The SAT is the college-admissions test created and administered by the College Board, and most U.S. colleges accept it as part of their admissions review. Today’s test is the Digital SAT, taken on a computer through the Bluebook app. This free practice test mirrors the real two-section exam, so working through it shows exactly where you stand.[1]
Click Start Test above to launch a full SAT simulation, run a single section, or drill one content domain. The hub groups everything under the exam’s two scored sections so your practice matches how the real test is built and timed.
The Digital SAT has a Reading and Writing section and a Math section. Each section is split into two separately timed modules, and your scores from both combine into the 400–1600 total.[1]
Every question is tagged to its official domain and includes a clear explanation, so you learn the reasoning behind each grammar rule, vocabulary choice, and math concept — not just the answer. To round out your prep, pair these with our free study guide, flashcards, and cheat sheet.[2]
SAT Exam at a Glance
The most important thing to understand about the Digital SAT is that it is two scored sections, each split into two separately timed modules. The table below lays out both sections, their question counts and clocks, and how the test is scored.
| Detail | Digital SAT |
|---|---|
| Administered By | College Board |
| Section 1 — Reading and Writing | 54 questions · two 32-minute modules (64 minutes) |
| Section 2 — Math | 44 questions · two 35-minute modules (70 minutes) |
| Total Scored Questions | 98 (plus a few unscored pretest questions) |
| Total Testing Time | 2 hours 14 minutes (134 minutes), plus a 10-minute break |
| Format | Section-adaptive; taken on a computer in the Bluebook app |
| Score Scale | 400–1600 total (Reading and Writing 200–800 + Math 200–800) |
| Guessing Penalty | None — rights-only scoring, so always answer every question |
| Calculator | A graphing calculator is allowed on the entire Math section |
| Essay | None — the Digital SAT has no essay; only selected-response and grid-in questions |
The Digital SAT is two sections — Reading and Writing (54 Q / 64 min) and Math (44 Q / 70 min) — for 98 scored questions in 134 minutes of testing time, scored on a 400–1600 scale with no penalty for guessing.
One note on this practice test: like the real Digital SAT, it has no essay. Every question here is multiple choice or short-answer, mirroring the scored, selected-response content you will actually face.
The Two SAT Sections and How They’re Weighted
The SAT covers eight content domains grouped under its two scored sections. Reading and Writing tests reading comprehension, vocabulary, rhetoric, and grammar. Math tests algebra, advanced math, data analysis, and geometry and trigonometry.[2]
Our full practice exam runs both sections, each on its own clock, weighted to the College Board blueprint. Here is how the questions break down within each section:
The bars above are each domain’s share of its own section, not of the whole exam. In Reading and Writing the four domains are fairly balanced, while in Math, Algebra and Advanced Math together carry roughly two-thirds of the questions.
Practice Questions by Domain
Use Start Test for a full weighted SAT simulation, run a single section on its own clock, or open the hub and drill one domain. After each section you get a per-domain breakdown so you know exactly where to focus.
Most students need the most reps on Algebra and Advanced Math, which dominate the Math section, and on Standard English Conventions, where the grammar and punctuation rules trip people up.
How the Adaptive Format Works
The Digital SAT is section-adaptive. Each section has two modules, and your performance on the first module determines whether the second module is harder or easier.[1]
This matters for your score: doing well on the first module routes you to a harder, higher-ceiling second module. The adaptivity is between modules, not question by question, so within a module you can still skip, flag, and return to questions freely.
Because the first module sets your path, treat the opening questions of each section with care — but never freeze on one item, since pacing across the whole module is what protects your score.
How Do You Register for the SAT?
You register for the SAT through your College Board account, where you pick a test date and a nearby test center, then pay the registration fee.[5]
Fee waivers are available for eligible students, and you download the Bluebook testing app and complete exam setup before test day. Registration deadlines fall several weeks before each test date, so check the current SAT calendar early.
Confirm current fees, deadlines, and accepted IDs on the College Board site before you register, because these details are updated each testing year.[5]
How Is the SAT Scored?
The SAT is scored on a 400–1600 scale, combining a Reading and Writing score and a Math score that each range from 200 to 800.[4]
There is no penalty for guessing— the test uses rights-only scoring, so a blank and a wrong answer both earn zero. Always answer every question, even a last-second guess.
There is no fixed passing score. What counts as a good SAT score depends on the colleges you are targeting, since each school publishes its own typical admitted-student range. Many colleges also superscore, taking your highest section scores across multiple test dates.
How Hard Is the SAT?
The SAT is challenging mainly because of its pace and the breadth of skills it tests in a short window. The Digital SAT is shorter than the old paper exam, but the time pressure per question is real.
The Math section trips up students on Advanced Math (quadratics, exponentials, and functions), while Reading and Writing punishes shaky grammar and weak vocabulary in context.[3] Timed, domain-weighted practice is the most reliable way to build the speed and accuracy the test rewards.
The takeaway: simulate each section under its own clock until your pacing and accuracy are consistent, then aim for the composite score your target colleges expect.
What to Expect on Test Day
The Digital SAT is taken on a computer or tablet in the Bluebook app, in two sections: Reading and Writing first, then Math, with a 10-minute break between them.[1]
Each section is split into two timed modules with an on-screen countdown, and the built-in tools include a graphing calculator (available the whole Math section) and an annotation feature. Bring an acceptable photo ID and arrive early for check-in.
Simulating each section’s module timing with full practice tests makes the pacing feel routine, so test day is about execution rather than surprises.
How to Use This SAT Practice Test
- Respect the clocks. Run each section timed to its real limit — 64 minutes for Reading and Writing, 70 for Math.
- Diagnose, then drill. Take a full simulation to find weak domains, then drill them one at a time.
- Prioritize the big domains. Algebra and Advanced Math carry the most math questions — bank the most reps there.
- Answer everything. With no guessing penalty, never leave a question blank.
- Learn the why. Read every explanation — understanding the rule beats memorizing the answer.
Why the SAT Still Matters
A strong SAT score strengthens college applications, can unlock merit scholarships, and helps you stand out even at test-optional schools.[4] These free SAT practice tests are the most efficient way to raise your score across both sections.
Conclusion
Raising your SAT score comes down to two things: command of the Reading and Writing domains and the Math domains, each scored on its own 200–800 scale. Use this free SAT practice test to find your weak domains, drill them to mastery under each section’s own clock, and reinforce them with our study guide, flashcards, and cheat sheet. Start with one full-length practice test to find your weakest section, then drill that section first.
SAT Practice Test FAQ
The SAT is created and administered by the College Board, a nonprofit organization. The current test is the Digital SAT, taken on a computer or tablet through the Bluebook application at a test center or school, and most U.S. colleges accept SAT scores as part of their admissions review.
The Digital SAT has two sections: Reading and Writing, then Math. Each section is split into two equal-length, separately timed modules. Reading and Writing has 54 questions across two 32-minute modules (64 minutes total), and Math has 44 questions across two 35-minute modules (70 minutes total).
There are 98 scored questions in total — 54 in Reading and Writing and 44 in Math. Total testing time is 2 hours and 14 minutes (134 minutes), not counting the 10-minute break between the two sections. A small number of unscored pretest questions are mixed in and do not affect your score.
Reading and Writing covers four content domains: Information and Ideas, Craft and Structure, Expression of Ideas, and Standard English Conventions. Each question is tied to a single short passage, and the domains test reading comprehension, vocabulary in context, rhetoric, and grammar and punctuation.
Math covers four content domains: Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry and Trigonometry. Most questions are multiple choice, but about a quarter are student-produced response (grid-in) items where you type the answer. A graphing calculator is allowed on the entire Math section.
Yes — it is section-adaptive. In each section, your performance on the first module determines whether the second module is harder or easier, and the difficulty mix you receive helps determine your score. The adaptivity is between modules, not question by question, so you can still move back and forth freely within a module.
The SAT is scored on a 400–1600 scale, combining a Reading and Writing score and a Math score that each range from 200 to 800. There is no fixed passing score; what counts as a good score depends on the colleges you are targeting, since each school publishes its own typical score range.
No. The SAT uses rights-only scoring, so you are not penalized for wrong answers. Because a blank and a wrong answer both earn zero points, you should always answer every question, even if you have to guess in the last seconds of a module.
The SAT is challenging mainly because of its pace and the breadth of math and reading skills it tests in a short window. The Digital SAT is shorter than the old paper test, but the time pressure per question is real, which is exactly why timed, domain-weighted practice is the most effective way to prepare.
There is no official lifetime limit, and many students take the SAT two or three times to improve their score. Most colleges accept your highest section scores across test dates (superscoring), so retaking the test strategically can raise your composite score.
SAT question bank
All 379 questions, by domain
A reference copy of every question in this practice test. Each answer stays hidden until you choose to show it. To practice with scoring, timing and your readiness score, use Start Test at the top of the page.
Craft and Structure (70)
On the Digital SAT Reading and Writing section, a "Words in Context" question primarily asks a student to do which of the following?
- A.Reorganize a list of student notes to achieve a stated goal
- B.Choose the word or phrase that best completes a passage based on its surrounding meaning
- C.Correct the punctuation between two independent clauses
- D.Identify a grammatical error in an underlined portion of a sentence
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Correct answer: Choose the word or phrase that best completes a passage based on its surrounding meaning
Choosing the word or phrase that best fits the passage based on surrounding meaning is correct because Words in Context items provide a short passage with a blank (or a key term) and ask which option most logically and precisely completes it given the sentence's overall sense. Identifying grammatical errors and correcting punctuation belong to Standard English Conventions, and reorganizing notes is Rhetorical Synthesis, an Expression of Ideas skill.
Read the following sentence: "Although the committee's report was praised for its thoroughness, several reviewers noted that its conclusions were ______, offering little guidance for the policymakers who had requested it." Which choice best completes the text?
- A.Exhaustive
- B.Optimistic
- C.Equivocal
- D.Definitive
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Correct answer: Equivocal
"Equivocal" is correct because the contrast signaled by "Although" and the phrase "offering little guidance" indicate the conclusions were unclear or noncommittal, which is exactly what equivocal means. "Definitive" and "exhaustive" would suggest clarity or completeness, contradicting the lack of guidance, and "optimistic" describes a hopeful tone rather than the vagueness the sentence requires.
Read the following sentence: "The novelist was known for her ______ prose: every sentence felt trimmed of excess, conveying complex ideas in remarkably few words." Which choice best completes the text?
- A.Economical
- B.Ornate
- C.Ambiguous
- D.Spontaneous
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Correct answer: Economical
"Economical" is correct because the colon's explanation describes prose "trimmed of excess" that conveys ideas "in remarkably few words," which defines economical writing. "Ornate" means heavily decorated and contradicts the trimming of excess, "spontaneous" addresses how writing is produced rather than its concision, and "ambiguous" means unclear, which conflicts with conveying complex ideas effectively.
Read the following sentence: "Far from being a passive observer, the journalist took a ______ role in the investigation, personally tracking down sources and filing public-records requests." Which choice best completes the text?
- A.Detached
- B.Active
- C.Minor
- D.Reluctant
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Correct answer: Active
"Active" is correct because "Far from being a passive observer" sets up a direct contrast with passivity, and the examples of personally tracking sources and filing requests demonstrate hands-on involvement. "Reluctant," "detached," and "minor" all describe limited or unwilling participation, which contradicts the energetic engagement the sentence describes.
Read the following sentence: "The senator's remarks were so ______ that even her allies struggled to determine which policy she actually intended to support." Which choice best completes the text?
- A.Candid
- B.Blunt
- C.Persuasive
- D.Nebulous
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Correct answer: Nebulous
"Nebulous" is correct because the result clause—allies could not tell which policy she supported—signals that the remarks were vague and lacking clear definition, which is what nebulous means. "Candid" and "blunt" both mean direct or frank, the opposite of confusing, and "persuasive" describes convincing speech rather than the unclear quality the sentence requires.
As used in the sentence "The treaty was meant to be a temporary measure, but its provisions proved remarkably durable, shaping diplomacy for decades," the word "durable" most nearly means
- A.Controversial
- B.Expensive
- C.Long-lasting
- D.Fragile
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Correct answer: Long-lasting
"Long-lasting" is correct because the clause "shaping diplomacy for decades" indicates the provisions endured over time, the precise sense of durable in this context. "Fragile" is the opposite of durable, while "expensive" and "controversial" introduce ideas of cost and dispute that the sentence's emphasis on longevity does not support.
As used in the sentence "The scientist offered only a cursory glance at the data before dismissing the findings," the word "cursory" most nearly means
- A.Curious and eager
- B.Careful and thorough
- C.Hasty and superficial
- D.Hostile and angry
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Correct answer: Hasty and superficial
"Hasty and superficial" is correct because pairing "cursory glance" with the quick dismissal of findings signals an examination that was brief and not thorough. "Careful and thorough" is the opposite of cursory, and "hostile and angry" and "curious and eager" describe emotional states that the sentence's focus on a quick, shallow review does not convey.
As used in the sentence "Critics initially derided the building's design as garish, but over time the public came to admire its bold colors," the word "derided" most nearly means
- A.Praised
- B.Ridiculed
- C.Ignored
- D.Redesigned
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Correct answer: Ridiculed
"Ridiculed" is correct because the contrast with the public later coming to "admire" the design, plus the negative descriptor "garish," shows critics were mocking it, which is what derided means. "Praised" is the opposite, "redesigned" describes altering the building rather than reacting to it, and "ignored" would not fit the active negative judgment the critics expressed.
A student reads a question that quotes a phrase from a passage and asks what it "most nearly means" as used in context. The most reliable strategy for answering is to
- A.Select the most common dictionary definition of the word regardless of the passage
- B.Substitute each answer choice into the sentence and judge which preserves the intended meaning
- C.Choose the most difficult or unfamiliar-sounding word among the options
- D.Pick the word that appears most often elsewhere in the passage
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Correct answer: Substitute each answer choice into the sentence and judge which preserves the intended meaning
Substituting each choice into the sentence to see which preserves the intended meaning is correct because Determining the Meaning questions hinge on contextual fit, not a word's most frequent definition. Choosing the most common dictionary meaning ignores context, selecting the hardest-sounding option rewards trickery rather than meaning, and frequency elsewhere in the passage does not establish what the phrase means in its specific sentence.
As used in the sentence "The mentor's feedback, though pointed, was ultimately benevolent, intended to help the young writer improve," the word "benevolent" most nearly means
- A.Mandatory
- B.Indifferent
- C.Sarcastic
- D.Well-meaning
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Correct answer: Well-meaning
"Well-meaning" is correct because the phrase "intended to help the young writer improve" clarifies that the feedback came from good intentions, the core sense of benevolent. "Sarcastic" and "indifferent" conflict with the supportive intent the sentence states, and "mandatory" addresses whether feedback was required rather than the kindly motive described.
A short passage explains a historical event, then closes by listing three ways later generations misunderstood it. On the Digital SAT, a "Text Structure and Purpose" question about this passage would most likely ask the student to
- A.Correct a verb-tense error in the final sentence
- B.Identify the main purpose of the text as a whole
- C.Compute a statistic mentioned in the passage
- D.Supply a transition word between two clauses
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Correct answer: Identify the main purpose of the text as a whole
Identifying the main purpose of the text as a whole is correct because Text Structure and Purpose questions ask how a passage is organized or what its overall function is. Correcting verb tense is a Standard English Conventions task, computing a statistic is a math skill, and supplying a transition is an Expression of Ideas task, none of which addresses the passage's structure or purpose.
A passage opens by describing a widely accepted belief, then presents new evidence that contradicts it, and ends by proposing a revised explanation. The overall structure of this passage is best described as
- A.A chronological narrative of a single person's life
- B.A problem introduced, complicated, and then reframed
- C.A list of unrelated facts in no particular order
- D.A step-by-step set of instructions for a procedure
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Correct answer: A problem introduced, complicated, and then reframed
"A problem introduced, complicated, and then reframed" is correct because the passage moves from an accepted belief to contradicting evidence to a revised explanation, a classic claim-challenge-revision arc. A chronological life narrative, an unordered list of facts, and a procedural set of instructions do not match the argumentative progression the passage actually follows.
