Advanced Practice Test 3

Correct choice: sanguine

Despite two failures, Field “immediately began raising money” and “assured investors that a stronger cable would succeed.” He remained sanguine—hopeful and confident about the outcome—before the 1866 success.

A: “Ambivalent” means having mixed feelings, but the text describes Field acting decisively and assuring investors of success.

B: Field was vindicated only when the 1866 cable worked; the blank describes his attitude after the 1865 failure, before any cable had lasted, so “remained vindicated” reverses the order of events.

C: “Complacent” means contentedly satisfied and so not inclined to act, but Field responded to the failures by immediately organizing another costly attempt.

Watch for: Place the blank in the story’s timeline: a word that fits the ending may not fit the moment the blank describes.

Correct choice: exposed

The boys’ tallness made them easier to notice in their conspicuous uniforms, which is why W—— wanted to slip through “alleys and blind ways” to “elude notice.” Being “obnoxious to observation” here means being exposed, or liable, to it—an older sense of the word.

B: The quarrels arose because W—— cared a great deal about being noticed, so the boys were not indifferent to observation; the sentence describes how noticeable they were.

C: W—— disliked being watched, but the sentence says that the boys’ tallness made them more “obnoxious to observation”; height could not make them hostile to being seen, only more exposed to it.

D: “Offensive” is the modern sense of “obnoxious,” but the narrator is describing how easily the boys could be seen, not any offense they gave; being tall does not make anyone offensive to observation.

Watch for: In older prose, “obnoxious to” something often means “exposed” or “liable” to it; check what the context says actually happens to the people described.

Correct choice: To suggest why some English letters seem to stand for the wrong sounds by placing two developments of the same period side by side

The text opens with a puzzle—English letters seem to stand for the “wrong” sounds—and then sets two developments side by side: the long vowels changed between 1400 and 1700 while spellings were being fixed. It leaves readers to infer that spellings settled on during or before the shift still record the older sounds, so its purpose is to account for the puzzle.

A: The text does not say that the vowel shift created a need for standard spellings; it presents the two developments as happening in the same period without making one the cause of the other.

B: The text never says that printers modeled their spellings on other languages; the comparison with Spanish, Italian and French only sets up the puzzle.

C: The text states what Middle English vowels sounded like but says nothing about how linguists learned this.

Watch for: When a text lays two facts side by side without stating their connection, the purpose is often to let that connection explain something raised at the start.

Correct choice: It gives a result implying that, if the first sentence were true, the liver would have to turn food into blood far faster than seems possible.

If the tissues used up the blood and the liver replaced it from food, the liver would have to make more than the body’s entire supply of blood every half hour. The underlined result shows how implausible that is, which is why Harvey concluded that the same blood must circulate.

A: The calculation concerns how much blood passes through the heart, but its role in the text is to undercut the view that the liver makes blood continually, not to explain the heart’s workload.

C: The sentence compares the heart’s output with the body’s total blood; it does not claim that the body contains less blood than anyone believed.

D: Harvey drew the opposite conclusion: the same blood returns again and again, so the tissues do not use it up.

Watch for: Ask what the number would mean if the first view were true; a calculation often works by showing that a rival view demands something impossible.

Correct choice: It exposes a contradiction in Scythrop’s ambition, since the liberty he means to establish would depend on his own unlimited power.

Scythrop means to found “a perfect republic” served by “dispensers of liberty,” yet he gives himself “absolute sovereignty” over them. The sentence quietly exposes the contradiction—liberty imposed by an absolute ruler—and so mocks his reforming passion.

B: The castles were imaginary from the start (“castles in the air”); the sentence does not explain why they stayed imaginary, and nothing says Scythrop distrusted others.

C: The sentence does name Scythrop’s goal, a republic, but the narrator pairs it with “absolute sovereignty” to mock the scheme rather than to approve of it.

D: Nothing suggests that Scythrop confused his imagined societies; the sentence concerns the power he gives himself over them.

Watch for: Satire often works by setting two words side by side that cannot both be true of the same scheme—here “republic” and “absolute sovereignty.”

Correct choice: By arguing that the experiment tested a kind of change that Lamarck’s proposal did not concern and so could not by itself refute it.

Text 1 treats the mice’s normal tails as a refutation of Lamarck. Text 2 explains that Lamarck’s idea concerned changes an animal produces in itself through use or disuse, not injuries imposed from outside. The author of Text 2 would therefore say that cutting off tails tested something Lamarck never proposed, so the experiment could not by itself refute him.

A: Text 2 does not claim that Lamarck’s proposal has been confirmed or that Weismann’s results were wrong; it questions only what the results can show.

B: The number of generations is a possible objection, but Text 2 does not raise it; its point is about what kind of change the experiment tested.

D: Text 2 distinguishes changes produced by an animal’s own use or disuse of its organs from injuries inflicted from outside, and it says Lamarck’s account concerns only the former, so the author would not treat a cut tail as a fair test.

Watch for: Before accepting that an experiment refutes a theory, check that it tests what the theory actually claims.

Correct choice: They would argue that it cannot account for why the advantage appeared only among later entrants who began signing as young children.

If longer use alone refined the language, everyone signing in the community later on should have benefited, including later entrants who started as teenagers. Instead, the advantage appeared only in later entrants who started as young children—and those signers outdid first-group signers who had been signing longer. The change depended on young children learning the language, which elapsed time alone cannot explain.

A: Text 2 reports that the later entrants who began young used locations more consistently, so the language grew more systematic, not less.

B: It is true that later entrants learned an older language, but older-starting later entrants learned the same language and showed no advantage, so the language’s age cannot be what made the difference.

D: Neither text mentions spoken language, so nothing suggests that Senghas and Coppola would raise it.

Watch for: When a second text reports only a comparison, ask which explanation the comparison rules out: here, time in use is the same for groups that differ in result.

Correct choice: They would argue that the drugs’ success does not show that acid is the underlying cause, since ulcers usually returned if the bacterium remained.

Text 1 reasons that because reducing acid healed ulcers, acid must be the cause. Text 2 shows that ulcers healed that way usually returned, while eliminating the bacterium kept them away. Relief from treating a symptom does not show what caused the disease; Warren and Marshall would point to the bacterium.

A: Text 2 says the antibiotics were given in addition to acid-reducing drugs, so it offers no evidence that antibiotics alone healed ulcers or that acid plays no part.

B: Text 2 grants that the drugs healed ulcers, but its trials show that the ulcers usually came back if the bacterium remained, which undercuts the inference that acid was the underlying cause.

C: Text 2 says nothing about stress or about how the bacterium grows; this links the two texts’ explanations in a way neither text supports.

Watch for: A treatment that relieves a condition does not by itself reveal the condition’s cause; look for evidence about what happens when the treatment stops.

Correct choice: Parry traced the fixed phrases in Homer’s epics to the needs of poets composing aloud, which explains why a phrase can appear where it does not suit the moment.

