omit the instruction even when its prose is fluent. Use human review for reasoning and language; local checks do not assign an official writing score.