In a short passage, the author devotes the first half to describing the benefits of a new technology and the second half to cataloging its hidden costs. The main rhetorical purpose of this two-part structure is most likely to
- A.Instruct readers on how to operate the technology
- B.Present a balanced assessment that weighs advantages against drawbacks
- C.Prove that the technology has no benefits whatsoever
- D.Provide a chronological history of the technology's invention
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Correct answer: Present a balanced assessment that weighs advantages against drawbacks
Presenting a balanced assessment is correct because devoting equal space to benefits and hidden costs signals an even-handed evaluation rather than one-sided advocacy. Claiming the technology has no benefits ignores the first half, a chronological history misreads the benefit/cost organization, and operating instructions are absent from a passage that weighs pros and cons.
A Text Structure and Purpose question presents a sentence in bold and asks, "Which choice best describes the function of the underlined portion in the text as a whole?" To answer well, a student should focus on
- A.Whether that portion contains any spelling errors
- B.How many words the underlined portion contains
- C.The literal dictionary meaning of every word in the portion
- D.How that portion relates to and serves the surrounding ideas
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Correct answer: How that portion relates to and serves the surrounding ideas
Focusing on how the portion relates to and serves the surrounding ideas is correct because function questions ask what role a sentence plays within the passage's larger argument or structure. Spelling, isolated dictionary definitions, and word count do not reveal a sentence's purpose, which is defined by its connection to the ideas around it.
A passage begins with a striking anecdote about a single patient before transitioning to a discussion of a nationwide medical trend. The most likely function of the opening anecdote is to
- A.List statistics about the patient's recovery
- B.Provide step-by-step treatment instructions
- C.Disprove the nationwide trend that follows
- D.Engage the reader and introduce the broader issue concretely
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Correct answer: Engage the reader and introduce the broader issue concretely
Engaging the reader and introducing the broader issue concretely is correct because a vivid individual story is a common strategy for drawing readers in before broadening to a general trend. The anecdote sets up rather than disproves the trend, it is illustrative rather than instructional, and a single narrative example is not a statistical list.
Which question best captures what an "Author's Point of View and Tone" item on the Digital SAT asks a reader to determine?
- A.The attitude or perspective the author conveys toward the subject
- B.The numerical answer to a word problem in the passage
- C.The number of paragraphs the passage contains
- D.Whether a comma is correctly placed in a sentence
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Correct answer: The attitude or perspective the author conveys toward the subject
Determining the author's attitude or perspective toward the subject is correct because point-of-view and tone questions ask readers to infer how the author feels about the topic from word choice and emphasis. Counting paragraphs, judging comma placement, and solving word problems address structure, conventions, and math rather than the author's stance.
A passage states: "The so-called breakthrough, trumpeted in press releases, amounted to little more than a modest tweak to an existing method." The author's tone toward the breakthrough is best described as
- A.Reverent
- B.Alarmed
- C.Neutral
- D.Skeptical
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Correct answer: Skeptical
"Skeptical" is correct because phrases like "so-called," "trumpeted," and "little more than a modest tweak" signal the author doubts the breakthrough's significance. "Reverent" suggests admiration that the dismissive language contradicts, "neutral" ignores the clearly evaluative diction, and "alarmed" implies fear that the passage's mocking tone does not express.
A passage reads: "Watching the volunteers rebuild the flooded library shelf by shelf, the author writes, restored her faith that ordinary people can accomplish extraordinary things." The author's attitude toward the volunteers is best described as
- A.Resentful
- B.Indifferent
- C.Dismissive
- D.Admiring
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Correct answer: Admiring
"Admiring" is correct because the phrase "restored her faith" and the praise that ordinary people can accomplish "extraordinary things" express clear approval and respect. "Dismissive" and "resentful" convey negativity that contradicts the uplifting language, and "indifferent" ignores the strong positive emotion the author voices.
When a passage uses words such as "unfortunately," "regrettably," and "a missed opportunity" to characterize an outcome, these word choices most directly help establish the passage's
- A.Verb tense
- B.Margin of error
- C.Sentence length
- D.Tone
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Correct answer: Tone
"Tone" is correct because evaluative, emotionally colored words like "unfortunately" and "regrettably" reveal the author's attitude, which is exactly what tone captures. Verb tense and sentence length are structural or grammatical features, and margin of error is a statistical concept unrelated to an author's emotional stance.
A passage about urban planning maintains an even-handed manner, presenting arguments on both sides without endorsing either. The author's point of view is best described as
- A.Impartial
- B.Contemptuous
- C.Despairing
- D.Celebratory
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Correct answer: Impartial
"Impartial" is correct because presenting both sides without endorsement signals a balanced, unbiased perspective. "Contemptuous" and "despairing" convey strong negative emotions absent from an even-handed treatment, and "celebratory" implies enthusiastic approval that contradicts the author's refusal to take a side.
A passage describes a scientist who insists, despite mounting contrary data, that her original hypothesis is correct. The author notes she "clings" to the idea and "waves away" objections. The author most likely intends to portray the scientist as
- A.Open-minded and flexible
- B.Stubbornly resistant to evidence
- C.Indifferent to her research
- D.Celebrated by her peers
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Correct answer: Stubbornly resistant to evidence
"Stubbornly resistant to evidence" is correct because "clings" and "waves away objections," set against "mounting contrary data," depict someone refusing to update her view. "Open-minded and flexible" is the opposite of clinging, "celebrated by her peers" addresses reputation rather than the described behavior, and "indifferent" conflicts with her active insistence.
On the Digital SAT, a "Cross-Text Connections" question is distinctive because it requires a student to
- A.Compare the views or claims of two related texts
- B.Calculate values from a data table
- C.Analyze a single passage in isolation
- D.Correct grammatical errors across two sentences
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Correct answer: Compare the views or claims of two related texts
Comparing the views or claims of two related texts is correct because Cross-Text Connections items pair two short passages and ask how their authors' perspectives relate. Analyzing one passage alone is a different Reading skill, correcting grammar is a Conventions task, and calculating from a table is a math skill, none of which involves comparing two texts.
Text 1 argues that remote work boosts employee productivity by eliminating commutes. Text 2 contends that remote work erodes productivity by weakening team collaboration. The relationship between the two texts is best described as
- A.Text 2 provides a real-world example that supports Text 1
- B.Text 2 is entirely unrelated to the topic of Text 1
- C.Text 2 restates Text 1's claim in different words
- D.Text 2 directly disagrees with Text 1's central claim
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Correct answer: Text 2 directly disagrees with Text 1's central claim
"Text 2 directly disagrees with Text 1's central claim" is correct because Text 1 says remote work boosts productivity while Text 2 says it erodes productivity, an explicit opposition. Restating would require agreement, supplying a supporting example would require alignment, and the two texts are clearly related since both address remote work's effect on productivity.
Text 1 claims a new diet improves heart health based on a small study. Text 2 notes that larger, long-term studies have found no measurable benefit. Based on Text 2, how would its author most likely respond to Text 1's claim?
- A.By cautioning that the supporting evidence is too limited to be reliable
- B.By endorsing it as conclusively proven
- C.By arguing the diet harms heart health
- D.By recommending an entirely different diet
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Correct answer: By cautioning that the supporting evidence is too limited to be reliable
Cautioning that the evidence is too limited is correct because Text 2 cites larger, long-term studies finding no benefit, implying its author would view Text 1's small study as insufficient. Endorsing it as proven contradicts Text 2's findings, claiming the diet is harmful overstates "no measurable benefit," and recommending a different diet introduces a stance Text 2 never takes.
Text 1 and Text 2 both examine a city's decision to build a new stadium. Text 1 emphasizes the jobs the project will create, while Text 2 emphasizes the public debt the city will incur. The two authors most clearly differ in their
- A.Use of semicolons
- B.Page length
- C.Spelling of the city's name
- D.Emphasis on which consequences of the project matter most
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Correct answer: Emphasis on which consequences of the project matter most
Differing in emphasis on which consequences matter most is correct because Text 1 highlights job creation while Text 2 highlights public debt, showing each author foregrounds different effects of the same decision. Spelling, semicolon use, and page length are surface features that do not capture the substantive difference in the authors' priorities.
Text 1 asserts that standardized testing fairly measures student achievement. Text 2 acknowledges that such tests measure some skills but argues they overlook creativity and critical thinking. The relationship between the texts is best characterized as Text 2
- A.Partially conceding Text 1's point while raising a significant limitation
- B.Completely agreeing with Text 1
- C.Praising Text 1 without reservation
- D.Ignoring the subject of standardized testing
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Correct answer: Partially conceding Text 1's point while raising a significant limitation
Partially conceding while raising a limitation is correct because Text 2 admits the tests "measure some skills" yet argues they miss creativity and critical thinking, a qualified disagreement. Complete agreement ignores the criticism, ignoring the subject is false since both discuss testing, and unreserved praise contradicts Text 2's stated concerns.
A Cross-Text Connections question asks: "Based on the texts, how would the author of Text 2 most likely respond to the underlined claim in Text 1?" To answer accurately, a student should base the response on
- A.Personal opinions about the topic
- B.The perspective and evidence actually expressed in Text 2
- C.The longer of the two texts only
- D.Whichever answer choice is most detailed
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Correct answer: The perspective and evidence actually expressed in Text 2
Basing the response on the perspective and evidence in Text 2 is correct because the question asks how Text 2's author would react, which must be grounded in that author's stated views. Personal opinions are irrelevant to the texts, favoring the longer passage ignores Text 2's actual content, and choosing the most detailed option rewards length over accuracy.
Read the following sentence: "Rather than dismissing the apprentice's unconventional idea outright, the master craftsman remained ______, willing to test it before passing judgment." Which choice best completes the text?
- A.Receptive
- B.Oblivious
- C.Hostile
- D.Dismissive
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Correct answer: Receptive
"Receptive" is correct because "Rather than dismissing" sets up a contrast, and "willing to test it before passing judgment" describes openness to the idea, which receptive means. "Dismissive" and "hostile" contradict that openness, and "oblivious" implies unawareness rather than the deliberate willingness the sentence describes.
Read the following sentence: "The film's plot was so ______ that viewers needed a second screening just to follow how the storylines connected." Which choice best completes the text?
- A.Intricate
- B.Predictable
- C.Brief
- D.Straightforward
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Correct answer: Intricate
"Intricate" is correct because needing a second screening to follow connected storylines indicates the plot was complex and detailed, the meaning of intricate. "Straightforward," "brief," and "predictable" all suggest simplicity or ease, which contradicts the difficulty viewers experienced in following the plot.
As used in the sentence "The negotiator's tactful phrasing prevented the tense meeting from collapsing into open conflict," the word "tactful" most nearly means
- A.Loud and forceful
- B.Sensitive and diplomatic
- C.Silent and withdrawn
- D.Careless and rude
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Correct answer: Sensitive and diplomatic
"Sensitive and diplomatic" is correct because phrasing that prevented a tense meeting from erupting into conflict reflects skillful, considerate communication, the essence of tactful. "Loud and forceful" and "careless and rude" would more likely escalate conflict, and "silent and withdrawn" describes inaction rather than the deft phrasing credited with keeping the peace.
A passage about a coral reef opens with lush description, shifts to documenting its rapid decline, and closes by urging conservation. The author's primary purpose is best described as
- A.To raise awareness of the reef's decline and motivate action
- B.To compare two competing scientific theories
- C.To provide instructions for scuba diving
- D.To entertain readers with an unrelated adventure story
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Correct answer: To raise awareness of the reef's decline and motivate action
Raising awareness and motivating action is correct because the passage moves from beauty to decline to an explicit call for conservation, an arc designed to persuade readers to care and act. An unrelated adventure story ignores the conservation focus, diving instructions are absent, and the passage advances one argument rather than comparing two theories.
Read the following sentence: "The biographer refused to ______ the inventor's flaws, instead presenting his selfishness and brilliance with equal honesty." Which choice best completes the text?
- A.Gloss over
- B.Celebrate
- C.Investigate
- D.Exaggerate
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Correct answer: Gloss over
"Gloss over" is correct because "refused to" combined with "presenting his selfishness ... with equal honesty" shows the biographer would not minimize or skip past the flaws, which is what gloss over means. "Exaggerate" would mean overstating the flaws, "celebrate" implies praising them, and "investigate" describes examining rather than downplaying, none of which fits the honest, unflinching treatment described.
Text 1 reports that a city's bike-share program reduced car traffic downtown. Text 2 presents survey data showing most bike-share users had previously walked or taken buses rather than driven. How does the information in Text 2 most directly relate to the claim in Text 1?
- A.It is unrelated to the effects of the bike-share program
- B.It confirms that the program eliminated all downtown traffic
- C.It restates Text 1's claim using different statistics
- D.It complicates Text 1's claim by suggesting the traffic reduction may stem from other factors
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Correct answer: It complicates Text 1's claim by suggesting the traffic reduction may stem from other factors
Complicating Text 1's claim is correct because if most users formerly walked or took buses rather than drove, the program may not have cut car traffic as Text 1 asserts, casting doubt on the causal link. Confirming the elimination of all traffic overstates both texts, restating would require agreement rather than challenge, and the survey is plainly relevant since it concerns the same program's users.
Read the sentence: "The professor's lectures were anything but ______; she peppered each class with jokes, role-play, and surprise demonstrations." Which choice best completes the text?
- A.Spontaneous
- B.Dull
- C.Interactive
- D.Lively
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Correct answer: Dull
"Dull" is the answer because "anything but" sets up a contrast with the lively jokes, role-play, and demonstrations that follow, so the blank needs a word meaning boring. "Lively," "interactive," and "spontaneous" all describe the energetic teaching itself, so they would break the contrast the sentence requires.
Read the sentence: "Investors grew wary as the company's projections became increasingly ______, promising returns that no rival had ever achieved." Which choice best completes the text?
- A.Grounded
- B.Modest
- C.Transparent
- D.Extravagant
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Correct answer: Extravagant
"Extravagant" is the answer because promising returns no rival had ever achieved describes claims that are excessive and unrealistic, which is what extravagant means here. "Modest" and "grounded" suggest restraint, the opposite of the inflated promises, and "transparent" would not explain why investors grew wary.
Read the sentence: "Once a ______ figure known only to specialists, the mathematician became a household name after the documentary aired." Which choice best completes the text?
- A.Controversial
- B.Celebrated
- C.Wealthy
- D.Obscure
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Correct answer: Obscure
"Obscure" is the answer because the contrast with becoming a "household name" signals the mathematician was previously little known, the meaning of obscure. "Celebrated" would contradict that contrast, while "controversial" and "wealthy" introduce ideas the sentence never supports.
Read the sentence: "The coach praised the rookie's ______, noting how readily she adjusted her strategy whenever the game shifted." Which choice best completes the text?
- A.Reluctance
- B.Fatigue
- C.Stubbornness
- D.Adaptability
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Correct answer: Adaptability
"Adaptability" is the answer because adjusting strategy readily as the game shifts is the very definition of being adaptable. "Stubbornness" and "reluctance" describe resistance to change, the opposite of the praised behavior, and "fatigue" addresses tiredness rather than flexible play.
Read the sentence: "The editor urged the writer to ______ the report, trimming its forty pages to a tight ten." Which choice best completes the text?
- A.Expand
- B.Condense
- C.Postpone
- D.Duplicate
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Correct answer: Condense
"Condense" is the answer because trimming forty pages to ten means making the report shorter and more compact, which condense describes. "Expand" is the opposite, and "duplicate" and "postpone" describe actions unrelated to shortening the text.
Read the sentence: "Critics found the sequel ______, a near-identical retread that added almost nothing to the original." Which choice best completes the text?
- A.Ambitious
- B.Inventive
- C.Derivative
- D.Divisive
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Correct answer: Derivative
"Derivative" is the answer because a near-identical retread that adds nothing new is, by definition, derivative of the original. "Inventive" and "ambitious" suggest originality, contradicting the retread description, and "divisive" would describe disagreement rather than the lack of novelty.
Read the sentence: "Far from being ______, the witness gave a remarkably precise account, recalling exact times and street names." Which choice best completes the text?
- A.Vague
- B.Reliable
- C.Articulate
- D.Thorough
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Correct answer: Vague
"Vague" is the answer because "far from being" contrasts the blank with the precise account of exact times and street names, so the blank needs a word meaning imprecise. "Reliable," "thorough," and "articulate" all align with precision, so they would not create the required contrast.
Read the sentence: "The new manager's ______ approach won over the staff, as she sought input from every department before deciding." Which choice best completes the text?