The text raises a puzzle (phrases that do not fit the moment), reports Parry’s finding that each phrase is tied to a position in the rhythmic line, and ends with his view that the poems come from a tradition of composing aloud. Together these explain the puzzle: a poet composing in performance reached for the phrase that fit the meter, whatever the character was doing.

A: The text attributes the mismatches to how the poems were composed, not to multiple authors; Parry’s finding that each phrase is tied to a metrical position points to the needs of composition.

B: This offers an explanation that the text does not give: Parry’s finding is that the choice of phrase depends on its position in the line, not on a quality the poet wished to stress.

D: The first part matches Parry’s finding, but the text says the poems grew out of a tradition of poets composing aloud, not that they prove a single poet working from memory.

Watch for: A main-idea choice must connect the puzzle and the explanation; choices that offer an explanation the text never gives are traps, however natural they sound.

Correct choice: “But nothing local, as one may say. / There couldn’t be,—for the Deacon’s art / Had made it so like in every part / That there wasn’t a chance for one to start.”

The claim has two parts: no single part wore out first, and the narrator traces this to the deacon’s building. These lines say the decay was “nothing local” and explain why: the deacon’s art had made the carriage “so like in every part” that no part could fail first.

A: These lines state the general rule that ordinary carriages have a weakest spot somewhere; they say nothing about this carriage’s condition or why it had no weak spot.

C: These lines describe the carriage’s sudden collapse, not its condition that morning or the cause the narrator gives for it.

D: These lines establish that the carriage had aged, and “general” decay at most hints at even wear; they do not state that no single part wore out first, and they give no cause.

Watch for: Match every part of the claim, including any cause it names; a vivid line about the same event can still leave half the claim unsupported.

Correct choice: Blindfolded patients reacted when told they were being treated even when no one was treating them, but did not react when treated without being told.

To support the hypothesis, expectation and treatment must be separated. Blindfolded patients reacted when they only believed they were being treated and did not react when they were treated without knowing it, so the reactions followed belief, not treatment.

A: Patients who expected effects also received the treatment, so their reactions fit a real fluid just as well as they fit expectation; this finding does not separate the two.

C: Mesmer might transmit more of the supposed fluid, or might inspire stronger expectations; this finding fits both explanations.

D: Improvement after repeated treatment fits either explanation and does not separate expectation from the supposed fluid.

Watch for: The strongest evidence separates two possible causes, showing the effect with one cause alone and its absence with the other alone.

Correct choice: conflict added 47.0 seconds to the time needed to name ink colors but only 2.3 seconds to the time needed to read color words.

The conclusion compares how much conflict slowed each task. Conflict raised color-naming time from 63.3 to 110.3 seconds, an increase of 47.0, but raised word-reading time only from 41.0 to 43.3 seconds, an increase of 2.3.

B: These figures are accurate, but they compare the two tasks under conflict only; without the times for each task without conflict, they cannot show how much conflict slowed either task.

C: This is accurate but gives only the small effect of conflict on word reading, without the large effect on color naming that the conclusion contrasts it with.

D: This has the right shape, but it misreads the table: 69.3 is the difference between color naming with conflict (110.3) and word reading without conflict (41.0); conflict added 47.0 seconds to color naming (63.3 to 110.3).

Watch for: When a claim compares the effect of a condition on two tasks, compare each task with itself (with versus without), not the tasks with each other.

Correct choice: students recalled less of the tested passage than of the reread one after five minutes (75 versus 81 percent) but more after two days (68 versus 54 percent) and after one week (56 versus 42 percent).

The conclusion has two parts: the benefit appeared only after a delay (the tested passage was recalled less well at five minutes, 75 versus 81 percent, but better later), and it did not shrink (the advantage was 14 points after two days, 68 versus 54, and still 14 after one week, 56 versus 42). Only this choice gives all three delays correctly.

A: These figures are accurate and show a lasting advantage for testing, but they omit the five-minute result, which is needed to show that the benefit appeared only after a delay.

B: This has the right shape and would support the conclusion, but it misreads the graph: after one week, students recalled 56 percent of the tested passage, not 58.

D: This shows that the benefit appeared only after a delay, but by skipping the two-day result it cannot show that the benefit held steady rather than shrinking over time.

Watch for: A claim about how an effect changes over time needs every time point the claim covers; a choice that skips one can look complete.

Correct choice: the gas obtained from air contained, in addition to nitrogen, a small amount of some heavier gas that had not been removed.

The gas from air was the denser one, so the difference points to something heavier in the air sample or something lighter in the chemical samples. Different reactions gave nitrogen of exactly the same density, which would be a remarkable coincidence if each reaction added its own impurity. The better explanation is that the air sample contained an unknown heavier gas. (Rayleigh and William Ramsay soon isolated it: argon.)

A: A heavier gas mixed into the chemical nitrogen would make it denser, but the chemical nitrogen was the lighter sample.

B: A lighter gas mixed into the air sample would make it less dense, but the air sample was the denser one.

C: This would explain the direction of the difference, but it is hard to square with the finding that several different reactions all gave nitrogen of exactly the same density.

Watch for: Check each explanation against both the direction of a difference and any result that stays constant across different methods.

Correct choice: owners of reliable cars, unwilling to sell below their cars’ worth, would withdraw them, lowering the average quality of cars for sale and so the price buyers offer.

An average price is too low for owners of reliable cars, so they keep their cars. That leaves a worse mix for sale, which lowers the average quality and so the price buyers will offer, which drives out still more good cars.

B: This reverses the incentives: an average price is more than an unreliable car is worth, so its owner is glad to sell; it is the owners of reliable cars who are shortchanged.

C: The text says buyers offer a price reflecting the average quality of cars for sale; if the mix stayed the same, that price would not keep falling, so this describes a spiral with no cause in the situation described.

D: Because buyers cannot tell reliable cars apart, cutting the price of a reliable car would not attract buyers to it; the decline comes from reliable cars leaving the market.

Watch for: For a process that “feeds on itself,” trace one step through: who gains or loses at the current price, what they do, and how that changes the next price.

Correct choice: The speaker sees in it a clue to a happiness that he has long wondered when he will reach, though he cannot say how it works.

The poem sets Cliff’s unexplained act—quietly taking the bitter glass and giving his guest the wine—beside the speaker’s long wondering “when I shall be / As happy as Cliff.” The juxtaposition implies that the speaker links Cliff’s happiness to that habit, which Cliff will not explain (“it was a way of his”), and that the speaker has not reached that happiness himself.

A: The speaker calls Cliff happy and wonders when he himself will be as happy; nothing suggests he thinks Cliff is secretly unhappy.

B: The speaker notes that he was given no choice and asks what Cliff meant, but he shows puzzlement and admiration, not resentment of the habit.

D: The speaker is still “a-wondering when I shall be / As happy” as Cliff; nothing says he has adopted the habit himself.

Watch for: In a short poem, the final lines often reinterpret an earlier act; ask why the speaker places that act beside the closing thought.

Correct choice: has

The subject is “more than one scholar.” Although it refers to several people, the phrase is built on the singular noun “scholar” and takes a singular verb: “more than one scholar has asked.”

B: “Have” is plural; “more than one” followed by a singular noun takes a singular verb.

C: “Were asked” is plural and passive, and it would make the scholars the ones being asked rather than the ones asking.

D: “Are asked” is plural and passive, and it reverses who does the asking.

Watch for: “More than one” + a singular noun takes a singular verb, even though the meaning is plural.

Correct choice: author, nor has anyone

“Nor” joins two independent clauses here, so a comma goes before it, and the clause it introduces inverts its subject and helping verb: “names no author, nor has anyone succeeded.”

A: When “nor” joins two independent clauses, a comma is needed before it; without one, the two clauses run together.

C: A clause that begins with “nor” must invert its subject and helping verb (“nor has anyone”); “nor anyone has” keeps ordinary word order.

D: “Nor” is a coordinating conjunction, so it takes no comma after it, and a semicolon before a coordinating conjunction is not needed here.

Watch for: A clause beginning with “nor” puts the helping verb before the subject: “nor has anyone,” “nor did they.”

Correct choice: corrected

The sentence reports in the past what the engineers knew: “unless they ______ the flaw, the … images would remain blurred.” The result clause takes “would,” and the condition takes the simple past: “unless they corrected the flaw.”

A: “Would” belongs in the result (“the images would remain blurred”), not in the “unless” clause that states the condition; the condition takes the simple past, “corrected.”

B: “Will correct” is future tense, but the sentence reports, in the past, what the engineers knew in 1990; the condition must shift to the past.

D: “Have corrected” is present perfect, which does not fit a condition reported from the engineers’ point of view in 1990.

Watch for: In a condition introduced by “if” or “unless,” do not use “would”; put “would” in the result clause.

Correct choice: tracing

In “was devoted to,” “to” is a preposition (compare “devoted to its readers”), so the verb that follows must take the -ing form: “devoted to tracing.”

A: In “devoted to,” “to” is a preposition, so it must be followed by a noun or a gerund, not the base form of a verb.

C: “Traced” is a past-tense or past-participle form, which cannot serve as the object of the preposition “to.”

D: “Have traced” is a finite verb phrase, which cannot follow the preposition “to” in “devoted to.”

Watch for: When “to” is part of a phrase such as “devoted to” or “committed to,” it is a preposition and takes an -ing form.

Correct choice: However

Here “However” means “in whatever way”: “However the builders lifted … into place, they left no … account.” It introduces a dependent clause, so no punctuation follows it; the comma after “into place” ends the dependent clause.

B: A semicolon cannot follow a single introductory word; outside of complex lists, a semicolon joins two independent clauses.

C: With a comma, “However” becomes a transition, and “the builders lifted so many of these blocks into place” becomes an independent clause joined to the next one by only a comma—a comma splice.

D: A colon must follow an independent clause, and “However” alone is not one.

Watch for: When “however” means “in whatever way,” it begins a dependent clause and takes no comma after it.

Correct choice: This kind

The subject names one kind of dye, and its verb is the singular “was,” so both the demonstrative and the noun must be singular: “This kind of dye … was.”

B: “These” is plural, but it modifies the singular noun “kind,” and the verb “was” is singular; the demonstrative must match the noun: “This kind.”

C: “These kinds” is plural, but the verb “was” and the singular reference to one dye require a singular subject.

D: “This” is singular and cannot modify the plural noun “kinds.”

Watch for: A demonstrative must agree with its noun—“this kind,” “these kinds”—and the pair must agree with the verb.

Correct choice: Alternatively,

The text presents two different possible causes for the same rings: rising methane and warm-water eddies. “Alternatively” introduces a second, competing explanation.

A: “Similarly” would present the eddies as another example of the same process, but they are offered as a different cause than the methane explanation.

B: “Conversely” signals the reverse of what came before, but the eddy explanation is not the opposite of the methane explanation; it is another possible cause of the same rings.

C: “In turn” signals a next step in a chain of effects, but the later study proposes a rival cause, not a consequence of the methane.

Watch for: When a second sentence offers a different answer to the same question, look for a transition that signals an alternative, not a reversal.

Correct choice: Rather,

The first sentence says what the images are not (woven); the second says what they are (embroidered). “Rather” introduces a statement that replaces the one just denied.

A: “Nevertheless” signals that something holds despite what came before, but being embroidered is not something true in spite of not being woven; it is the correct description in place of the rejected one.

B: “Similarly” signals a likeness, but embroidery is presented as the alternative to weaving, not as something like it.

D: “Consequently” would make the embroidery a result of the images not being woven, but the text is correcting a description, not tracing a cause.

Watch for: After a sentence that denies one description, “Rather” introduces the accurate one.

Correct choice: Zhang Heng’s device reportedly signaled an earthquake that no one in the capital had felt, though the official history does not describe how its internal mechanism worked.

The goal has two parts: what the device reportedly accomplished (signaling an earthquake no one in the capital felt) and uncertainty about how it worked (the official history does not describe its mechanism). Only this choice gives both.

A: This acknowledges the uncertainty about the mechanism but omits what the device is reported to have accomplished.

B: This gives the reported accomplishment but says nothing about uncertainty over how the device worked.

C: This gives the reported accomplishment and describes the device’s outward signal, but it acknowledges no uncertainty about how the device worked—if anything, it makes the mechanism sound known.

Watch for: Describing what a device visibly did is not the same as acknowledging uncertainty about how it worked inside.

Correct choice: Having photographed more than 5,000 snow crystals, Wilson Bentley reported that he had never found two that were alike.

The goal pairs Bentley’s claim (no two crystals alike) with the extent of the evidence behind it (more than 5,000 crystals photographed). Only this choice gives both.

A: This presents the claim together with Bentley’s method, but it gives no indication of how many crystals he examined, which is what “how much evidence” asks for.

C: This describes Bentley’s first photograph and his method but not his claim about snow crystals.

D: This reports a number of published photographs but not Bentley’s claim, and the published selection is not the full body of evidence behind it.

Watch for: “How much evidence” asks for a scale—here, the number of crystals examined—not simply a method or a publication.

Correct choice: Because most characters in Sequoyah’s writing system stand for whole Cherokee syllables, a speaker could sound out most words, and within a few years thousands of Cherokee could read and write.

The goal has two parts: why the system could be learned quickly (most characters are whole syllables, so learning them lets a speaker sound out most words) and evidence that it was (thousands could read and write within a few years). Only this choice gives both.

A: This describes features of the system, but the resemblance to Latin letters does not explain quick learning, and the choice gives no evidence of how fast people learned it.

B: This gives the evidence of rapid learning but not the reason the system could be learned quickly.

C: This gives the reason but not the evidence that people in fact learned the system quickly.

Watch for: A goal that asks “why” and “evidence that” needs both a cause and a result; either one alone meets only half the goal.

Correct choice: equivocal

The colon introduces the explanation: in Italian, canali can mean either natural channels or artificial canals. A term open to two interpretations is equivocal.