- A.Autocratic
- B.Inclusive
- C.Secretive
- D.Hasty
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Correct answer: Inclusive
"Inclusive" is the answer because seeking input from every department reflects an approach that brings many voices in, the meaning of inclusive. "Autocratic" and "secretive" describe top-down or hidden decision-making, and "hasty" would clash with the deliberate consultation described.
As used in the sentence "The diplomat's prudent silence at the summit kept a fragile alliance from unraveling," the word "prudent" most nearly means
- A.Loud and aggressive
- B.Cowardly and weak
- C.Sudden and impulsive
- D.Wise and cautious
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Correct answer: Wise and cautious
"Prudent" most nearly means wise and cautious because a silence that kept a fragile alliance from unraveling reflects careful, sensible judgment. "Cowardly and weak" misreads strategic restraint as fear, while "sudden and impulsive" and "loud and aggressive" contradict the calm caution the silence shows.
As used in the sentence "The historian's meticulous footnotes documented every source she had consulted," the word "meticulous" most nearly means
- A.Openly biased
- B.Brief and incomplete
- C.Hastily written
- D.Extremely careful and precise
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Correct answer: Extremely careful and precise
"Meticulous" most nearly means extremely careful and precise because documenting every source consulted shows exacting thoroughness. "Brief and incomplete" and "hastily written" contradict that thoroughness, and "openly biased" addresses fairness rather than the carefulness the sentence describes.
As used in the sentence "The proposal met with tepid applause, a few scattered claps that quickly died away," the word "tepid" most nearly means
- A.Hostile
- B.Prolonged
- C.Enthusiastic
- D.Lukewarm and unenthusiastic
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Correct answer: Lukewarm and unenthusiastic
"Tepid" most nearly means lukewarm and unenthusiastic because a few scattered claps that quickly died away signal a halfhearted response. "Enthusiastic" and "prolonged" overstate the weak applause, and "hostile" implies active opposition that mere faint clapping does not convey.
As used in the sentence "The candidate gave an evasive reply, dodging the reporter's question with a vague joke," the word "evasive" most nearly means
- A.Lengthy and detailed
- B.Avoiding a direct answer
- C.Honest and direct
- D.Openly hostile
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Correct answer: Avoiding a direct answer
"Evasive" most nearly means avoiding a direct answer because dodging the question with a vague joke is precisely sidestepping a straight reply. "Honest and direct" is the opposite, "lengthy and detailed" contradicts the vagueness, and "openly hostile" describes aggression rather than avoidance.
As used in the sentence "The town's resilient residents rebuilt their homes within a year of the flood," the word "resilient" most nearly means
- A.Easily discouraged
- B.Able to recover quickly
- C.Wealthy and comfortable
- D.Slow and reluctant
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Correct answer: Able to recover quickly
"Resilient" most nearly means able to recover quickly because rebuilding homes within a year of a flood demonstrates a fast bounce-back from hardship. "Easily discouraged" and "slow and reluctant" contradict that swift recovery, and "wealthy and comfortable" introduces a financial idea the sentence does not support.
As used in the sentence "Her gregarious nature made her the center of every gathering she attended," the word "gregarious" most nearly means
- A.Selfish and cold
- B.Shy and withdrawn
- C.Sociable and outgoing
- D.Quiet and serious
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Correct answer: Sociable and outgoing
"Gregarious" most nearly means sociable and outgoing because being the center of every gathering reflects someone who enjoys and seeks out company. "Shy and withdrawn" is the opposite, "quiet and serious" misses the social energy, and "selfish and cold" contradicts the welcoming presence described.
A passage first defines a rare genetic condition, then describes three patients who live with it, and closes by explaining a promising new therapy. Which choice best describes the overall structure of the passage?
- A.It compares two unrelated diseases
- B.It argues against pursuing any treatment
- C.It narrates the discovery of the gene in chronological order
- D.It defines a condition, illustrates it with cases, and points to a treatment
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Correct answer: It defines a condition, illustrates it with cases, and points to a treatment
The structure defines a condition, illustrates it with cases, and points to a treatment, matching the passage's move from definition to three patient examples to a new therapy. Arguing against treatment contradicts the promising therapy, and the passage neither compares two diseases nor narrates a gene's discovery.
A passage states: "The factory did boost local employment. Yet within a decade, the river it polluted had killed off the fishing trade that once supported the same town." What is the function of the second sentence?
- A.To list the chemicals the factory released
- B.To restate the first sentence in stronger words
- C.To compare the town with a neighboring city
- D.To present a long-term consequence that complicates the initial benefit
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Correct answer: To present a long-term consequence that complicates the initial benefit
The second sentence presents a long-term consequence that complicates the initial benefit, because "Yet" signals that the lost fishing trade undercuts the employment gain. It does not restate the first sentence, list chemicals, or compare towns, none of which appears in the passage.
A passage opens by quoting a folk saying about the weather before reviewing modern forecasting science. What is the most likely purpose of beginning with the folk saying?
- A.To prove the saying is scientifically accurate
- B.To list the steps of a forecasting procedure
- C.To provide the passage's final recommendation
- D.To introduce the topic in a familiar, attention-getting way
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Correct answer: To introduce the topic in a familiar, attention-getting way
The purpose is to introduce the topic in a familiar, attention-getting way, because a well-known folk saying eases readers into a discussion of forecasting. Proving the saying accurate is not the aim of an opening hook, and stating a recommendation or listing procedural steps would come later, not at the start.
A passage devotes its opening to common myths about a deep-sea creature and its remainder to the facts that overturn each myth. Which choice best describes how the passage is organized?
- A.It gives instructions for diving safely
- B.It tells the life story of a single researcher
- C.It presents misconceptions and then corrects them
- D.It ranks sea creatures from largest to smallest
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Correct answer: It presents misconceptions and then corrects them
The passage presents misconceptions and then corrects them, since it opens with myths and then supplies facts that overturn each one. Ranking creatures, telling a researcher's life story, and giving diving instructions describe organizations the passage does not use.
A passage about a city park ends with the line, "Perhaps it is time we asked not what the park costs us, but what we would lose without it." What is the function of this closing line?
- A.To correct an error made earlier in the passage
- B.To introduce a brand-new topic unrelated to the park
- C.To reframe the issue and prompt the reader to reconsider its value
- D.To supply a precise budget figure for the park
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Correct answer: To reframe the issue and prompt the reader to reconsider its value
The closing line reframes the issue and prompts the reader to reconsider the park's value, shifting from cost to what would be lost without it. It offers no budget figure, introduces no unrelated topic, and corrects no earlier error.
A passage traces how a single invention moved from a laboratory curiosity, to a luxury product, to an everyday household item. Which choice best describes the overall structure?
- A.It compares the invention to a competing device
- B.It follows the invention's progression through distinct stages over time
- C.It refutes the value of the invention
- D.It lists random facts about the inventor's childhood
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Correct answer: It follows the invention's progression through distinct stages over time
The structure follows the invention's progression through distinct stages over time, moving from lab curiosity to luxury to household staple. Refuting the invention's value contradicts the passage, and it neither compares devices nor lists childhood facts.
A passage about volunteer firefighting begins with a tense rescue scene, then steps back to explain how volunteer crews are trained and funded. What is the function of opening with the rescue scene?
- A.To provide the statistical conclusion of the passage
- B.To dramatize the stakes before explaining how the system works
- C.To list the equipment used in firefighting
- D.To define the word 'volunteer' for readers
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Correct answer: To dramatize the stakes before explaining how the system works
Opening with the rescue scene dramatizes the stakes before the passage explains how the system works, drawing readers in with urgency before shifting to training and funding. It is not a statistical conclusion, an equipment list, or a definition.
Text 1 argues that wind farms are essential for cutting carbon emissions. Text 2 contends that poorly placed wind farms endanger migratory birds. Which statement best describes the relationship between the texts?
- A.The two texts discuss unrelated energy sources
- B.They reach identical conclusions about wind farms
- C.Text 2 raises a concern that qualifies Text 1's enthusiasm
- D.Text 2 denies that wind farms exist
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Correct answer: Text 2 raises a concern that qualifies Text 1's enthusiasm
Text 2 raises a concern that qualifies Text 1's enthusiasm, because it points to a drawback (danger to birds) that tempers Text 1's claim that wind farms are essential. The texts do not reach identical conclusions, deny that wind farms exist, or address unrelated sources.
Text 1 reports that a school's new later start time improved students' test scores. Text 2 finds that the same schools simultaneously hired more tutors. How does Text 2 most directly affect a reader's view of Text 1?
- A.It denies that test scores rose at all
- B.It changes the topic to school lunches
- C.It confirms the start time alone caused the gains
- D.It suggests another factor may explain the improved scores
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Correct answer: It suggests another factor may explain the improved scores
Text 2 suggests another factor may explain the improved scores, since hiring more tutors at the same time offers a competing cause for the gains Text 1 credits to the later start. It does not confirm a single cause, deny the score increase, or change the subject.
Text 1 maintains that ancient ruins should be left undisturbed to preserve them. Text 2 argues that careful excavation is the only way to learn what such sites can teach. On what underlying point do both texts agree, even as they differ?
- A.That the sites should be destroyed
- B.That the ruins should be sold to private collectors
- C.That the ruins have value worth protecting or studying
- D.That archaeology has nothing to offer
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Correct answer: That the ruins have value worth protecting or studying
Both texts agree that the ruins have value worth protecting or studying, even though they disagree about whether to leave them alone or excavate them. Selling, dismissing archaeology, and destroying the sites are positions neither text shares.
Text 1 claims that switching to electric buses will sharply reduce a city's air pollution. Text 2 notes that the city's electricity still comes largely from coal-fired plants. How would the author of Text 2 most likely respond to Text 1?
- A.By arguing the topic has nothing to do with pollution
- B.By insisting electric buses cannot be built
- C.By cautioning that the pollution benefit may be smaller than claimed
- D.By fully endorsing the pollution claim without qualification
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Correct answer: By cautioning that the pollution benefit may be smaller than claimed
The author of Text 2 would most likely caution that the pollution benefit may be smaller than claimed, because coal-generated electricity shifts emissions rather than eliminating them. Full endorsement ignores that complication, and denying the buses or the topic's relevance misreads both texts.
Text 1 presents a famous explorer as a fearless pioneer. Text 2 argues that the same explorer relied heavily on local guides whose contributions went unrecorded. Which statement best captures how Text 2 relates to Text 1?
- A.It complicates Text 1's heroic portrait by adding overlooked context
- B.It denies that the explorer ever traveled
- C.It confirms Text 1's account in every respect
- D.It discusses a completely different explorer
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Correct answer: It complicates Text 1's heroic portrait by adding overlooked context
Text 2 complicates Text 1's heroic portrait by adding overlooked context, crediting unrecorded local guides for what Text 1 framed as solo heroism. It does not confirm Text 1 fully, switch to another explorer, or deny the travels.
Text 1 and Text 2 both evaluate a proposal to ban gas-powered leaf blowers. Text 1 stresses the noise and emissions the ban would eliminate, while Text 2 stresses the added costs landscaping crews would face. The two authors most clearly differ in
- A.How many sentences each text contains
- B.Whether the proposal is being debated at all
- C.Which consequences of the ban they treat as most important
- D.The spelling of the word 'blower'
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Correct answer: Which consequences of the ban they treat as most important
The authors most clearly differ in which consequences they treat as most important, with Text 1 foregrounding noise and emissions and Text 2 foregrounding costs to crews. Whether the proposal exists, spelling, and sentence count are surface matters that miss the substantive difference.
Read the sentence: "The novel's narrator is utterly ______; readers can never be sure whether to trust a single thing he reports." Which choice best completes the text?
- A.Wealthy
- B.Dependable
- C.Unreliable
- D.Cheerful
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Correct answer: Unreliable
"Unreliable" is the answer because readers never being sure whether to trust the narrator's reports describes a narrator who cannot be depended on, the meaning of unreliable. "Dependable" is the opposite, and "cheerful" and "wealthy" describe traits the sentence never connects to trust.
As used in the sentence "The committee reached a consensus only after hours of patient negotiation," the word "consensus" most nearly means
- A.A final vote against
- B.A deep disagreement
- C.A written contract
- D.A general agreement
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Correct answer: A general agreement
"Consensus" most nearly means a general agreement because reaching one after patient negotiation describes the group arriving at a shared position. "A deep disagreement" is the opposite, "a final vote against" implies rejection, and "a written contract" names a document rather than the agreement itself.
Read the sentence: "What had seemed an ______ obstacle dissolved the moment the two sides agreed to share the data." Which choice best completes the text?
- A.Minor
- B.Insurmountable
- C.Imaginary
- D.Expected
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Correct answer: Insurmountable
"Insurmountable" is the answer because the contrast between an obstacle that "seemed" impossible and one that quickly "dissolved" requires a word meaning seemingly impossible to overcome. "Minor" would undercut the surprise of its dissolving, and "imaginary" and "expected" do not capture the sense of a daunting barrier.
A passage about a riverside town opens with a serene description of daily life, then recounts a sudden flood, and ends by detailing how the community rebuilt. Which choice best describes the overall structure?
- A.It moves from calm, to disruption, to recovery
- B.It lists flood-safety rules in numbered order
- C.It compares two rival towns throughout
- D.It argues the town should be abandoned
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Correct answer: It moves from calm, to disruption, to recovery
The structure moves from calm, to disruption, to recovery, tracing serene daily life, the flood, and the rebuilding in sequence. It does not compare rival towns, list safety rules, or argue for abandonment.
As used in the sentence "The chef's signature dish was deceptively simple, hiding a dozen subtle techniques beneath its plain appearance," the phrase "deceptively simple" most nearly means
- A.Appearing easy but actually complex
- B.Unpleasant to look at
- C.Genuinely effortless to make
- D.Famous and expensive
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Correct answer: Appearing easy but actually complex
"Deceptively simple" most nearly means appearing easy but actually complex because the dish hides a dozen subtle techniques beneath a plain appearance. "Genuinely effortless" ignores the hidden complexity, while "unpleasant to look at" and "famous and expensive" address qualities the phrase does not describe.
Text 1 argues that requiring uniforms reduces distractions and bullying in schools. Text 2 contends that uniforms suppress students' individuality and self-expression. The relationship between the texts is best described as
- A.Text 2 supports Text 1 with an example
- B.Text 2 opposes Text 1 by stressing a competing value
- C.Text 2 addresses an entirely different policy
- D.Text 2 repeats Text 1's argument in new words
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Correct answer: Text 2 opposes Text 1 by stressing a competing value
Text 2 opposes Text 1 by stressing a competing value, weighing individuality and self-expression against the order and safety Text 1 emphasizes. It does not repeat or support Text 1, and both texts address the same uniform policy.
A passage begins by describing the cramped, smoke-filled offices of early newspapers before tracing how the industry moved online. What is the most likely function of the opening description?
- A.To set a vivid historical scene before charting the industry's change
- B.To list the names of every early editor
- C.To provide the passage's concluding statistic
- D.To argue that print newspapers were superior
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Correct answer: To set a vivid historical scene before charting the industry's change
The opening description sets a vivid historical scene before charting the industry's change, grounding the later shift online in a concrete picture of the past. It does not argue print's superiority, list editors, or deliver a concluding statistic.
Read the sentence: "The treaty's language was intentionally ______, broad enough that each nation could claim it favored their position." Which choice best completes the text?
- A.Trivial
- B.Elastic
- C.Hostile
- D.Precise
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Correct answer: Elastic
"Elastic" is the answer because language broad enough for each nation to read it in its own favor is flexible and stretchable in meaning, which elastic captures. "Precise" is the opposite of broad, while "hostile" and "trivial" describe tone or importance rather than flexibility of meaning.
Text 1 reports that a popular fitness app helped users exercise more often. Text 2 observes that most of those users were already regular exercisers before downloading it. How does Text 2 most directly relate to Text 1's claim?
- A.It shifts the subject to dieting apps
- B.It complicates the claim by questioning whether the app caused the change
- C.It strengthens the claim with new participants
- D.It denies that the app was ever released
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Correct answer: It complicates the claim by questioning whether the app caused the change
Text 2 complicates the claim by questioning whether the app caused the change, since users who already exercised regularly may have done so regardless of the app. It does not add participants, deny the app's release, or switch topics.
Read the sentence: "The veteran reporter's questions were ______, cutting straight to the contradictions the official had hoped to hide." Which choice best completes the text?
- A.Irrelevant
- B.Incisive
- C.Rambling
- D.Timid
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Correct answer: Incisive
"Incisive" is the answer because questions that cut straight to hidden contradictions are sharp and penetrating, which incisive means. "Rambling" and "irrelevant" suggest the opposite of pointed questioning, and "timid" would not fit questions that expose what someone hoped to hide.
Information and Ideas (60)
Within the Information and Ideas domain on the Digital SAT Reading and Writing section, which official question type asks a student to choose the quotation or detail that best backs up a stated claim?
- A.Words in Context
- B.Command of Evidence
- C.Transitions
- D.Text Structure and Purpose
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Correct answer: Command of Evidence
Command of Evidence is correct because it is the Information and Ideas question type that asks which quotation or detail most directly supports a given claim. Words in Context and Text Structure and Purpose belong to Craft and Structure, and Transitions belongs to Expression of Ideas.
On the Digital SAT, an Inferences question typically asks a student to do which of the following?
- A.Define an underlined word using its surrounding context
- B.Choose the most logical word or phrase to complete a passage's reasoning
- C.Correct a punctuation error at a clause boundary
- D.Combine bulleted notes to accomplish a writing goal
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Correct answer: Choose the most logical word or phrase to complete a passage's reasoning
Choosing the most logical word or phrase to complete a passage's reasoning is correct because Inferences items present a short passage ending in a blank and ask which conclusion follows from the information given. Correcting punctuation is a Conventions skill, combining notes is Rhetorical Synthesis, and defining a word is a Craft and Structure task.
In an Information and Ideas question that supplies a graph or table, the credited Command of Evidence answer must be one that is ______.
- A.Placed at the end of the passage
- B.Supported by the data shown in the graphic
- C.The shortest of the four choices
- D.Written in the most formal tone
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Correct answer: Supported by the data shown in the graphic
Supported by the data shown in the graphic is correct because a quantitative Command of Evidence item rewards the choice the data actually backs. Length, tone, and position within the passage have no bearing on whether a choice is justified by the data.
A Central Ideas and Details question on the Digital SAT most often asks a student to identify which of the following?
- A.The grammatical subject of an underlined sentence
- B.The connotation of a single underlined word
- C.The most concise way to phrase a wordy sentence
- D.The main point a passage conveys or a specific detail it provides
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Correct answer: The main point a passage conveys or a specific detail it provides
The main point a passage conveys or a specific detail it provides is correct because Central Ideas and Details items test comprehension of a text's overall idea and its supporting specifics. Identifying a grammatical subject and condensing wordy phrasing are Conventions and Expression of Ideas tasks, and judging a word's connotation is a Craft and Structure skill.
A short passage explains that a museum began offering free admission on weekday evenings, and attendance during those hours tripled within a month, drawing many first-time visitors. Which choice best states the main idea?
- A.Offering free evening hours greatly increased the museum's attendance
- B.The museum's daytime attendance fell sharply after the change
- C.Most evening visitors had been to the museum many times before
- D.The museum lost money because of the free evening hours
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Correct answer: Offering free evening hours greatly increased the museum's attendance
Offering free evening hours greatly increasing the museum's attendance is correct because the passage states attendance tripled and drew many first-timers. A drop in daytime attendance and a loss of money are not mentioned, and saying most visitors were repeat guests contradicts the note about first-time visitors.
A passage describes a city that planted rows of trees along a hot downtown street; afterward, surface temperatures on the street dropped, foot traffic to nearby shops rose, and residents reported the area felt more inviting. Which detail most directly supports the claim that the trees cooled the street?
- A.Foot traffic to nearby shops rose
- B.The trees were planted in rows downtown
- C.Surface temperatures on the street dropped
- D.Residents reported the area felt more inviting
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Correct answer: Surface temperatures on the street dropped
Surface temperatures on the street dropping is correct because cooling is most directly shown by lower temperatures. Increased foot traffic and a more inviting feel describe effects on people's behavior and impressions, and the planting arrangement describes what was done rather than its cooling effect.
A passage notes that a popular novelist published her early books under a male pen name because, at the time, editors rarely accepted adventure stories credited to women. According to the passage, which detail most directly explains why she used the pen name?
- A.She wrote in several different genres over her career
- B.Her novels were eventually adapted into successful films
- C.Her books were printed in unusually large quantities
- D.Editors at the time rarely accepted adventure stories credited to women
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Correct answer: Editors at the time rarely accepted adventure stories credited to women
Editors rarely accepting adventure stories credited to women is correct because the passage gives this bias as the reason she adopted a male pen name. The film adaptations, her range of genres, and her print runs describe her success and output rather than the motivation behind the pen name.
A passage argues that, while a new highway shortened commute times, it also encouraged so many people to drive that within a few years traffic was as heavy as before, suggesting that adding road capacity does not permanently relieve congestion. What is the central idea?
- A.Commute times became far longer immediately after the highway opened
- B.The highway permanently eliminated traffic congestion
- C.Adding road capacity may not provide lasting relief from congestion
- D.Drivers stopped using the highway soon after it was built
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Correct answer: Adding road capacity may not provide lasting relief from congestion
Adding road capacity not providing lasting relief from congestion is correct because the passage shows traffic returned to former levels after a short improvement. Claiming the highway permanently ended congestion contradicts that return, and saying commutes lengthened immediately or that drivers abandoned the road conflicts with the described initial improvement and heavy use.
A passage reports that a school replaced its long midday break with two shorter recess periods spread through the day, and afterward teachers observed that students were more focused during afternoon lessons than they had been under the single break. The central idea is that ______.
- A.Longer single breaks are always better for student attention
- B.The school eliminated recess entirely to save time
- C.Spreading shorter breaks through the day improved students' afternoon focus
- D.Students lost all interest in afternoon lessons after the change
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Correct answer: Spreading shorter breaks through the day improved students' afternoon focus
Spreading shorter breaks through the day improving students' afternoon focus is correct because teachers observed better afternoon focus after the schedule changed. Eliminating recess contradicts the two new recess periods, losing all interest reverses the observed improvement, and praising longer single breaks contradicts the passage's finding.
A passage states that a coastal town built a system of sand dunes and native grasses in front of its beachfront homes, and during the next major storm those homes suffered far less flooding than nearby homes left unprotected. Which choice most logically completes the inference that the dunes and grasses most likely ______?
- A.Helped shield the homes behind them from storm flooding
- B.Caused the protected homes to flood more severely
- C.Were destroyed before the storm arrived
- D.Had no effect on flooding during the storm
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Correct answer: Helped shield the homes behind them from storm flooding
Helping shield the homes behind them from storm flooding is correct because protected homes flooded far less than unprotected ones, indicating the dunes and grasses reduced flooding. Claiming no effect and being destroyed beforehand contradict the reduced flooding, and causing worse flooding reverses the comparison.
A passage explains that a bookstore began hosting weekly author readings, and although the readings themselves were free, the store's book sales on those evenings rose well above its usual nightly totals. Which choice most logically completes the inference that the readings most likely ______?
- A.Encouraged attendees to buy books while at the store
- B.Drove away the store's regular customers
- C.Reduced the store's overall book sales
- D.Had no connection to the store's sales figures
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Correct answer: Encouraged attendees to buy books while at the store
Encouraging attendees to buy books while at the store is correct because sales rose above usual totals on reading nights, linking the free events to higher purchases. Driving away customers and reducing sales contradict the rise, and claiming no connection ignores the clear increase on those evenings.
A passage describes a desert plant whose seeds will sprout only after being soaked by a heavy rain, a condition that occurs rarely in its dry habitat. Which choice most logically completes the inference that this trait most likely helps the plant by ______?
- A.Preventing its seeds from ever sprouting at all
- B.Ensuring seedlings appear only when enough water is available to survive
- C.Forcing the seeds to sprout during the driest part of the year
- D.Allowing the plant to grow without any water
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Correct answer: Ensuring seedlings appear only when enough water is available to survive
Ensuring seedlings appear only when enough water is available to survive is correct because requiring a heavy soaking means seeds sprout only after substantial rain, when young plants have a better chance. Never sprouting and growing without water contradict the plant's reproduction, and sprouting during the driest period would leave seedlings without the rain the trait depends on.
A passage notes that a factory installed new machines that use less electricity per item produced, yet the factory's total electricity bill rose because the cheaper operation led managers to greatly increase production. Which choice most logically completes the inference that, in this case, the energy savings per item ______?
- A.Were the sole reason the total bill stayed unchanged
- B.Made each item far more expensive to produce
- C.Caused the factory to shut down production entirely
- D.Were more than offset by the rise in total output
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Correct answer: Were more than offset by the rise in total output
Being more than offset by the rise in total output is correct because per-item savings were outweighed when production increased enough to raise the total bill. Shutting down contradicts the increased production, an unchanged bill conflicts with the stated rise, and per-item costs fell rather than rose under the more efficient machines.
A passage reports that researchers played recorded predator calls near a flock of feeding birds, and the birds immediately stopped eating, fell silent, and crouched low until the calls ended. Which choice most logically completes the inference that the birds most likely ______?
- A.Continued feeding without interruption
- B.Treated the recorded calls as a sign of nearby danger
- C.Failed to notice the recorded calls
- D.Were attracted toward the source of the calls
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Correct answer: Treated the recorded calls as a sign of nearby danger
Treating the recorded calls as a sign of nearby danger is correct because the birds froze, silenced, and crouched, behaviors typical of a response to threat. Failing to notice the calls and continuing to feed contradict their reaction, and being attracted toward the source conflicts with their crouching and falling silent.
A passage explains that a region's only newspaper closed, and in the following years voter turnout in local elections declined and fewer residents could name their town officials. Which choice most logically completes the inference that the newspaper had most likely been ______?
- A.Unrelated to residents' awareness of local government
- B.Preventing residents from learning who their officials were
- C.Discouraging residents from voting in local elections
- D.An important source of information about local civic affairs
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Correct answer: An important source of information about local civic affairs
Being an important source of information about local civic affairs is correct because turnout fell and residents knew their officials less after the paper closed, suggesting it had informed them. Discouraging voting and preventing residents from learning about officials reverse this role, and calling it unrelated ignores the changes that followed its closure.
A passage describes a remote island where, after rats were accidentally introduced by ships, several ground-nesting bird species that had thrived for centuries vanished within a few decades. Which choice most logically completes the inference that the rats most likely ______?
- A.Contributed to the disappearance of the ground-nesting birds
- B.Arrived only after the birds had already vanished
- C.Helped the ground-nesting birds increase in number
- D.Had no impact on the island's bird populations
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Correct answer: Contributed to the disappearance of the ground-nesting birds
Contributing to the disappearance of the ground-nesting birds is correct because long-thriving birds vanished soon after rats arrived, pointing to the rats as a cause. Claiming no impact and helping the birds increase contradict the decline, and arriving only after the birds vanished reverses the order the passage gives.
A passage states that a clothing company published the locations and wages of every factory it used, and although some rivals predicted the disclosure would scare off customers, the company's sales grew as shoppers praised its openness. Which choice most logically completes the inference that, for these shoppers, the disclosure ______?
- A.Made the company seem less trustworthy
- B.Had no effect on their purchasing decisions
- C.Increased their willingness to buy from the company
- D.Convinced them the company mistreated its workers
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Correct answer: Increased their willingness to buy from the company
Increasing their willingness to buy from the company is correct because sales grew and shoppers praised the openness after the disclosure. Seeming less trustworthy and believing workers were mistreated contradict that praise, and claiming no effect ignores the rise in sales tied to the disclosure.
A student wants to support the claim that a particular composer drew heavily on folk melodies from her home region. Which piece of evidence would most directly support that claim?
- A.She composed steadily for more than forty years
- B.She studied at a prestigious conservatory abroad
- C.Her concerts regularly sold out in major cities
- D.Several of her symphonies quote traditional songs sung in her region
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Correct answer: Several of her symphonies quote traditional songs sung in her region
Several of her symphonies quoting traditional songs from her region is correct because borrowing those melodies directly shows she drew on regional folk music. Her conservatory training, sold-out concerts, and long career describe her education, popularity, and output rather than her use of folk melodies.
A researcher hypothesizes that a city's new nighttime bus service reduced drunk-driving incidents. Which finding would most strongly support this hypothesis?
- A.Reported drunk-driving incidents fell sharply only after the night service began
- B.The buses were funded by a local business association
- C.The buses were painted in bright, eye-catching colors
- D.The city already had a large network of daytime buses
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Correct answer: Reported drunk-driving incidents fell sharply only after the night service began
Reported drunk-driving incidents falling sharply only after the night service began is correct because the timing directly links the new service to fewer incidents. The buses' color, their funding, and the existing daytime network describe features and context rather than a change in drunk-driving rates.
A passage proposes that an explorer kept detailed records mainly to settle future disputes over who reached a region first. Which detail would most directly support this proposal?
- A.His journals carefully note the exact date and time he arrived at each landmark
- B.He was admired by other explorers of his generation
- C.His expeditions were financed by a wealthy patron
- D.He carried lightweight, durable equipment on his journeys
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Correct answer: His journals carefully note the exact date and time he arrived at each landmark
His journals carefully noting the exact date and time of each arrival is correct because precise timing records are what would settle disputes over who arrived first. His equipment, his reputation, and his financing concern how he traveled, how he was regarded, and who paid, not his aim of documenting priority.
A geologist claims that a layer of rock formed underwater rather than on dry land. Which observation would most directly support this claim?
- A.The rock layer contains the fossilized shells of marine organisms
- B.The rock layer is exposed at the top of a tall cliff
- C.The rock layer is a dark gray color
- D.The rock layer is unusually thick
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Correct answer: The rock layer contains the fossilized shells of marine organisms
The rock layer containing fossilized shells of marine organisms is correct because the presence of sea creatures directly indicates the rock formed in water. The layer's thickness, its current exposure on a cliff, and its color describe its size, location today, and appearance rather than the setting in which it formed.
A passage hypothesizes that a playwright wrote her comedies primarily to criticize the social customs of her time. Which detail would most directly support this hypothesis?
- A.She wrote both comedies and tragedies over her career
- B.Her comedies repeatedly mock specific manners and rules her society took seriously
- C.Her plays featured large casts of characters
- D.Her plays were performed in the largest theaters of the era
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Correct answer: Her comedies repeatedly mock specific manners and rules her society took seriously
Her comedies repeatedly mocking specific manners and rules her society took seriously is correct because ridiculing those customs directly shows an aim to criticize them. The size of the theaters, her work in two genres, and her large casts describe staging and scope rather than a critical purpose.
A historian argues that a medieval town grew wealthy chiefly because it controlled a key river crossing used by traders. Which piece of evidence would most strongly support this argument?
- A.The town's buildings were made mostly of timber
- B.The town held a yearly religious festival
- C.The town was surrounded by a high stone wall
- D.The town's surviving records show large tolls collected from merchants crossing the river
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Correct answer: The town's surviving records show large tolls collected from merchants crossing the river
Surviving records showing large tolls collected from merchants crossing the river is correct because toll income from the crossing directly ties the town's wealth to its control of the route. The wall, the festival, and the building material describe its defenses, traditions, and construction rather than the source of its wealth.
A passage claims that a beekeeper's hives stayed healthier than others in the area because she planted a wide variety of flowers nearby. Which observation would most directly support this claim?
- A.Her bees had access to many flower types and showed lower rates of disease than bees with limited forage
- B.Her property was located far from the nearest town
- C.She had kept bees for many years
- D.Her hives were painted in several different colors
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Correct answer: Her bees had access to many flower types and showed lower rates of disease than bees with limited forage
Her bees having access to many flower types and showing lower disease rates than bees with limited forage is correct because linking the varied flowers to healthier bees directly supports the claim. The hive colors, her years of experience, and her property's distance from town describe appearance, background, and location rather than the effect of varied forage on hive health.
A passage claims that a tutoring program improved students' grades, and a table shows that the share of participating students earning a B or higher rose from 40 percent before the program to 68 percent after it. Which choice best uses the data to support the claim?
- A.Every student earned a B or higher after the program
- B.The share earning a B or higher rose after the program
- C.The share of students earning a B or higher decreased after the program
- D.The share earning a B or higher was unchanged by the program
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Correct answer: The share earning a B or higher rose after the program
The share earning a B or higher rising after the program is correct because 68 percent is greater than the earlier 40 percent, supporting the claim of improved grades. A decrease and no change contradict the rise, and saying every student earned a B or higher conflicts with the 68 percent figure, which leaves nearly a third below that mark.