A: Nothing suggests that canali was out of date; the text says it is still a word with two meanings in Italian.

B: The term was not wrong; the problem was that it could be read two ways and translators chose one reading.

D: “Evasive” implies a deliberate attempt to avoid being pinned down, but the text says only that the Italian word has two meanings, not that Schiaparelli chose it to dodge the question.

Watch for: A colon after the blank usually defines the missing word; match the definition exactly, and do not add intent that the text does not state.

Correct choice: irrelevant

Hazlitt holds that everything should be referred to “a source and principle of its own … and not to the principles of other things,” and that heroes, legislators and explorers cannot be ranked by a scholar’s standard. Comparisons are “impertinent” in the older sense of “not pertinent”: they judge one kind of achievement by a standard that does not apply to it.

B: Hazlitt does say that comparisons lead to “the discovery of defects,” but he gives that as a second, separate charge (“and lead only to …”); “impertinent” names a different fault.

C: “Insolent” reflects the modern sense of “impertinent” (rude), but Hazlitt’s complaint is that comparisons judge things by “the principles of other things,” not that they are rude.

D: Nothing in the text suggests that comparisons are tedious; the objection concerns the standard they apply.

Watch for: When a sentence gives two charges joined by “and,” the word in the first charge usually means something different from the second; look back for the principle the writer has just stated.

Correct choice: graciousness

The narrator excepts from his satire high-ranking people who, by a well-guided condescension, make their superiority easy to bear for those below them, and he gives an example of a man who is both “patron and … companion.” “Condescension” here is used in its older, approving sense: gracious, friendly courtesy shown to people of lower rank.

A: “Arrogance” reflects the modern, disapproving sense of “condescension,” but the narrator praises these people as a credit to their rank who make their superiority “as easy as possible” for those below them.

B: “Deference” is courteous respect shown to someone of higher rank, but here the courtesy runs downward, from high people to “those whom fortune … hath placed below them.”

D: “Discretion” means careful judgment or tact, which suggests restraint; the passage stresses warm, open kindness—conversing “with freedom” and being “a patron and a companion.”

Watch for: Some words once carried praise that they no longer do; check whether the passage approves or disapproves of the quality it names.

Correct choice: It shows that the islands’ return to their earlier species counts did not depend on the return of the same species that had lived there before.

Each island’s species count came back to about its old level, yet many of the species were new. The underlined sentence separates the number of species from their identity: the islands returned to a similar number without regaining the same residents, which suggests that something about each island, not the particular species, set the number.

A: The sentence says only that the species were different; nothing in the text suggests the newcomers were better suited than the earlier residents.

C: The final sentence links the number of species to nearness to the mainland, but the underlined sentence concerns which species arrived, not what set the islands’ counts.

D: The species count had already returned to its earlier level, so the islands were not still waiting to fill up; the earlier species had been replaced, not merely delayed.

Watch for: When a “however” sentence follows a result, ask what it rules out: here, that the count recovered because the same species came back.

Correct choice: It backs up the children’s judgment that the elders wasted their freedom, with examples revealing what the children themselves counted as pleasures.

The children accuse the elders of wasting their “absolute licence to indulge in the pleasures of life.” The underlined list names the pleasures the elders neglect—dabbling in the pond, chasing chickens, firing cannons—which are a child’s pleasures. The examples support the charge while showing, with the adult narrator’s humor, that it rests on the children’s own standard.

B: The adult narrator’s humor is gentle; he presents the children’s view with evident affection, and nothing calls their pleasures foolish or destructive.

C: The text presents these freedoms as belonging to adults generally—“absolute licence to indulge in the pleasures of life”—and says nothing about the elders’ lack of children.

D: Nothing in the excerpt says the narrator now shares the elders’ view; the underlined portion presents the children’s judgment.

Watch for: When an adult narrator reports a child’s judgment, check whose standard the examples reflect; the humor often lies in that standard.

Correct choice: They would argue that the lower death rate may reflect the more dangerous causes of low birth weight among nonsmokers’ babies, not a benefit of smoking.

Text 1 compares only babies born small. Text 2 explains that among these babies, the nonsmokers’ babies are more likely to be small for a more dangerous reason. Their higher death rate can therefore come from those other conditions, so the comparison does not show that smoking protects anyone.

B: That smoking-related low birth weight is less dangerous does not mean smoking protects anyone; Text 2 uses this point to explain the comparison, not to endorse the observers’ interpretation.

C: Text 2 does not say how common each cause of low birth weight is; it explains why comparing only small babies can mislead.

D: Text 2 says nothing to suggest that birth weight is unimportant; it concerns why small babies of nonsmokers fare worse.

Watch for: When a comparison is limited to cases that share an outcome, ask whether the groups reached that outcome for different reasons.

Correct choice: He would argue that the contrast fits a link between usefulness and exchange value once usefulness is judged by one more unit of a good, not by the good as a whole.

Smith compares the usefulness of water as a whole with that of diamonds. Menger measures value by the least important use of the supply one has—in effect, by what one more unit adds. Because water is plentiful, one more bucket adds little; because diamonds are scarce, one more diamond adds much. The contrast is therefore consistent with usefulness governing value, which is exactly what Smith denied.

A: Menger ties value to the use served by one more unit, so the plentiful supply of water explains its low value through usefulness rather than showing that usefulness plays no part.

C: Text 2 says nothing about labor; it explains value through the use served by one more unit.

D: Scarcity matters in Text 2, but only because it changes the use that one more unit serves; Menger does not set scarcity against usefulness—he explains value through usefulness at the margin.

Watch for: A paradox can dissolve when a second text measures the key quantity differently; compare what each text is measuring—the whole supply or one more unit.

Correct choice: They would argue that the link may largely reflect children’s backgrounds and early abilities rather than an effect of waiting itself.

If waiting itself caused later achievement, the link should remain after accounting for other differences among children. Instead, it largely disappeared once family background and early cognitive ability were taken into account, which suggests that those factors shape both waiting and achievement. The researchers would therefore doubt that waiting is a major cause.

A: Text 2 says nothing about behavior outside the laboratory; its evidence concerns other differences among the children.

B: This overstates Text 2: the link shrank sharply but did not disappear entirely, so the findings do not show that there is no connection at all.

C: They did find a link, but a weaker one that shrank sharply once other factors were considered, which undercuts the claim that waiting is a major cause.

Watch for: When a link weakens after other factors are “taken into account,” those factors may explain the link; that is a challenge to a causal claim, not an endorsement.

Correct choice: Imlac maintains that a poet should study nature widely but depict only those features that any reader, attentive or not, would recognize.

Imlac says he studied “all the appearances of nature,” and when the Prince objects that he must have missed much, Imlac explains that a poet does not record minute details but presents “prominent and striking features” that recall the original “to every mind.” The main idea combines both: study widely, depict the general.

A: The remark about leaving “much unobserved” is the Prince’s, and Imlac answers it by explaining what a poet should depict, not by conceding that his study was careless.

C: Imlac says he was “careful to study” all the appearances of nature; he does not treat study as unimportant, only the recording of minute details.

D: This reverses Imlac’s view: he says a poet “does not number the streaks of the tulip” and “must neglect the minuter discriminations.”

Watch for: Keep track of who says what in a dialogue, and look for how a speaker’s answer reconciles two claims that seem to conflict.

Correct choice: Metal chips from the boring could absorb as much heat per pound as solid metal from the same cannon.

The defenders’ explanation rests on the premise that chips hold less caloric than solid metal. If chips could absorb just as much heat per pound as solid metal, that premise fails, and the explanation loses its basis. (Rumford reported just such a result.)

A: On the defenders’ view, longer boring produces more chips and so releases more caloric; a cannon growing hotter over time fits their explanation.

B: Water boiling shows that a great deal of heat was produced, but the defenders’ explanation already accounts for heat being released; it does not test their premise.

D: If caloric is squeezed out of the chips, the chips should be hot; this finding fits the defenders’ explanation.

Watch for: To weaken an explanation, attack the premise it depends on; findings that the explanation already predicts do not weaken it.

Correct choice: When one person in the group gave a wrong answer different from everyone else’s, participants went along with the rest of the group far less often.

To show that unanimity itself matters, the group must stop being unanimous without the participant gaining an ally. A dissenter who gives a different wrong answer does exactly that: the group is no longer unanimous, yet no one shares the participant’s view. If conformity still falls sharply, unanimity is what mattered.

A: This concerns the number of people giving the wrong answer, a factor Asch’s proposal set aside, and it does not involve breaking unanimity.

B: A dissenter who gives the correct answer breaks unanimity but also gives the participant an ally, so this result cannot show which of the two mattered.

D: This shows that the task was easy without group pressure, but it does not distinguish unanimity from the other possible sources of that pressure.

Watch for: Look for the condition that changes only the factor in question; a result that changes two factors at once cannot show which one mattered.

Correct choice: only 2 of the 649 men with lung cancer were nonsmokers, compared with 27 of the 649 men with other diseases.

Because nearly everyone smoked, the telling comparison is among the few nonsmokers: 27 of the men with other diseases were nonsmokers, but only 2 of the men with lung cancer were. Nonsmokers were far rarer among the cancer patients, which is what an association between smoking and lung cancer predicts.

A: This is accurate, but on its own it cannot show an association: 622 of the 649 men with other diseases were also smokers, so a high count of smokers among the cancer patients is expected either way.

C: This is accurate (647 of 649 and 622 of 649), but it stresses the similarity between the groups rather than the difference that supports the conclusion.

D: This has the right shape and would support the conclusion, but it overstates the table: 27 nonsmokers against 2 is a ratio of 13.5, not more than twenty.

Watch for: When almost everyone in both groups shares a trait, compare the groups on the rare cases—those without the trait.

Correct choice: Reported cases fell from 441,703 in 1960 to 86 in 2000, although they rose from 13,506 in 1980 to 27,786 in 1990.

The claim has two parts: cases fell dramatically (441,703 in 1960 to 86 in 2000) and the fall was not continuous (cases rose from 13,506 in 1980 to 27,786 in 1990). Only this choice supports both parts accurately.

B: This is accurate and shows that the decline was interrupted, but it omits the dramatic overall fall that the claim also asserts.

C: This is accurate and supports the claim’s first part, the dramatic fall, but not its second part, that the decline was not continuous.

D: This has the right shape but misreads the table: cases fell from 47,351 in 1970 to 13,506 in 1980; the rise came between 1980 and 1990.

Watch for: When a claim has an “although” clause, the evidence must support both parts, and each number must be matched to the right year.

Correct choice: each male would, on average, have more offspring than each female, so parents inclined to produce sons would leave more grandchildren, and the share of males would rise.

Every offspring has one father and one mother, so males as a group and females as a group have the same total number of offspring. If males were fewer, that total would be divided among fewer individuals, so the average male would have more offspring than the average female. Parents producing sons would then leave more grandchildren, and the share of males would rise until the sexes were about equal.

A: This reverses the reasoning: if males were the rarer sex, the shared total of offspring would be divided among fewer males, so each male, not each female, would have more offspring.

B: The first part is right, but it draws the wrong consequence: the advantage of the rarer sex favors parents who produce sons, so the share of males would rise rather than stay the same.

D: The text says that a population with few males could produce as many young as one with equal numbers, so the population would not produce fewer young.

Watch for: When two groups must share a fixed total equally, the smaller group has the higher average; trace which parents that favors.

Correct choice: wine made with the labor of 80 Portuguese workers could be traded for cloth that would have taken the labor of 90 Portuguese workers to make.

What matters to Portugal is what it gives up and what it gets. By making wine with 80 workers’ labor and trading it, Portugal obtains cloth that would have cost it 90 workers’ labor to make itself, a saving of 10 workers’ labor on each trade—even though Portugal makes cloth more cheaply than England does.

A: This compares English cloth with Portuguese wine, but Portugal’s gain depends on comparing the labor it spends on wine with the labor it would spend making cloth itself.

C: Subtracting the two countries’ advantages does not measure Portugal’s gain; Portugal saves the difference between 90 and 80 workers’ labor, or 10, on each trade.

D: This concerns England’s costs and misstates the comparison: English cloth took 100 workers’ labor, more than the 90 needed in Portugal.

Watch for: A country gains from trade when it can obtain a good more cheaply by trading for it than by making it; compare costs within one country, not across countries.

Correct choice: It is a claim about perception: the fogs were present, but no one truly saw them until art taught people to notice their beauty.

Vivian grants that “there may have been fogs for centuries” but says “no one saw them.” By “exist” he means being truly seen—noticed for their beauty—which, he claims, art made possible. The paradox is about perception, not about records, taste or weather.

A: This makes the change one of taste, but Vivian says that “no one saw” the earlier fogs at all, not that people saw them and disliked them.

C: Vivian does say “we do not know anything about them,” but only because “no one saw them”; his point concerns seeing, not the lack of records.

D: Vivian concedes that “there may have been fogs for centuries,” so he is not claiming that the weather changed to match art.

Watch for: When a speaker states a paradox, look for a nearby concession (“I dare say there were”) that shows the claim cannot be meant literally.

Correct choice: each other’s

Each man relied on the other’s expertise, so the reciprocal pronoun must be possessive: “each other’s expertise.” Because “each other” is singular in form, the apostrophe comes before the s.

A: “Each other” is singular in form, so its possessive is “each other’s”; “each others’” treats it as a plural noun.

B: The expertise belongs to the two men, so a possessive is needed, and “each other” has no plural form “each others.”

C: Without an apostrophe, “each other” is not possessive, so “relied on each other expertise” does not say whose expertise it was.

Watch for: The possessive of “each other” is “each other’s,” never “each others’.”

Correct choice: woman to reach

“To reach the summit of Mount Everest” identifies what Tabei was the first woman to do; it is essential to “the first woman,” so no punctuation separates them.