A passage about recycling includes a table showing a town's recycling rate as 22 percent in 2022, 31 percent in 2023, and 45 percent in 2024. Which choice best uses the data to complete the claim that the town's recycling rate ______ over the period?
- A.Fell each year
- B.Stayed the same each year
- C.Increased each year
- D.Reached its highest value in 2022
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Correct answer: Increased each year
Increasing each year is correct because the rate rises from 22 percent to 31 percent to 45 percent across the three years. Falling and staying the same contradict the climbing figures, and calling 2022 the highest reverses the data, since 45 percent in 2024 is the largest value.
A passage claims that a wildlife reserve's elephant population recovered after poaching was banned, and a table shows the population was 120 in 2018 and 210 in 2024. Which choice best uses the data to support the claim?
- A.The elephant population shrank between 2018 and 2024
- B.The elephant population was the same in both years
- C.The elephant population grew between 2018 and 2024
- D.The reserve had no elephants by 2024
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Correct answer: The elephant population grew between 2018 and 2024
The elephant population growing between 2018 and 2024 is correct because 210 is greater than the earlier 120, supporting the claim of recovery. A shrinking or unchanged population contradicts the increase, and having no elephants by 2024 conflicts with the figure of 210.
A passage reports that a hospital began offering nurses a quiet room for short breaks during long shifts, and over the next six months reported medication errors on those units dropped noticeably. Which inference is best supported?
- A.The break room likely helped reduce errors by easing fatigue
- B.Medication errors rose after the break room opened
- C.Nurses stopped working long shifts altogether
- D.The hospital hired no additional staff that year
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Correct answer: The break room likely helped reduce errors by easing fatigue
It is best supported that the break room likely helped reduce errors by easing fatigue, because errors dropped after the room was introduced. Saying errors rose contradicts the reported decline, and the passage says nothing about ending long shifts or hiring levels.
A passage about a glassblower returns again and again to how she revived a nearly forgotten regional technique and taught it to dozens of younger artisans. Which choice best states the central idea?
- A.She preferred working in solitude
- B.Her studio was located far from any city
- C.She kept the technique secret from others
- D.She rescued and passed on a fading regional craft technique
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Correct answer: She rescued and passed on a fading regional craft technique
The central idea is that she rescued and passed on a fading regional craft technique, because the passage stresses both reviving the method and teaching it to others. Keeping it secret directly contradicts her teaching, and her location or preference for solitude are not the unifying point.
A passage claims that adding shaded shelters at bus stops increased ridership. A table shows average daily boardings at the improved stops rose from 320 to 470, while boardings at unimproved stops stayed near 300. Which choice best uses the data to support the claim?
- A.No boardings were recorded at any stop
- B.Boardings rose equally at improved and unimproved stops
- C.Boardings rose at improved stops but stayed flat at unimproved stops
- D.Boardings at improved stops fell after the shelters were added
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Correct answer: Boardings rose at improved stops but stayed flat at unimproved stops
Boardings rising at improved stops but staying flat at unimproved stops best supports the claim, because the increase appears only where shelters were added. A decline contradicts the figures, equal rises ignore the flat unimproved total, and no boardings conflicts with both numbers.
A passage explains that a vineyard owner suspects that a row of lavender planted between her grapevines attracts insects that prey on grape pests. Which finding, if true, would most strongly support this idea?
- A.The vineyard has operated for over twenty years
- B.The lavender produces a strong, pleasant scent
- C.The lavender blooms later than the grapevines
- D.Rows bordered by lavender show fewer grape pests than rows without it
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Correct answer: Rows bordered by lavender show fewer grape pests than rows without it
Rows bordered by lavender showing fewer grape pests than rows without it would most strongly support the idea, because that contrast links the lavender to pest reduction. The lavender's scent, its bloom timing, and the vineyard's age describe traits and background rather than an effect on pests.
A short passage states that a community center began offering evening childcare, and afterward attendance at its adult night classes rose sharply, with many new students saying they could finally attend. Which choice best states the main idea?
- A.The evening childcare enabled more adults to attend night classes
- B.The night classes were canceled for lack of interest
- C.Adults stopped attending daytime programs entirely
- D.The community center reduced its class offerings
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Correct answer: The evening childcare enabled more adults to attend night classes
The evening childcare enabling more adults to attend night classes is correct because attendance rose and new students credited the childcare. Canceling classes and reducing offerings contradict the rise, and abandoning daytime programs is never mentioned.
A passage describes a lake where, after a town stopped using a phosphate-heavy fertilizer on nearby lawns, the thick algae blooms that had clouded the water each summer grew far smaller. Which choice most logically completes the inference that the phosphate runoff most likely ______?
- A.Cleared the lake of all algae
- B.Had been feeding the lake's summer algae blooms
- C.Arrived only after the blooms disappeared
- D.Had no relationship to the algae blooms
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Correct answer: Had been feeding the lake's summer algae blooms
Having been feeding the lake's summer algae blooms is correct because the blooms shrank once the phosphate fertilizer stopped, pointing to it as a cause. Clearing the lake reverses the effect, no relationship ignores the change, and arriving after the blooms disappeared misorders the events.
A passage argues that a chef's restaurant earned its reputation mainly for reinventing humble street foods as elaborate dishes. Which detail would most directly support that argument?
- A.The menu features refined versions of dishes once sold by sidewalk vendors
- B.The restaurant occupies a renovated warehouse
- C.The chef trained in several countries
- D.The restaurant is booked weeks in advance
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Correct answer: The menu features refined versions of dishes once sold by sidewalk vendors
The menu featuring refined versions of dishes once sold by sidewalk vendors is correct because transforming street foods into elaborate dishes directly shows the reinvention the argument describes. The building, the chef's training, and the wait for a table describe setting, background, and popularity rather than the reinvention itself.
A passage about a frontier settlement keeps returning to how its survival depended on an agreement with a neighboring Indigenous nation that taught settlers which local crops would endure the climate. Which choice best states the central idea?
- A.The settlement's houses were built of stone
- B.The settlers preferred imported foods
- C.The settlement was the largest in the region
- D.An alliance and shared agricultural knowledge were key to the settlement's survival
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Correct answer: An alliance and shared agricultural knowledge were key to the settlement's survival
An alliance and shared agricultural knowledge being key to the settlement's survival is correct because the passage ties survival to the agreement and the crop knowledge it provided. The houses' material, a preference for imported foods, and the settlement's size are incidental or contradicted details.
A passage notes that an airline began boarding passengers from the back rows toward the front, and average boarding time per flight afterward fell by several minutes. Which inference is best supported?
- A.The airline reduced the number of flights
- B.Passengers stopped bringing carry-on bags
- C.Boarding time increased after the change
- D.The back-to-front order likely sped up boarding
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Correct answer: The back-to-front order likely sped up boarding
The back-to-front order likely speeding up boarding is correct because boarding time fell after the new order was adopted. An increase contradicts the reported drop, and the passage mentions nothing about carry-ons or fewer flights.
A passage claims that a clinic's free flu-shot drive lowered flu cases in its town. A table shows confirmed flu cases fell from 540 the previous winter to 310 the winter after the drive. Which choice best uses the data to support the claim?
- A.Flu cases were the same in both winters
- B.Confirmed flu cases rose after the drive
- C.Confirmed flu cases dropped by 230 after the drive
- D.No flu cases were recorded after the drive
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Correct answer: Confirmed flu cases dropped by 230 after the drive
Confirmed flu cases dropping by 230 after the drive is correct because a decrease aligns with the claim that the drive lowered cases. A rise and an unchanged total contradict the figures, and zero cases conflicts with the 310 recorded.
A passage describes an orchard owner who suspects that hanging reflective ribbons in his trees keeps fruit-eating birds away. Which observation would most directly support this suspicion?
- A.The ribbons sway in the wind
- B.The orchard sits on a gentle slope
- C.The ribbons are inexpensive to buy
- D.Trees with ribbons lose far less fruit to birds than trees without them
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Correct answer: Trees with ribbons lose far less fruit to birds than trees without them
Trees with ribbons losing far less fruit to birds than trees without them is correct because that comparison directly links the ribbons to reduced bird damage. The ribbons' movement, the orchard's slope, and the cost of the ribbons describe features and circumstances rather than an effect on the birds.
A passage states that a publisher of language-learning apps removed all timers from its lessons, and afterward the share of users who completed a full course rose substantially. Which choice most logically completes the inference that the timers had most likely been ______?
- A.The only reason users started the app
- B.Discouraging some users from completing courses
- C.Unrelated to whether users finished
- D.Helping users finish their courses faster
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Correct answer: Discouraging some users from completing courses
Discouraging some users from completing courses is correct because completion rose once the timers were removed, suggesting they had been a barrier. Helping users finish reverses the result, no relationship ignores the rise, and being the only reason users started the app is unsupported.
A passage about a documentary filmmaker emphasizes repeatedly that he spent years living within the communities he filmed so that his subjects would speak candidly on camera. Which choice best states the central idea?
- A.His long immersion in communities earned the candor that defined his films
- B.He completed each film in under a month
- C.He used the most advanced cameras available
- D.He rarely spoke with the people he filmed
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Correct answer: His long immersion in communities earned the candor that defined his films
His long immersion in communities earning the candor that defined his films is correct because the passage ties his years of living among subjects to their willingness to speak openly. Rarely speaking with subjects contradicts that immersion, and his equipment or pace are not the unifying point.
A passage describes a study comparing two groups of office workers, one given adjustable standing desks and one kept at fixed seated desks, with all other conditions matched, and finds the standing-desk group reported less back pain. To best support the claim that the desks caused the difference, which detail matters most?
- A.The two groups were alike except for the type of desk
- B.The desks came in two colors
- C.The office had large windows
- D.The study lasted several weeks
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Correct answer: The two groups were alike except for the type of desk
The two groups being alike except for the type of desk is correct because that controlled comparison is what lets the difference in back pain be attributed to the desks. The study's length, the desk colors, and the office windows do not establish that the desks caused the result.
A passage notes that a city converted an abandoned rail line into an elevated walking trail, and property values for homes along the route rose faster than the citywide average in the following years. Which inference is best supported?
- A.The trail was removed within a year
- B.The trail likely made nearby homes more desirable
- C.Residents stopped using the trail immediately
- D.Homes along the route lost all their value
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Correct answer: The trail likely made nearby homes more desirable
The trail likely making nearby homes more desirable is correct because values along the route rose faster than the city average after the trail opened. Losing all value contradicts the rise, and the passage says nothing about removing the trail or residents abandoning it.
A passage about a marine biologist returns repeatedly to how her decades of tagging a single sea-turtle population revealed migration routes no one had known about. Which choice best states the central idea?
- A.She worked mainly from a laboratory
- B.The turtles she studied were unusually large
- C.Her long-term tagging uncovered previously unknown turtle migration routes
- D.Sea turtles are difficult to capture
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Correct answer: Her long-term tagging uncovered previously unknown turtle migration routes
Her long-term tagging uncovering previously unknown turtle migration routes is correct because the passage stresses how her decades of work revealed those routes. The difficulty of capturing turtles, working from a lab, and the turtles' size are incidental or contradicted details.
A passage claims that a school's new attendance policy reduced tardiness. A table shows late arrivals fell from 210 per month to 95 per month after the policy began, while enrollment stayed the same. Which choice best uses the data to support the claim?
- A.Late arrivals more than halved after the policy began
- B.Late arrivals stayed at 210 per month
- C.Late arrivals increased after the policy began
- D.Enrollment dropped sharply, explaining the figures
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Correct answer: Late arrivals more than halved after the policy began
Late arrivals more than halving after the policy began is correct because the drop from 210 to 95 supports the claim of reduced tardiness. A rise and an unchanged total contradict the figures, and dropping enrollment is ruled out by the note that enrollment stayed the same.
A passage describes a baker who believes that letting her dough rest overnight in the refrigerator gives her bread a more complex flavor. Which observation would most directly support this belief?
- A.Her bakery opens early each morning
- B.The refrigerator is a recent purchase
- C.She bakes several kinds of bread
- D.Tasters consistently rate the overnight-rested loaves as more flavorful than same-day loaves
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Correct answer: Tasters consistently rate the overnight-rested loaves as more flavorful than same-day loaves
Tasters consistently rating the overnight-rested loaves as more flavorful than same-day loaves is correct because that comparison directly links the long rest to better flavor. The bakery's hours, the new refrigerator, and the variety of breads describe operations rather than an effect on flavor.
A passage states that after a national park began requiring visitors to reserve timed entry slots, reports of crowded trails and trampled vegetation dropped sharply. Which choice most logically completes the inference that the timed-entry system most likely ______?
- A.Drew far more visitors than before
- B.Caused the vegetation to die off faster
- C.Had no effect on trail conditions
- D.Had spread out visitors and eased pressure on the trails
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Correct answer: Had spread out visitors and eased pressure on the trails
Having spread out visitors and eased pressure on the trails is correct because crowding and trampling fell after timed entry began. Drawing more visitors and worsening the vegetation contradict the improvement, and no effect ignores the reported drop.
A short passage explains that a town fixed a long-broken streetlight at a dim intersection, and in the following months nighttime traffic collisions there fell while daytime collisions stayed the same. Which choice best states the main idea?
- A.Daytime collisions rose after the repair
- B.Restoring the light appears to have reduced nighttime collisions at the intersection
- C.Drivers avoided the intersection entirely
- D.The intersection was closed to traffic
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Correct answer: Restoring the light appears to have reduced nighttime collisions at the intersection
Restoring the light appearing to have reduced nighttime collisions is correct because nighttime crashes fell while daytime crashes were unchanged after the repair. A rise in daytime collisions contradicts the passage, and closing the intersection or drivers avoiding it are never mentioned.
A passage hypothesizes that an author wrote her travel essays chiefly to challenge the stereotypes outsiders held about her home country. Which detail would most directly support this hypothesis?
- A.Her books were translated into many languages
- B.She traveled by train whenever possible
- C.Her essays repeatedly contrast common misconceptions with everyday realities she observed
- D.Her essays were published over many years
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Correct answer: Her essays repeatedly contrast common misconceptions with everyday realities she observed
Her essays repeatedly contrasting common misconceptions with everyday realities she observed is correct because directly confronting stereotypes shows the purpose the hypothesis names. The translations, her mode of travel, and her publishing span describe reach and habits rather than a corrective aim.
A passage claims that switching a fleet of delivery vans to electric models cut the company's fuel spending. A table shows monthly fuel-and-charging costs fell from $18,000 to $7,500 after the switch. Which choice best uses the data to support the claim?
- A.Monthly energy costs were unchanged by the switch
- B.The fleet stopped making deliveries
- C.Monthly energy costs dropped by $10,500 after the switch
- D.Monthly energy costs rose after the switch
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Correct answer: Monthly energy costs dropped by $10,500 after the switch
Monthly energy costs dropping by $10,500 after the switch is correct because a decrease aligns with the claim of reduced fuel spending. A rise and an unchanged cost contradict the figures, and halting deliveries is never indicated.
A passage about a cartographer emphasizes repeatedly that she interviewed local fishers and elders to record place-names that printed maps had long omitted. Which choice best states the central idea?
- A.She worked only in coastal regions
- B.She recorded overlooked local place-names by consulting community members
- C.Her maps were drawn entirely from satellite images
- D.She charged high prices for her maps
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Correct answer: She recorded overlooked local place-names by consulting community members
Her recording overlooked local place-names by consulting community members is correct because the passage stresses her interviews with fishers and elders to capture omitted names. Drawing maps from satellite images contradicts those interviews, and her pricing or where she worked are not the unifying point.
A passage notes that a coffee shop began composting its grounds and food scraps, and over a year the weight of trash it sent to the landfill dropped by nearly half. Which inference is best supported?
- A.Composting added to the shop's landfill waste
- B.The shop produced more trash than before
- C.Composting likely diverted much of the shop's waste from the landfill
- D.The shop closed for half the year
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Correct answer: Composting likely diverted much of the shop's waste from the landfill
Composting likely diverting much of the shop's waste from the landfill is correct because landfill trash fell by nearly half after composting began. Producing more trash and adding to landfill waste contradict the drop, and closing for half the year is never stated.
A passage describes a researcher testing whether a new typeface improves reading speed for beginning readers, who gives matched groups the same story set in either the new typeface or a standard one and times their reading. What is the main purpose of using matched groups and the same story?
- A.To isolate the effect of the typeface by holding other factors constant
- B.To shorten the reading session
- C.To ensure the story is easy to understand
- D.To make the test more enjoyable
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Correct answer: To isolate the effect of the typeface by holding other factors constant
Isolating the effect of the typeface by holding other factors constant is correct because matched groups reading an identical story let differences in speed be attributed to the typeface. Making the test enjoyable, shortening the session, and easing the story do not establish what the typeface alone does.
A passage states that a city replaced a confusing five-way intersection with a roundabout, and the number of injury crashes there fell while the volume of traffic passing through stayed about the same. Which choice most logically completes the inference that the roundabout most likely ______?
- A.Made the intersection safer without reducing traffic flow
- B.Sharply cut the amount of traffic using the intersection
- C.Increased injury crashes at the intersection
- D.Had no effect on crashes at the intersection
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Correct answer: Made the intersection safer without reducing traffic flow
Making the intersection safer without reducing traffic flow is correct because injury crashes fell while traffic volume held steady. Cutting traffic conflicts with the unchanged volume, increasing crashes reverses the result, and no effect ignores the reported drop.
A passage about an architect returns again and again to how she designed buildings to channel natural breezes so they needed little air conditioning even in hot climates. Which choice best states the central idea?
- A.She used only locally quarried stone
- B.Her buildings were unusually tall
- C.She rarely finished a project on schedule
- D.Her designs relied on natural airflow to stay cool with minimal air conditioning
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Correct answer: Her designs relied on natural airflow to stay cool with minimal air conditioning
Her designs relying on natural airflow to stay cool with minimal air conditioning is correct because the passage repeatedly emphasizes channeling breezes to reduce cooling needs. Her materials, building height, and schedule are incidental or contradicted details.
A passage describes a winemaker who believes that aging her wine in barrels made from a particular oak gives it a distinctive vanilla note. Which observation would most directly support this belief?
- A.The barrels are stored in a cool cellar
- B.The barrels are imported from far away
- C.She has made wine for fifteen years
- D.Wine aged in that oak consistently shows a vanilla note absent from wine aged in steel
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Correct answer: Wine aged in that oak consistently shows a vanilla note absent from wine aged in steel
Wine aged in that oak consistently showing a vanilla note absent from wine aged in steel is correct because the contrast directly links the oak to the vanilla flavor. The barrels' origin, her years of experience, and the cellar conditions describe circumstances rather than an effect on flavor.
A passage notes that a stadium switched from disposable cups to refillable ones that fans returned for a deposit, and the amount of cup litter left in the stands after games fell dramatically. Which inference is best supported?
- A.Fans bought far fewer drinks than before
- B.The deposit system likely encouraged fans to return cups instead of discarding them
- C.The stadium banned all beverages
- D.Cup litter increased after the switch
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Correct answer: The deposit system likely encouraged fans to return cups instead of discarding them
The deposit system likely encouraging fans to return cups instead of discarding them is correct because cup litter fell sharply after the refillable cups and deposit were introduced. Buying fewer drinks and a litter increase are not supported, and banning beverages contradicts the use of cups.
A passage about a public-health worker emphasizes repeatedly that she translated medical instructions into several local languages so that more patients could follow their treatment correctly. Which choice best states the central idea?
- A.The clinic served a small number of patients
- B.She preferred written notes to spoken ones
- C.Translating instructions into local languages helped more patients follow treatment
- D.She worked the longest hours in her clinic
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Correct answer: Translating instructions into local languages helped more patients follow treatment
Translating instructions into local languages helping more patients follow treatment is correct because the passage ties the translations to better patient adherence. Her hours, her preference for written notes, and the clinic's size are incidental or contradicted details.
A passage claims that adding a crossing guard at a school improved pedestrian safety. A table shows reported near-misses involving children fell from 22 per month to 6 per month after the guard was hired. Which choice best uses the data to support the claim?
- A.Reported near-misses stayed at 22 per month
- B.Reported near-misses dropped from 22 to 6 per month after the guard was hired
- C.Reported near-misses rose after the guard was hired
- D.No children crossed after the guard was hired
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Correct answer: Reported near-misses dropped from 22 to 6 per month after the guard was hired
Reported near-misses dropping from 22 to 6 per month after the guard was hired is correct because the decrease supports the claim of improved safety. A rise and an unchanged total contradict the figures, and no children crossing is never indicated.
A passage describes a study testing whether a meditation course lowers stress, comparing volunteers who took the course with a similar group who joined a waitlist, with both groups measured the same way before and after. To best support the claim that the course reduced stress, which detail is most important?
- A.The two groups were similar and measured identically, differing only in taking the course
- B.The course lasted eight weeks
- C.The volunteers were paid to participate
- D.The course met in a comfortable room
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Correct answer: The two groups were similar and measured identically, differing only in taking the course
The two groups being similar and measured identically, differing only in taking the course, is correct because that controlled comparison lets the change in stress be attributed to the course. The room, the course length, and the payment do not establish that the course caused the difference.
A passage states that a grocery chain moved its fresh produce to the front entrance, and over the next months the share of shoppers who bought fruits and vegetables rose noticeably. Which choice most logically completes the inference that the new placement most likely ______?
- A.Had no influence on what shoppers bought
- B.Prompted more shoppers to buy produce by making it more visible
- C.Caused shoppers to avoid the produce section
- D.Reduced the amount of produce the store stocked
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Correct answer: Prompted more shoppers to buy produce by making it more visible
Prompting more shoppers to buy produce by making it more visible is correct because produce purchases rose after it was moved to the entrance. No influence ignores the rise, avoiding the section reverses the result, and reducing stock is never mentioned.
Standard English Conventions (26)
Which choice completes the sentence so that it conforms to the conventions of Standard English? "The flock of geese ______ across the lake each morning before settling near the reeds."
- A.Are gliding
- B.Glide
- C.Glides
- D.Have glided
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Correct answer: Glides
"Glides" is correct because the singular subject "flock" governs the verb, while the plural phrase "of geese" merely modifies it without changing the subject's number. "Glide," "are gliding," and "have glided" are plural forms that wrongly agree with "geese" instead of the true singular subject "flock."
Which choice completes the sentence so that it conforms to the conventions of Standard English? "Each of the volunteers ______ assigned a separate section of the trail to clear."
- A.Are
- B.Have been
- C.Were
- D.Was
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Correct answer: Was
"Was" is correct because the subject is the singular indefinite pronoun "each," not the plural "volunteers" inside the prepositional phrase, so a singular verb is required. "Are," "were," and "have been" are plural forms that mistakenly agree with "volunteers" rather than with "each."
Which choice completes the sentence so that it conforms to the conventions of Standard English? "Neither the gallery owner nor the visiting artists ______ aware that the alarm had been disabled."
- A.Was
- B.Were
- C.Is
- D.Has been
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Correct answer: Were
"Were" is correct because in a "neither...nor" construction the verb agrees with the nearer subject, and the closer subject "artists" is plural. "Has been," "is," and "was" are singular forms that ignore the plural noun standing nearest the verb.
Which choice completes the sentence so that it conforms to the conventions of Standard English? "Politics ______ a subject that the two neighbors carefully avoid at family gatherings."
- A.Have been
- B.Were
- C.Is
- D.Are
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Correct answer: Is
"Is" is correct because "politics," though it ends in -s, names a single subject of discussion here and takes a singular verb. "Are," "were," and "have been" treat the word as plural, but a field named as one topic remains grammatically singular.
Which choice completes the sentence so that it conforms to the conventions of Standard English? "Both the violinist and the cellist ______ rehearsing the difficult passage long after the others left."
- A.Is
- B.Was
- C.Has been
- D.Were
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Correct answer: Were
"Were" is correct because two singular subjects joined by "both...and" form a plural compound subject that requires a plural verb. "Was," "is," and "has been" are singular forms that fail to agree with the two performers acting together.
Which choice completes the sentence so that the pronoun agrees with its antecedent? "Every applicant must list ______ previous experience on the second page of the form."
- A.Its
- B.Whose
- C.His or her
- D.Their
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Correct answer: His or her
"His or her" is correct because the singular distributive word "every" makes "every applicant" grammatically singular and calls for a singular possessive pronoun. "Their" is plural and disagrees in number, "its" refers to a thing rather than a person, and "whose" is a relative pronoun unsuited to the simple possessive needed here.
Which choice completes the sentence so that the pronoun agrees with its antecedent? "The committee published ______ findings in a lengthy report that the public could download for free."
- A.There
- B.It's
- C.Their
- D.Its
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Correct answer: Its
"Its" is correct because the collective noun "committee" acts as a single unit here and therefore takes the singular possessive "its." "Their" is plural and mismatches the unified group, "it's" means "it is," and "there" indicates location rather than possession.
Which choice completes the sentence so that the pronoun agrees with its antecedent? "Somebody on the night shift left ______ access badge in the break room again."
- A.Their
- B.His or her
- C.They're
- D.Its
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Correct answer: His or her
"His or her" is correct because the indefinite pronoun "somebody" is grammatically singular and requires a singular possessive pronoun. "Their" is plural and disagrees in number, "its" refers to a thing rather than a person, and "they're" is a contraction of "they are" rather than a possessive.
Which choice completes the sentence so that the pronoun agrees with its antecedent? "When a citizen registers to vote, ______ should verify the registration deadline well in advance."
- A.It
- B.He or she
- C.One's
- D.They
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Correct answer: He or she
"He or she" is correct because the singular antecedent "a citizen" requires a singular subject pronoun. "They" is plural and disagrees in number, "it" refers to a thing rather than a person, and "one's" is a possessive form that cannot serve as the subject of the verb "should verify."
Which choice completes the sentence so that it conforms to the conventions of Standard English? "By the time the gallery closes tonight, more than five hundred visitors ______ the new exhibit."
- A.Have toured
- B.Were touring
- C.Toured
- D.Will have toured
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Correct answer: Will have toured
"Will have toured" is correct because the future perfect tense describes an action completed before a future point, signaled by "By the time the gallery closes tonight." "Toured" and "were touring" are past forms, and "have toured" is present perfect, none of which fits the future benchmark the sentence establishes.
Which choice completes the sentence so that it conforms to the conventions of Standard English? "If the city ______ the warnings earlier, the blackout might have been prevented."
- A.Would heed
- B.Heeds
- C.Had heeded
- D.Heeded
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Correct answer: Had heeded
"Had heeded" is correct because the result clause "might have been prevented" signals a past contrary-to-fact condition, which requires the past perfect in the "if" clause. "Heeded," "would heed," and "heeds" fail to express the hypothetical past condition the sentence demands.
Which choice completes the sentence so that it conforms to the conventions of Standard English? "The teacher requested that each student ______ a printed copy of the rough draft to class."
- A.Bring
- B.Brings
- C.Brought
- D.Will bring
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Correct answer: Bring
"Bring" is correct because the subjunctive trigger "requested that" requires the base form of the verb, producing "that each student bring." "Brings" and "brought" are indicative forms the subjunctive does not allow, and "will bring" supplies a future form rather than the required base form.
Which choice completes the sentence so that it conforms to the conventions of Standard English? "By the time the archaeologists arrived, looters ______ already removed several artifacts from the site."
- A.Have
- B.Were
- C.Had
- D.Will have
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Correct answer: Had
"Had" is correct because the past perfect shows that the removing happened before the later past action of the archaeologists' arrival. "Have" is present perfect, "will have" is future perfect, and "were" cannot pair with the participle "removed" to express the earlier-than-past sequence the sentence requires.
Which choice completes the sentence so that it conforms to the conventions of Standard English? "The orchestra rehearses on Tuesdays, performs on Fridays, and ______ on weekends to study new scores."
- A.Meets
- B.Met
- C.Is meeting
- D.Will have met
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Correct answer: Meets
"Meets" is correct because the sentence lists a series of present-tense actions—"rehearses," "performs," and "meets"—and the third verb must match that consistent present tense. "Met," "is meeting," and "will have met" shift tense or aspect and break the established present-tense pattern.
Which choice correctly punctuates the boundary between the two independent clauses? "The prototype passed every safety test ______ the engineers were still reluctant to approve it for production."
- A.; but
- B., and
- C.;
- D.,
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Correct answer: ;
A semicolon is correct because it links two closely related independent clauses when no coordinating conjunction is present, as in "safety test; the engineers." A lone comma produces a comma splice, "; but" overpunctuates by pairing a semicolon with a coordinating conjunction, and ", and" wrongly signals addition rather than the contrast the sentence implies.
Which choice correctly uses a colon in the sentence? "The recipe depends on a single secret ingredient ______ a pinch of smoked paprika stirred in at the end."
- A.,
- B.-
- C.:
- D.;
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Correct answer: :
A colon is correct because it follows a complete independent clause and introduces the explanation of the "single secret ingredient." A comma is too weak to introduce the explanatory element formally, a semicolon must join two independent clauses rather than introduce a phrase, and a single hyphen joins words rather than introducing an explanation.
Which choice correctly punctuates the list, whose items contain internal commas? "The judges came from Austin, Texas ______ Reno, Nevada; and Tampa, Florida, to evaluate the entries."
- A.,
- B.And
- C.:
- D.;
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Correct answer: ;
A semicolon is correct because when items in a series already contain commas, semicolons separate the items to avoid confusion, matching the sentence's later "Nevada; and Tampa." A comma would blur the city-and-state boundaries, a colon does not separate items in a series, and "and" alone fails to set off the first complex item from the second.
Which choice correctly punctuates the sentence? "After repairing the leaky roof in record time ______ the carpenter packed up her tools and drove home."
- A.No punctuation
- B.,
- C.;
- D.:
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Correct answer: ,
A comma is correct because the introductory phrase "After repairing the leaky roof in record time" must be separated from the main clause by a comma. Omitting punctuation runs the introductory element into the main clause, and a semicolon and a colon are both too strong to follow an introductory phrase before an independent clause.
Which choice correctly punctuates the sentence? "The lead researcher ______ a specialist in marine ecosystems ______ presented her results to a packed auditorium."
- A.No commas
- B.; ;
- C.: :
- D., , (a pair of commas around the appositive)
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Correct answer: , , (a pair of commas around the appositive)
A pair of commas around the appositive is correct because "a specialist in marine ecosystems" is a nonrestrictive appositive renaming "The lead researcher" and must be enclosed by commas. Using no punctuation fuses the appositive into the sentence, and semicolons or colons cannot set off a nonrestrictive renaming phrase.
Which choice correctly punctuates the sentence? "The shipment arrived a week early ______ so the warehouse crew scrambled to clear space for it."
- A.;
- B.,
- C.No punctuation
- D.:
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Correct answer: ,
A comma is correct because a comma precedes the coordinating conjunction "so" when it joins two independent clauses. Omitting punctuation creates a run-on, a semicolon paired with a coordinating conjunction is redundant, and a colon does not link two independent clauses joined by "so."
Which choice correctly punctuates the sentence? "She packed a sturdy ______ waterproof tent for the week-long trek through the mountains."
- A.:
- B.;
- C.No punctuation
- D.,
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Correct answer: ,
A comma is correct because "sturdy" and "waterproof" function as coordinate adjectives independently modifying "tent" and should be separated by a comma. Omitting punctuation improperly fuses the two adjectives, while a semicolon and a colon are clause- and list-level marks that do not belong between two adjectives describing one noun.
Which choice correctly forms the possessive in the sentence? "The ______ break room was relocated to the second floor after the renovation."
- A.Nurses's
- B.Nurses
- C.Nurses'
- D.Nurse's
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Correct answer: Nurses'
"Nurses'" is correct because the break room belongs to multiple nurses, and a plural noun ending in -s takes only an apostrophe after the s to show possession. "Nurses" is a plain plural with no possessive marking, "nurse's" indicates a single nurse, and "nurses's" wrongly adds an extra s to an already plural possessive.
Which choice correctly punctuates the sentence? "The startup secured its first major client ______ moreover, two investors offered additional funding the same week."
- A.No punctuation
- B.;
- C.:
- D.,
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Correct answer: ;
A semicolon is correct because it separates two independent clauses joined by the conjunctive adverb "moreover," which is followed by its own comma. A comma alone before "moreover" would create a comma splice, a colon does not link two complete clauses in this way, and omitting punctuation produces a run-on.