A: The infinitive phrase “to reach the summit …” is essential to the meaning (it says in what way she was first), so no comma should separate it from “woman.”

B: A colon would introduce an explanation or list, but the infinitive phrase is a necessary part of the noun phrase “the first woman to reach the summit.”

C: A semicolon must join two independent clauses, but “to reach the summit of Mount Everest” is only a phrase.

Watch for: After “the first,” “the only” or “the last,” an infinitive phrase is usually essential and takes no comma.

Correct choice: unearthed

The main verb of the sentence is “record,” so the blank must be a modifier of “Clay tablets.” The tablets were dug up by others, so the passive participle “unearthed” is correct: “Clay tablets unearthed at … Ugarit … record a script.”

A: “That unearthed” makes the tablets the ones that did the digging, and it leaves the relative clause without an object.

B: “Were unearthed” adds a second main verb, leaving “record” without a grammatical connection to the sentence.

D: “Unearthing” would mean that the tablets themselves did the digging; the tablets were dug up, so a passive form is needed.

Watch for: Find the sentence’s main verb first; a word describing the subject must then be a modifier, and its form must match who did the action.

Correct choice: Zealand, in

When a town is followed by its country, the country is set off by commas on both sides. The comma after “Brightwater” is already present, so a comma must follow “New Zealand” as well.

A: A colon would introduce an explanation or list, but “in 1871” simply continues the sentence.

B: A semicolon must join two independent clauses, but “in 1871 …” is not a clause.

C: Without a comma after “New Zealand,” the pair of commas that sets off the country is left incomplete.

Watch for: In “Town, Country,” the country is set off like an appositive: comma before and comma after.

Correct choice: viscous that

“So viscous that …” is a single construction: “so” sets up a degree, and the “that” clause gives its result. No punctuation separates “viscous” from “that.”

B: A semicolon joins independent clauses, but “that … only nine drops fell …” is a dependent clause that completes “so viscous.”

C: A colon would introduce an explanation after a complete statement, but “one so viscous” is not complete until the “that” clause finishes it.

D: A comma would split the “so … that” construction, separating the degree from the result it introduces.

Watch for: Keep “so … that” and “such … that” together without punctuation; the “that” clause completes the construction.

Correct choice: an ostrich’s

The sentence compares a kiwi’s egg with another bird’s egg, so the comparison must name the egg: “an ostrich’s” means “an ostrich’s egg.”

A: This compares the kiwi’s egg with an ostrich itself rather than with an ostrich’s egg, an illogical comparison.

B: “An ostrich does” compares the egg with something an ostrich does rather than with an ostrich’s egg.

C: “Ostriches” names the birds, not their eggs, so the comparison is still illogical.

Watch for: In a comparison, make sure like is compared with like: an egg with an egg, not an egg with a bird.

Correct choice: Ultimately,

The first sentence lists obstacles that arose over a decade of work; the second reports the final outcome, the canal’s opening in 1869. “Ultimately” introduces the end result of a long process.

A: “Initially” would place the opening at the start of the process, but the opening in 1869 came after the obstacles of 1859–1865.

B: “Consequently” would make the canal’s opening a result of the opposition, labor shortages and disease, but the canal opened despite them.

D: “Meanwhile” signals something happening at the same time as the preceding events, but the opening came after them.

Watch for: Check where the second sentence falls on the timeline: an end result after a series of events calls for a word like “Ultimately.”

Correct choice: By the same token,

The distant train and the highway are two effects of the same cause: sound bent back toward the ground by the warm layer. “By the same token” introduces a second consequence that follows for the same reason.

A: “Nevertheless” signals that something holds despite what came before, but the highway’s noise carrying farther follows from the same bending of sound, not in spite of it.

B: “In other words” would introduce a restatement of the train example, but the highway example is a second, separate consequence.

C: “Instead” signals a replacement, but the highway example adds to the train example rather than replacing it.

Watch for: When a second example follows from the same explanation as the first, look for a transition of parallel reasoning.

Correct choice: The Green Belt Movement paid the women who planted its seedlings for each one that survived, and by the early 2000s its groups had planted tens of millions of trees.

The goal has two parts: the incentive (payment for each seedling that survived) and the movement’s accomplishment (tens of millions of trees). Only this choice gives both.

A: This describes the movement and the benefits of its trees but neither how planters were encouraged to keep seedlings alive nor how much the movement accomplished.

C: This explains the incentive for keeping seedlings alive but does not indicate how much the movement accomplished.

D: This indicates the movement’s scale and Maathai’s honor but not how planters were encouraged to keep their seedlings alive.

Watch for: Match each part of the goal to a specific note; an honor such as a prize is not the same as a measure of what was accomplished.

Correct choice: Though Ramanujan had little formal training and no degree, Hardy, persuaded by theorems in Ramanujan’s letter that seemed beyond invention, arranged for him to come to Cambridge.

To show that Hardy’s judgment rested on the letter rather than on credentials, a choice must set Ramanujan’s lack of credentials against Hardy’s response to the letter’s theorems. Only this choice does both.

B: This describes the letter’s contents but not Hardy’s judgment or Ramanujan’s lack of credentials.

C: This mentions Ramanujan’s lack of credentials and Hardy’s response, but it presents them only as a sequence of events; it does not say that Hardy’s judgment rested on the letter’s content.

D: This reports Hardy’s response but not what it was based on or Ramanujan’s lack of credentials.

Watch for: “Rather than” goals need both sides: what the judgment rested on and what it did not rest on.

Correct choice: Like the eighth-century Kailasa temple at Ellora, the churches of Lalibela, made some four centuries later, were carved downward into solid rock.

The goal has two parts: a similarity in method (both carved downward into solid rock) and a difference in date (the eighth century versus some four centuries later). Only this choice gives both.

A: This stresses the similarity in method, but the added detail about the Ellora complex is not a difference in when the sites were made.

B: This gives the similarity in method and a difference, but the difference is in religion, not in when the sites were made.

C: This gives the difference in date but not the similarity in how the sites were made.

Watch for: When a goal says to stress one point “while noting” another, the stressed point belongs in the main clause; also check that each part is the right kind of information.

Answer: D.

The coefficient of h is the model’s increase for one more minute, so it predicts 6.5 additional grooves inspected per minute.

Check: $R(h+1)-R(h)=6.5$ for any allowed h.

A: That value is 21.

B: That would be the reciprocal rate.

C: That value is 27.5.

Watch for: A model may have a noninteger average rate even when it counts whole objects.

Percent change is the change divided by the original value. Sales fell by $5{,}040 - 4{,}200 = 840$, and $840 \div 5{,}040 \approx 0.167$, so the decrease is about $16.7\%$. Dividing the change by the new value instead of the original gives $840 \div 4{,}200 = 20\%$. Dividing June by May gives $83.3\%$, the fraction remaining rather than the fraction lost. Dividing May by June gives $120\%$.

Answer: D.

Above the first line means $y\ge-x+6$ because the boundary is solid. Below the horizontal line means $y<8$ because that boundary is dashed.

Check: A point such as (2,6) satisfies both inequalities and lies inside the shaded region.

A: This places the region below the sloped line.

B: This places the region above the horizontal line.

C: This reverses which boundaries are included.

Watch for: Translate the side of each boundary and its inclusion style independently.

Answer: 32

Let m be missed checkpoints. Then 2m are skipped, so 50 - 3m are successful. The score equation is $6(50-3m)-4m=168$. Thus $300-22m=168$, m = 6, and the successful count is $50-18=32$.

Check: The counts are 32 successful, 6 missed, and 12 skipped. They sum to 50 and give score $192-24=168$.

Watch for: The skipped category consumes checkpoints even though it contributes zero points.

Answer: C.

Taking $12\%$ off saves more than $15 exactly when $0.12p>15$, that is, when the price $p$ is greater than $125.

If the $15 discount were used, the price would be $132+15=147$ dollars. But at $147, $12\%$ off saves $0.12\cdot147=17.64$ dollars, which is more than $15, so the coupon would have taken $12\%$ off instead. This case is impossible.

If $12\%$ off was used, then $0.88p=132$, so $p=150$. At $150, $12\%$ off saves $18, more than $15, so this case is consistent. The price was $150.00.

Check: For a $150 bicycle, the two discounts are $15 and $18; the coupon takes the larger, $18, and the customer pays $132.

A: $147.00 undoes a $15 discount, but at that price $12\%$ off saves $17.64, so the coupon would not have taken $15.

B: $147.84 adds $12\%$ of the amount paid, $1.12\cdot132$, instead of dividing by $0.88$.

D: $151.80 adds $15\%$ of the amount paid, $1.15\cdot132$, mixing the two offers.

Watch for: Check each possible discount against the rule “whichever saves more”; a price that fits one discount is valid only if that discount is actually the larger one.

Answer: 13.

Squaring both sides gives $2x+10=x^2-14x+49$, so $x^2-16x+39=0$, which factors as $(x-3)(x-13)=0$. Squaring can create roots that do not satisfy the original equation, so both candidates must be tested. At $x=13$, $\sqrt{36}=6$ and $13-7=6$, which agree; at $x=3$, $\sqrt{16}=4$ but $3-7=-4$, so $x=3$ must be discarded.

Check: The left side $\sqrt{2x+10}$ is never negative, so any solution needs $x-7\ge 0$, that is $x\ge 7$. The candidate $x=3$ fails that requirement outright, and $x=13$ meets it.

Watch for: After squaring a radical equation, substitute every candidate back into the original equation, because a principal square root can never equal a negative number.

Answer: C.

The correspondence is 6 model centimeters to 24 actual meters, or 4 actual meters per model centimeter. The actual long side is 9(4) = 36 meters. The area is 24(36) = 864 square meters.

Check: The actual dimensions retain the model’s 2:3 ratio.

A: This applies a length factor to the model area only once.

B: This uses an incorrect long-side conversion.

D: This treats the plaza as a 36-meter square.

Watch for: Convert both corresponding lengths consistently before multiplying; a length scale is squared for area.

Answer: D.

At $x=-1$ the line gives $y=4(-1)+3=-1$, and the parabola must give the same value there: $2(-1)^2+k(-1)-5=-3-k$, so $-3-k=-1$ and $k=-2$. The graphs then meet where $2x^2-2x-5=4x+3$, that is, where $2x^2-6x-8=0$, or $2(x+1)(x-4)=0$. The intersections sit at $x=-1$ and $x=4$, and the line puts the second one at $y=4(4)+3=19$.

Check: The parabola reaches that same height at $x=4$: $2(4)^2-2(4)-5=32-8-5=19$, so $(4,19)$ lies on both graphs.

A: $-1$ is the $y$-coordinate of the point the stem already supplies, not of the second intersection.

B: 4 is the $x$-coordinate of the second intersection; the stem asks for its $y$-coordinate.

C: 11 reflects $x=-1$ across the parabola’s own axis of symmetry, $x=0.5$, landing at $x=2$. The two intersections are symmetric about the axis of the combined equation, $x=1.5$, not about the parabola’s axis.

Watch for: One stated intersection is enough to pin down an unknown constant; with the constant known the system collapses to a quadratic that factors, and the second root is read straight off.

Answer: C.

Using all $3{,}000$ milliliters of blue paint would need $\frac25(3{,}000)=1{,}200$ milliliters of white paint, but only $1{,}000$ milliliters are available. So the white paint limits the mixture.

With all $1{,}000$ milliliters of white paint, the blue paint is $\frac52(1{,}000)=2{,}500$ milliliters, and the total is $2{,}500+1{,}000=3{,}500$ milliliters.

Check: The ratio $2{,}500:1{,}000$ is $5:2$, and $2{,}500$ milliliters of blue paint is less than the $3{,}000$ milliliters available.

A: $1{,}400$ inverts the ratio: with $1{,}000$ milliliters of white paint it uses $\frac25(1{,}000)=400$ milliliters of blue paint.

B: $2{,}500$ is the volume of the blue paint only.

D: $4{,}200$ treats the blue paint as the paint that runs out first: all $3{,}000$ milliliters of blue paint would need $1{,}200$ milliliters of white paint, but only $1{,}000$ milliliters are available.

Watch for: Check which paint runs out first; convert liters to milliliters before comparing.

The equation is quadratic in $x^2$: substituting $u = x^2$ gives $u^2 - 13u + 36 = 0$, which factors as $(u - 4)(u - 9) = 0$, so $x^2 = 4$ or $x^2 = 9$. Each yields two solutions, $x = \pm 2$ and $x = \pm 3$, so the positive ones are $2$ and $3$ and their product is $6$. Multiplying all four solutions instead gives $36$, since the two negatives cancel. Answering $2$ or $3$ reports one of the pair rather than their product.

Answer: D.

Graph the two lines and read their intersection, $(4.5,8)$. The requested display is $D=3(4.5)+8=21.5$.

Check: $1.7(4.5)+2.3(8)=26.05$ and $2.6(4.5)-0.8(8)=5.3$. Both original equations are satisfied.

A: This is x, not the requested display.

B: This is y, not the requested display.

C: This gives x+y instead of 3x+y.

Watch for: An intersection gives two variables; calculate the expression actually requested.

Answer: B.

Silo A empties at $2.5(60)=150$ bushels per hour, so after $t$ hours it holds $6{,}400-150t$ bushels while silo B holds $2{,}800+90t$ bushels. Solving $2{,}800+90t>6{,}400-150t$ gives $240t>3{,}600$, so $t>15$, and the least whole number of hours is $16$.

Check: At $t=15$ each silo holds $4{,}150$ bushels, so B is not yet ahead. At $t=16$, silo A holds $4{,}000$ bushels and silo B holds $4{,}240$.

A: $15$ hours is when the two silos hold equal amounts. The question asks when B holds more, so $15$ is one hour too early.

C: $24$ hours is when silo A alone falls to silo B’s starting level of $2{,}800$ bushels. It ignores the grain being added to silo B.