Which choice correctly punctuates the sentence? "The discovery ______ which overturned a century of accepted theory ______ earned the young physicist international recognition."
- A.One comma only
- B.No dashes
- C.; ;
- D.— — (a pair of dashes around the clause)
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Correct answer: — — (a pair of dashes around the clause)
A pair of dashes around the clause is correct because dashes can set off the nonessential clause "which overturned a century of accepted theory" for emphasis. Using no marks runs the clause into the sentence, semicolons cannot enclose a nonrestrictive clause, and a single comma fails to close off the inserted element, which needs matching punctuation on both sides.
Which choice corrects the modifier so that it logically describes the right subject? "______, the abandoned factory now houses a thriving community art center."
- A.Touring the exhibits on opening night
- B.Recently converted by a nonprofit group
- C.Painting bold murals on the walls
- D.Having sketched the original blueprints
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Correct answer: Recently converted by a nonprofit group
"Recently converted by a nonprofit group" is correct because the introductory modifier must logically describe the subject that follows, and only the factory, not an artist or a visitor, could have been converted. The other options describe an action that cannot sensibly apply to the factory, producing a dangling modifier.
Which choice completes the sentence so that the items in the series are parallel? "A good mentor listens patiently, advises honestly, and ______."
- A.To challenge respectfully
- B.With respectful challenges
- C.Is challenging in a respectful way
- D.Challenges respectfully
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Correct answer: Challenges respectfully
"Challenges respectfully" is correct because the series uses parallel present-tense verbs with adverbs—"listens patiently," "advises honestly," and "challenges respectfully"—and the final item must match that pattern. The other options shift to a prepositional phrase, a different verb construction, or an infinitive, breaking the parallel structure of the list.
Expression of Ideas (23)
Which choice completes the text with the most logical transition? "The library extended its hours during final exams to give students more time to study. ______, it opened a quiet reading room that had previously been reserved for staff."
- A.Otherwise
- B.Moreover
- C.In contrast
- D.However
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Correct answer: Moreover
"Moreover" is correct because opening the quiet room is a second accommodation the library made for students, building on the extended hours and calling for an additive transition. "However" and "In contrast" signal opposition, and "Otherwise" presents an alternative condition, none of which fits a second supportive measure added to the first.
Which choice completes the text with the most logical transition? "Early maps depicted California as a large island separated from the mainland. ______, later surveys proved it was firmly attached to the continent."
- A.In fact
- B.Therefore
- C.Eventually
- D.For example
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Correct answer: Eventually
"Eventually" is correct because the later surveys came after the early maps and gradually overturned the island belief, signaling a passage of time leading to a new finding. "For example" introduces an illustration, "In fact" emphasizes a prior claim, and "Therefore" marks a logical result, none of which captures the temporal progression from old maps to later proof.
Which choice completes the text with the most logical transition? "The trail to the summit is steep and exposed to high winds. ______, hikers are advised to begin their climb before dawn to avoid the afternoon storms."
- A.For these reasons
- B.Nonetheless
- C.Conversely
- D.Meanwhile
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Correct answer: For these reasons
"For these reasons" is correct because the advice to start early follows from the steepness and the wind hazards just described, signaling cause and effect. "Nonetheless" and "Conversely" signal contrast, and "Meanwhile" marks simultaneous events, none of which links the trail's dangers to the recommended early start.
Which choice completes the text with the most logical transition? "Octopuses can change color to blend into nearly any background. ______, many of them are color-blind, relying on light-sensing cells in their skin instead of their eyes."
- A.Consequently
- B.For instance
- C.Likewise
- D.Surprisingly
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Correct answer: Surprisingly
"Surprisingly" is correct because an animal mastering camouflage while being color-blind is an unexpected contrast, and the transition flags that the second fact runs against what the first would lead a reader to assume. "Consequently" marks a result, "For instance" introduces an example, and "Likewise" signals similarity, none of which captures the counterintuitive tension between the two facts.
Which choice completes the text with the most logical transition? "The orchestra rehearsed the new symphony for only three days before its premiere. ______, the performance drew a standing ovation from the audience."
- A.Specifically
- B.In addition
- C.As a result
- D.Even so
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Correct answer: Even so
"Even so" is correct because the warm reception came despite the very short rehearsal time, signaling that the result defied the expectation set by the limited preparation. "As a result" wrongly treats the ovation as a consequence of brief rehearsal, "Specifically" narrows a detail, and "In addition" adds a point, none of which conveys the contrast between scant practice and success.
Which choice completes the text with the most logical transition? "The new policy required every department to submit a budget by March. ______, the finance office offered training sessions to help managers prepare their figures."
- A.In contrast
- B.Nevertheless
- C.By comparison
- D.To that end
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Correct answer: To that end
"To that end" is correct because the training sessions were provided in order to help departments meet the March budget requirement, signaling a purpose tied to the stated goal. "Nevertheless" and "In contrast" signal opposition, and "By comparison" introduces a comparison, none of which expresses the supporting purpose behind the training.
Which choice completes the text with the most logical transition? "Some economists predicted that the minimum-wage increase would cause widespread layoffs. ______, employment in the affected sector held steady in the year after the change."
- A.Contrary to these forecasts
- B.For example
- C.Accordingly
- D.In the same way
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Correct answer: Contrary to these forecasts
"Contrary to these forecasts" is correct because steady employment directly opposes the economists' prediction of layoffs, signaling that the outcome contradicted the expectation. "Accordingly" marks a result that follows, "In the same way" signals similarity, and "For example" introduces an illustration, none of which captures the reversal of the predicted outcome.
Which choice completes the text with the most logical transition? "The novelist set her story in a fictional town modeled on her childhood home. ______, she insisted that none of the characters were based on real people she had known."
- A.Therefore
- B.For this reason
- C.Nonetheless
- D.Likewise
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Correct answer: Nonetheless
"Nonetheless" is correct because the author's insistence that the characters were invented runs counter to the real-life inspiration for the setting, signaling a concession that contrasts with the preceding fact. "Therefore" and "For this reason" mark results, and "Likewise" signals similarity, none of which fits the tension between a real-place setting and fictional characters.
Which choice completes the text with the most logical transition? "Volunteers collected water samples from forty points along the river over the summer. ______, they recorded the temperature and flow rate at each location."
- A.Instead
- B.However
- C.In contrast
- D.At each site
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Correct answer: At each site
"At each site" is correct because recording temperature and flow rate is an additional measurement taken at the same sampling points, and the phrase ties the second task to the locations already mentioned. "However" and "In contrast" signal opposition, and "Instead" marks a replacement, none of which fits the additional data gathered alongside the water samples.
Which choice completes the text with the most logical transition? "The first printing of the dictionary contained several errors in its pronunciation guide. ______, the publisher recalled the copies and issued a corrected edition the following month."
- A.Even so
- B.By contrast
- C.Similarly
- D.In response
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Correct answer: In response
"In response" is correct because the recall and corrected edition were the publisher's reaction to the discovered errors, signaling an action taken because of the preceding problem. "By contrast" and "Even so" signal opposition, and "Similarly" signals likeness, none of which expresses the corrective action prompted by the errors.
Which choice completes the text with the most logical transition? "Wind turbines produce the most power when blades face directly into the wind. ______, modern turbines use sensors and motors that rotate the entire unit to track shifting wind direction."
- A.For this reason
- B.Meanwhile
- C.Regardless
- D.On the contrary
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Correct answer: For this reason
"For this reason" is correct because the tracking mechanism exists because turbines generate the most power when aimed into the wind, signaling cause and effect. "On the contrary" and "Regardless" signal opposition or dismissal, and "Meanwhile" marks simultaneous events, none of which links the power principle to the design solution it motivates.
Which choice completes the text with the most logical transition? "The film's special effects were widely praised when it premiered in 1982. ______, its script and pacing were criticized as confusing by many of the same reviewers."
- A.Furthermore
- B.At the same time
- C.As a result
- D.Therefore
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Correct answer: At the same time
"At the same time" is correct because the criticism of the script coexists with and offsets the praise for the effects, signaling that two opposing judgments held simultaneously. "Furthermore" adds a supporting point, and "Therefore" and "As a result" mark consequences, none of which captures the contrast between praised visuals and a faulted script.
On the Digital SAT Reading and Writing section, the two question types that make up the Expression of Ideas content domain are which of the following?
- A.Words in Context and Text Structure and Purpose
- B.Central Ideas and Details and Command of Evidence
- C.Transitions and Rhetorical Synthesis
- D.Boundaries and Form, Structure, and Sense
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Correct answer: Transitions and Rhetorical Synthesis
Transitions and Rhetorical Synthesis is correct because these are the two skills the College Board groups under the Expression of Ideas domain on the digital test. Words in Context and Text Structure and Purpose are Craft and Structure skills, Boundaries and Form, Structure, and Sense are Standard English Conventions skills, and Central Ideas and Details and Command of Evidence are Information and Ideas skills.
In a Digital SAT Transitions item, what role does the underlined or blank-marked transition word play within the passage?
- A.It identifies the grammatical subject of the following sentence
- B.It signals the logical relationship between the idea before it and the idea after it
- C.It marks where a quotation from a source must be inserted
- D.It corrects a punctuation error in the surrounding text
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Correct answer: It signals the logical relationship between the idea before it and the idea after it
Signaling the logical relationship between the idea before it and the idea after it is correct because a transition's job is to tell the reader how two ideas connect, such as by addition, contrast, or cause. A transition does not identify a grammatical subject, mark where a quotation belongs, or correct punctuation, which are tasks tied to other skills.
A Rhetorical Synthesis question on the Digital SAT will always specify which element that a student must use to judge the answer choices?
- A.A specific rhetorical goal the sentence must accomplish
- B.A grammar rule that the answer must demonstrate
- C.A passage from a second author to compare
- D.A required word count for the response
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Correct answer: A specific rhetorical goal the sentence must accomplish
A specific rhetorical goal the sentence must accomplish is correct because every Rhetorical Synthesis prompt states a purpose, and the best answer is the one that best meets that purpose using the given notes. The questions do not impose a word count, test a particular grammar rule, or supply a second author's passage, which belong to other question types.
Why does a transition such as "therefore" express a different relationship than a transition such as "nevertheless"?
- A."Therefore" signals a result that follows logically, while "nevertheless" signals a contrast or concession
- B."Therefore" introduces an example, while "nevertheless" introduces a definition
- C."Therefore" is used only in questions, while "nevertheless" is used only in statements
- D."Therefore" joins two nouns, while "nevertheless" joins two verbs
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Correct answer: "Therefore" signals a result that follows logically, while "nevertheless" signals a contrast or concession
The explanation that "therefore" signals a result that follows logically while "nevertheless" signals a contrast or concession is correct because these words mark opposite kinds of connections, one consequence and one opposition, which is exactly why Transitions items test them. The other options misdescribe the words as introducing examples or definitions, joining parts of speech, or being limited to questions or statements.
A student studying the human body has these notes: the inner ear contains tiny hair cells; these cells convert sound vibrations into nerve signals; loud noise can permanently damage them; the body cannot regrow them once lost. The student wants to explain why noise-induced hearing loss is permanent. Which choice most effectively uses the notes to accomplish this goal?
- A.Loud noise can be harmful to the hair cells in the inner ear.
- B.Because the body cannot regrow the inner ear's hair cells, the damage loud noise causes them is permanent.
- C.The inner ear's hair cells turn sound vibrations into nerve signals.
- D.The inner ear contains tiny hair cells.
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Correct answer: Because the body cannot regrow the inner ear's hair cells, the damage loud noise causes them is permanent.
The choice stating that the damage is permanent because the body cannot regrow the hair cells is correct because it explains the permanence of noise-induced hearing loss, which is exactly the goal. The other options describe the cells' function, the general harm of noise, or the cells' existence without explaining why the loss cannot be reversed.
A student has compiled these notes: the writer Zora Neale Hurston trained as an anthropologist; she collected folklore across the rural South in the 1920s and 1930s; she wove that folklore into her novels; her fiction preserved speech patterns that scholars later studied. The student wants to emphasize how Hurston's research shaped her fiction. Which choice best uses the notes to meet this goal?
- A.Zora Neale Hurston was both a writer and an anthropologist.
- B.Scholars have studied the speech patterns in Hurston's fiction.
- C.Drawing on the folklore she collected as an anthropologist, Hurston filled her novels with authentic Southern speech.
- D.Zora Neale Hurston collected folklore in the rural South.
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Correct answer: Drawing on the folklore she collected as an anthropologist, Hurston filled her novels with authentic Southern speech.
The choice noting that Hurston drew on her collected folklore to fill her novels with authentic speech is correct because it ties her anthropological research to her fiction, exactly the connection the goal asks for. The other options state her dual roles, her folklore collecting, or scholars' later interest without showing how the research shaped the novels.
A student has gathered these notes: a coastal city built an artificial reef from sunken concrete blocks; fish populations near the reef increased within two years; local fishing improved; divers now visit the reef as a tourist attraction. The student wants to present the reef project's economic benefits to the city. Which choice most effectively uses the notes to accomplish this goal?
- A.The reef was made from concrete blocks placed on the seabed.
- B.The artificial reef boosted local fishing and drew diving tourists, bringing new income to the city.
- C.Fish populations near the reef grew within two years of its construction.
- D.A coastal city built an artificial reef from sunken concrete blocks.
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Correct answer: The artificial reef boosted local fishing and drew diving tourists, bringing new income to the city.
The choice noting that the reef boosted fishing and drew tourists, bringing new income, is correct because it presents the project's economic benefits, the goal of the question. The other options describe the reef's construction, the rise in fish populations, or the materials used, none of which directly conveys the economic gains.
A student researching technology has these notes: a town installed smart streetlights that dim when no one is nearby; the lights brighten when they detect motion; energy use for street lighting fell by a third; residents reported feeling just as safe. The student wants to address a concern readers might have about dimming the lights at night. Which choice most effectively uses the notes to accomplish this goal?
- A.Even with the dimming system, residents reported feeling just as safe at night.
- B.A town installed smart streetlights to save energy.
- C.The smart streetlights dim automatically when no one is nearby.
- D.The streetlights brighten as soon as they detect motion.
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Correct answer: Even with the dimming system, residents reported feeling just as safe at night.
The choice stating that residents felt just as safe despite the dimming is correct because it directly addresses a reader's likely worry about safety, which the goal calls for. The other options describe how the lights dim, the reason for installing them, or how they respond to motion, none of which speaks to the safety concern.
A student has assembled these notes: a chef opened a restaurant that serves only surplus food from local farms; the produce would otherwise be discarded for cosmetic flaws; meals are sold at low prices; the restaurant has diverted tons of food from landfills. The student wants to highlight the restaurant's environmental impact. Which choice most effectively uses the notes to accomplish this goal?
- A.The restaurant sells its meals at low prices.
- B.A chef opened a restaurant that cooks with surplus farm produce.
- C.The produce used by the restaurant has minor cosmetic flaws.
- D.By cooking with produce that would otherwise be thrown away, the restaurant has kept tons of food out of landfills.
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Correct answer: By cooking with produce that would otherwise be thrown away, the restaurant has kept tons of food out of landfills.
The choice noting that using would-be-discarded produce has kept tons of food out of landfills is correct because it highlights the restaurant's environmental impact, the stated goal. The other options describe the restaurant's concept, its low prices, or the produce's flaws, none of which emphasizes the reduction in food waste.
A student is combining two notes for a report. The notes read: "A team mapped the genome of a deep-sea bacterium" and "The bacterium produces an enzyme that remains stable at near-freezing temperatures." The student wants a single sentence that emphasizes the practical value of the discovery for cold-water industries. Which choice best accomplishes this goal?
- A.A team mapped the genome of a deep-sea bacterium.
- B.The deep-sea bacterium's cold-stable enzyme, revealed by genome mapping, could prove useful for cold-water industrial processes.
- C.An enzyme from a deep-sea bacterium stays stable in the cold.
- D.The deep-sea bacterium's genome was recently sequenced.
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Correct answer: The deep-sea bacterium's cold-stable enzyme, revealed by genome mapping, could prove useful for cold-water industrial processes.
The choice describing the cold-stable enzyme revealed by genome mapping as useful for cold-water industry is correct because it combines both notes into one sentence and emphasizes the practical value the goal requires. The other options preserve only one note or state the enzyme's stability without connecting it to any industrial use.