D: This never converts $2.5$ bushels per minute into an hourly rate, solving $2{,}800+90t>6{,}400-2.5t$ instead, which gives $t>38.9$.

Watch for: Put both rates in the same time unit first, and remember that the boundary value of a strict inequality is not itself a solution.

Answer: B.

At a depth of $12$ meters, the temperature is $18-0.4(12)=13.2^\circ\text{C}$, which is still above $12^\circ\text{C}$. The remaining drop of $1.2^\circ\text{C}$ happens at $0.1^\circ\text{C}$ per meter, which takes $\dfrac{1.2}{0.1}=12$ more meters. The depth is $12+12=24$ meters.

Check: At $24$ meters the temperature is $18-0.4(12)-0.1(12)$, which is $18-4.8-1.2=12$ degrees Celsius.

A: $15$ uses $0.4^\circ\text{C}$ per meter for the whole drop, $\dfrac{6}{0.4}$, ignoring the change in rate at $12$ meters.

C: $60$ uses $0.1^\circ\text{C}$ per meter for the whole drop, $\dfrac{6}{0.1}$.

D: $72$ applies $0.1^\circ\text{C}$ per meter to the whole $6$-degree drop and then adds the first $12$ meters.

Watch for: Find the temperature at $12$ meters first; only the drop below that depth uses the second rate.

Answer: A.

Reflecting across the $x$-axis changes $(x,y)$ to $(x,-y)$. The reflected line passes through $(5,2)$, so line $\ell$ passes through $(5,-2)$. Line $\ell$ also passes through $(2,7)$, so its slope is

$m=\dfrac{-2-7}{5-2}=-3$

Check: Line $\ell$ is $y=-3x+13$, which contains $(2,7)$. Its reflection is $y=3x-13$, which contains $(5,2)$.

B: $-\dfrac{5}{3}$ uses $(5,2)$ itself as a point on line $\ell$, ignoring the reflection.

C: $\dfrac{5}{7}$ reflects $(5,2)$ across the $y$-axis to $(-5,2)$ instead of across the $x$-axis.

D: $3$ is the slope of the reflected line, not of line $\ell$.

Watch for: The point on the reflected line corresponds to $(5,-2)$ on line $\ell$.

Answer: B.

The vertex of the graph is $(2,k)$. The graph meets the $x$-axis where $|2x-4|=k$, that is, where $x=2\pm\dfrac k2$. So the enclosed triangle has base $\dfrac k2-\left(-\dfrac k2\right)=k$ and height $k$, and its area is $\tfrac12k^2$. Then $\tfrac12k^2=25$, so $k^2=50$ and $k=5\sqrt2$.

Check: With $k=5\sqrt2\approx7.07$, the intercepts are about $-1.54$ and $5.54$, a base of about $7.07$, and $\tfrac12(7.07)(7.07)\approx25$.

A: $5$ solves $|x-2|=k$ instead of $|2x-4|=k$, which doubles the base to $2k$ and gives an area of $k^2$.

C: $10$ uses $\tfrac k2$, the distance from the vertex to one intercept, as the whole base, so the area is $\tfrac{k^2}{4}$.

D: $\tfrac{25}{2}$ uses the $4$ in $|2x-4|$ as the height: $\tfrac12\cdot k\cdot4=25$.

Watch for: The coefficient $2$ inside the absolute value halves the distance from the vertex to each intercept.

Answer: C.

Because $\overline{AB}\parallel\overline{DC}$, triangles $ABE$ and $CDE$ are similar. Their area ratio is $27:12=9:4$, so their similarity ratio is $3:2$, and $AE:EC=3:2$. Triangles $ABE$ and $CBE$ share the height from $B$ to line $AC$, so their areas are in the ratio $AE:EC=3:2$:

$\text{area}(CBE)=\tfrac23(27)=18$

Similarly, triangles $ADE$ and $CDE$ share the height from $D$, so $\text{area}(ADE)=\tfrac32(12)=18$. The trapezoid’s area is $27+12+18+18=75$.

Check: Triangles $ABD$ and $ABC$ have the same base $\overline{AB}$ and the same height, so they have equal areas: $27+18=45$ each. Also, $18^2=27\cdot12$, as the two side triangles require.

A: This adds only the two given triangles and leaves out triangles $ADE$ and $BCE$.

B: This assumes each of triangles $ADE$ and $BCE$ has the same area as triangle $CDE$, $12$: $27+12+12+12=63$.

D: This uses the area ratio $9:4$ as the ratio $AE:EC$, which makes the side triangles $\tfrac49(27)=12$ and $\tfrac94(12)=27$, for a total of $78$.

Watch for: Similar triangles have areas in the ratio of the squares of their sides; take the square root before using the ratio for lengths.

Answer: 23 Let x and y count the 9- and 14-kilogram containers. Then 9x + 14y = 235, with x, y nonnegative integers. To maximize x, minimize y. Testing y = 0, 1, 2 gives remainders 235, 221, 207 for the 9-kilogram containers. The first divisible by 9 is 207 = 9(23), so y = 2 and x = 23. Larger y only decreases x.

Watch for: The continuous bound 235/9 is not attainable with whole containers of the given sizes.

Answer: -3

The zeros of $p$ are $0$, $3$, and $-a$. The graph touches the $x$-axis without crossing it at a zero that is repeated an even number of times, so one of the three zeros must repeat.

If $-a=0$, then $a=0$ and $p(x)=x^2(x-3)$, which touches the axis at $x=0$ and crosses it at $x=3$.

If $-a=3$, then $a=-3$ and $p(x)=x(x-3)^2$, which touches the axis at $x=3$ and crosses it at $x=0$.

For any other value of $a$, the three zeros are different and the graph crosses the axis at each of them. The possible values are $0$ and $-3$, so the least is $-3$.

Check: For $a=-3$, $p(2)=2(-1)^2=2$ and $p(4)=4(1)^2=4$ are both positive, so the graph stays above the axis on both sides of $x=3$.

Watch for: The most common wrong value is $0$, which makes $x=0$ the repeated zero; the zero $x=3$ can also repeat, when $a=-3$.

Answer: 3

Let the intersection point be $(t,0)$. The slope of $\ell$ is $\dfrac{2-0}{4-t}$, and the slope of $m$ is $\dfrac{3-0}{-1-t}$. Perpendicular lines have slopes whose product is $-1$:

$\dfrac{2}{4-t}\cdot\dfrac{3}{-1-t}=-1$

$6=(4-t)(1+t)$

$t^2-3t+2=0$

So $(t-1)(t-2)=0$, and $t=1$ or $t=2$. Both work, and the sum of the possible $x$-coordinates is $3$.

Check: For $t=1$, the slopes are $\dfrac23$ and $-\dfrac32$; for $t=2$, they are $1$ and $-1$. Each pair has product $-1$.

Watch for: The most common wrong values are $1$ or $2$ alone; the condition gives a quadratic equation, and both of its solutions work.