A student has the following notes: a city replaced a four-lane road with two car lanes and a wide bike-and-bus corridor; car traffic did not increase as critics feared; bus ridership rose; the redesign won a national planning award. The student wants to present the outcome that contradicted critics' predictions. Which choice most effectively uses the notes to accomplish this goal?
- A.A city redesigned a four-lane road to add a bike-and-bus corridor.
- B.The redesigned road won a national planning award.
- C.Despite critics' warnings, car traffic on the redesigned road did not increase.
- D.Bus ridership rose after the road was redesigned.
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Correct answer: Despite critics' warnings, car traffic on the redesigned road did not increase.
The choice stating that, despite critics' warnings, car traffic did not increase is correct because it presents the outcome that contradicted the predictions, exactly the goal. The other options describe the redesign, the award, or the rise in bus ridership, none of which directly addresses the critics' specific fear about traffic.
Algebra (70)
Solve for : .
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Correct answer:
The correct answer is . Subtract from both sides to get , then subtract to get , and divide by to get . The value before dividing leads some students to other choices, but only satisfies the original equation.
In the linear equation , what does the coefficient describe about the graph of the line?
- A.The point where the line crosses the x-axis
- B.The point where the line crosses the y-axis
- C.The constant rate of change, or slope, of the line
- D.The horizontal distance from the origin
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Correct answer: The constant rate of change, or slope, of the line
The correct answer is the constant rate of change, or slope, of the line. In , is the slope, telling how much changes for each unit increase in . The constant is the y-intercept where the line crosses the y-axis, and the x-intercept is a separate point found by setting .
A linear equation written in standard form looks like . What is one defining feature of the standard form of a linear equation?
- A.It can only represent vertical lines
- B.The variables appear on the same side with integer coefficients and the constant alone on the other
- C.It always shows the slope directly as a coefficient
- D.It requires the y-variable to be isolated
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Correct answer: The variables appear on the same side with integer coefficients and the constant alone on the other
The correct answer is that the variables appear on the same side with integer coefficients and the constant alone on the other. Standard form keeps both variable terms together equal to a constant, and the slope is not shown directly; you must rearrange to slope-intercept form to read the slope. It can represent vertical, horizontal, and slanted lines.
A car rental company charges a $25 base fee plus $0.20 per mile driven. Which equation gives the total charge , in dollars, for driving miles?
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Correct answer:
The correct answer is . The per-mile rate of $0.20 multiplies the number of miles to form the slope, and the fixed $25 base fee is the constant added regardless of distance, giving the y-intercept. The form swaps the rate and the base fee.
Solve for : .
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Correct answer:
The correct answer is . Distribute to get , subtract to get , and divide by to get . Forgetting that dividing by a negative reverses the sign gives the incorrect value .
What is the slope of the line passing through the points and ?
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Correct answer:
The correct answer is . Slope is the change in over the change in : . Inverting the ratio gives , and dropping the negative sign gives , both of which are common errors.
Solve the system and for the value of .
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Correct answer:
The correct answer is . Substitute into the first equation: , giving , so and . Substituting back gives , which is a distractor for the x-value here.
A line has a slope of and passes through the point . What is its equation in slope-intercept form?
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Correct answer:
The correct answer is . The slope is the coefficient of , and the point is the y-intercept, so , giving . The form swaps the slope and intercept, and inverts the slope.
Which value of satisfies the inequality ?
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Correct answer:
The correct answer is . Subtract from both sides to get , then divide by and flip the inequality sign to get . Of the choices, only is greater than ; the values , , and are not greater than .
The cost to join a streaming service is modeled by , where is the number of months and is the total cost in dollars. What does the constant represent in this context?
- A.The monthly subscription rate
- B.A one-time sign-up fee charged regardless of the number of months
- C.The number of free months included
- D.The total cost after 5 months
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Correct answer: A one-time sign-up fee charged regardless of the number of months
The correct answer is a one-time sign-up fee charged regardless of the number of months. In a linear function , the constant is the value when , representing a fixed fee independent of months. The coefficient is the monthly rate, the slope of the function.
A line is written as . What is the y-intercept of the line?
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Correct answer:
The correct answer is . The y-intercept occurs where , so gives and , the point . The point is the x-intercept, and drops the negative sign.
Solve for : .
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Correct answer:
The correct answer is . Multiply both sides by to get , add to get , and divide by to get . The value is before the final division, and other choices come from skipping the multiplication by .
Solve the system and by elimination. What is the value of ?
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Correct answer:
The correct answer is . Subtracting the second equation from the first eliminates : gives , so . Substituting back gives , which is the x-value, not the requested y-value.
Two perpendicular lines intersect on a coordinate plane. If one line has a slope of , what is the slope of the line perpendicular to it?
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Correct answer:
The correct answer is . Perpendicular lines have slopes that are negative reciprocals of each other, so the perpendicular slope is the reciprocal of with the sign flipped, giving . A slope of would describe a parallel line, and omits the required negative sign.
A delivery driver earns a fixed daily wage plus a fee per package. The function gives the driver's daily earnings , in dollars, for delivering packages. How much does the driver earn for each additional package?
- A.$68
- B.$52
- C.$8
- D.$60
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Correct answer: $8
The correct answer is $8. In the linear function , the coefficient of is the slope, representing the change in earnings per additional package, which is $8. The constant is the fixed daily wage earned with zero packages, not the per-package amount.
Which inequality describes all values of for which is greater than or equal to ?
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- D.
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Correct answer:
The correct answer is . Subtract from both sides to get , then divide by to get . The endpoint is included because the original statement uses 'greater than or equal to,' so a strict inequality would incorrectly exclude .
A line passes through the points and . Which equation represents this line in slope-intercept form?
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Correct answer:
The correct answer is . The slope is , and substituting into gives , so , producing . The choice uses the wrong sign for the intercept.
How many solutions does the equation have?
- A.No solutions
- B.Exactly two solutions
- C.Exactly one solution
- D.Infinitely many solutions
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Correct answer: Infinitely many solutions
The correct answer is infinitely many solutions. Distributing the left side gives , an identity that is true for every value of , so any real number is a solution. A contradiction such as would give no solutions, and a unique value would give exactly one.
A small business models its monthly profit with , where is the number of units sold and is the profit in dollars. How many units must be sold for the business to break even, meaning ?
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Correct answer:
The correct answer is . Setting gives , so and units. Selling fewer than units produces a loss, and selling more produces a profit; the value and others come from arithmetic errors in dividing by .
The graph of a linear inequality is drawn on a coordinate plane. Which describes how the boundary line and shading should appear?
- A.A dashed line with shading below it
- B.A solid line with shading above it
- C.A solid line with shading below it
- D.A dashed line with shading above it
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Correct answer: A dashed line with shading above it
The correct answer is a dashed line with shading above it. The strict symbol excludes points on the line itself, so the boundary is dashed rather than solid, and 'greater than' the expression means the region above the line is shaded. A solid line would represent , and shading below would represent or .
Solve for : .
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Correct answer:
The correct answer is . Distribute the right side to get , subtract to get , add to get , and divide by to get . The value is before the final division.
A system of two linear equations is graphed and the two lines coincide exactly, lying on top of each other. How many solutions does this system have?
- A.No solution
- B.Exactly one solution
- C.Infinitely many solutions
- D.Exactly two solutions
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Correct answer: Infinitely many solutions
The correct answer is infinitely many solutions. When two lines coincide, every point on one line is also on the other, so each shared point is a solution and there are infinitely many. Lines crossing once give exactly one solution, and parallel lines that never meet give no solution.
The equation of a line is . Which of the following lines is parallel to it?
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Correct answer:
The correct answer is . Parallel lines have identical slopes but different y-intercepts, and shares the slope of while having a different intercept. The line has the opposite slope, and the half-slope choices are not parallel.
A linear function passes through the points and . What is true about this function?
- A.Its slope is undefined
- B.Its slope is positive
- C.Its slope is negative
- D.Its slope is 0 and it is a constant function
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Correct answer: Its slope is 0 and it is a constant function
The correct answer is its slope is and it is a constant function. The two points have the same y-value of , so the change in is , making the slope and the function constant at . An undefined slope would require equal x-values instead.
Solve for : .
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Correct answer:
The correct answer is . Subtract from both sides to get , subtract to get , and divide by to get . Dividing by is the key step; mistakenly dividing by instead of gives , a value far too small.
A landscaper's total charge is modeled by a linear function. The charge is $90 for 2 hours of work and $190 for 6 hours of work. What is the hourly rate?
- A.$25 per hour
- B.$30 per hour
- C.$50 per hour
- D.$45 per hour
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Correct answer: $25 per hour
The correct answer is $25 per hour. The hourly rate is the slope: dollars per hour. Dividing the total charge by total hours without accounting for a fixed fee gives misleading values such as $45 or $50.
For which value of is the inequality NOT satisfied?
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Correct answer:
The correct answer is . Subtract 4 to get , then multiply by and flip the inequality to get , so values above 6 fail the inequality. Since 8 is greater than 6, it does not satisfy the inequality, while 0, 4, and 6 all do.
A line has an x-intercept at and a y-intercept at . What is the slope of the line?
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Correct answer:
The correct answer is . Using the two intercepts as points, the slope is . The negative signs in both the numerator and denominator cancel, so the slope is positive; dropping one sign produces the incorrect .
When solving the system and , what can be concluded?
- A.The lines are perpendicular
- B.The two equations represent the same line, so there are infinitely many solutions
- C.The system has no solution
- D.The system has exactly one solution at
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Correct answer: The two equations represent the same line, so there are infinitely many solutions
The correct answer is that the two equations represent the same line, so there are infinitely many solutions. Rearranging gives , or , which is identical to the first equation. Because both describe the same line, every point on it is a solution.
A theater charges $12 per adult ticket and $8 per child ticket. One evening, 150 tickets were sold for a total of $1,480. How many adult tickets were sold?
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Correct answer:
The correct answer is 70. Let be adult tickets and be child tickets: and . Substituting gives , so , , and . The remaining 80 are child tickets.
A line is graphed so that it rises steeply from left to right and crosses the y-axis well above the origin. Which equation is most consistent with this description?
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Correct answer:
The correct answer is . Rising from left to right requires a positive slope, and a steep rise requires a large slope magnitude, so 5 fits better than 0.2; crossing above the origin requires a positive y-intercept, which the provides. A negative slope would fall, and would cross below the origin.
Solve for : , and identify the largest integer value of that satisfies it.
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Correct answer:
The correct answer is 4. Subtract from both sides to get , subtract 2 to get , and divide by 2 to get . Because can equal at most 4, the largest integer satisfying the inequality is 4; the value 5 exceeds the limit.
A function is defined by . For what value of does equal 0?
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Correct answer:
The correct answer is 4. Set to get , so and . This value is the x-intercept of the line, where the function output equals zero; the value results from a sign error when isolating .
Solve for x: .
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Correct answer:
The correct answer is . Subtract from both sides to get , add 4 to get , and divide by 4 to get . The value before dividing can mislead toward other choices.
A line passes through the points and . What is the slope of this line?
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Correct answer:
The correct answer is . The slope is the change in y over the change in x: . Inverting the ratio gives , and using only the change in y gives , both common errors.
Solve for x: .
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Correct answer:
The correct answer is . Distribute to get , subtract to get , add 15 to get , and divide by 3 to get . The value is before the final division.
A printing service charges a $12 setup fee plus $0.05 per page. Which equation gives the total cost C, in dollars, for printing p pages?
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Correct answer:
The correct answer is . The per-page rate of $0.05 multiplies the number of pages p to form the slope, and the fixed $12 setup fee is the constant added regardless of page count, giving the y-intercept. The form swaps the rate and the fee.
Solve the system and for the value of x.
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Correct answer:
The correct answer is . Substitute into the second equation: , giving , so and . Substituting back gives , which is a distractor for the requested x-value.
Which value of x satisfies the inequality ?
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- D.
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Correct answer:
The correct answer is . Add 4 to both sides to get , then divide by 3 to get . Of the choices, only 4 is less than 5; the values 5, 6, and 7 are not less than 5.
A line has a slope of and passes through the point . What is its equation in slope-intercept form?
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- B.
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- D.
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Correct answer:
The correct answer is . The slope is the coefficient of , and the point is the y-intercept, so , giving . The form swaps the slope and intercept.
A subscription box costs a $15 joining fee plus $25 per shipment. The function gives the total cost , in dollars, for shipments. What does the slope of this function represent?
- A.The cost added for each shipment
- B.The total cost of all shipments
- C.The one-time joining fee
- D.The number of free shipments
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Correct answer: The cost added for each shipment
The correct answer is the cost added for each shipment. In , the slope is , the rate at which the cost increases for every additional shipment. The constant is the fixed joining fee, which is the y-intercept rather than the slope.
A line is written as . What is the y-intercept of the line?
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- B.
- C.
- D.
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Correct answer:
The correct answer is . The y-intercept occurs where , so gives and , the point . Setting instead would give the x-intercept .
Solve for : .
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- D.
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Correct answer:
The correct answer is . Multiply both sides by to get , subtract to get , and divide by to get . The value is before the final division.
Solve the system and by elimination. What is the value of ?
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- B.
- C.
- D.
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Correct answer:
The correct answer is . Subtracting the second equation from the first eliminates : gives , so . Substituting back gives , but the requested value is .
Two perpendicular lines intersect on a coordinate plane. If one line has a slope of , what is the slope of the line perpendicular to it?
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- B.
- C.
- D.
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Correct answer:
The correct answer is . Perpendicular lines have slopes that are negative reciprocals of each other, so the perpendicular slope is the reciprocal of with the sign flipped, giving . A slope of would describe a parallel line, and keeps the wrong sign.
A freelancer charges a flat project fee plus an hourly rate. The function gives the freelancer's earnings , in dollars, for working hours. How much does the freelancer earn for each additional hour?
- A.$145
- B.$100
- C.$45
- D.$55
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Correct answer: $45
The correct answer is $45. In the linear function , the coefficient of is the slope, representing the change in earnings per additional hour, which is $45. The constant is the flat project fee earned with zero hours, not the per-hour amount.
Which inequality describes all values of for which is greater than or equal to ?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . Add to both sides to get , then divide by to get . The endpoint is included because the original statement uses 'greater than or equal to,' so a strict inequality would incorrectly exclude .
A line passes through the points and . Which equation represents this line in slope-intercept form?
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- C.
- D.
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Correct answer:
The correct answer is . The slope is , and substituting into gives , so , producing . The choice uses the wrong sign for the intercept.
How many solutions does the equation have?
- A.Exactly two solutions
- B.Infinitely many solutions
- C.No solutions
- D.Exactly one solution
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Correct answer: Infinitely many solutions
The correct answer is infinitely many solutions. Distributing the left side gives , an identity that is true for every value of , so any real number is a solution. A contradiction such as would give no solutions, and a unique value would give exactly one.
A bakery models its daily profit with , where is the number of dozens of cookies sold and is the profit in dollars. How many dozens must be sold for the bakery to break even, meaning ?
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Correct answer:
The correct answer is . Setting gives , so and dozens. Selling fewer than dozens produces a loss, and selling more produces a profit; the other choices come from arithmetic errors in dividing by .
The graph of a linear inequality is drawn on a coordinate plane. Which describes how the boundary line and shading should appear?
- A.A solid line with shading above it
- B.A dashed line with shading below it
- C.A solid line with shading below it
- D.A dashed line with shading above it
Show answerHide answer
Correct answer: A solid line with shading below it
The correct answer is a solid line with shading below it. The symbol includes points on the line itself, so the boundary is solid rather than dashed, and 'less than or equal to' the expression means the region below the line is shaded. A dashed line would represent a strict inequality, and shading above would represent or .
Solve for : .
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- D.
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Correct answer:
The correct answer is . Subtract from both sides to get , subtract to get , and divide by to get . The value is before the final division.
The equation of a line is . Which of the following lines is parallel to it?
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- B.
- C.
- D.
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Correct answer:
The correct answer is . Parallel lines have identical slopes but different y-intercepts, and shares the slope of while having a different intercept. The line has the opposite slope, and the fifth-slope choices are not parallel.
A snack stand sells hot dogs for $4 each and sodas for $2 each. One game, items were sold for a total of $620. How many hot dogs were sold?
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Correct answer:
The correct answer is . Let be hot dogs and be sodas: and . Substituting gives , so , , and . The remaining are sodas.
A function is defined by . For what value of does equal ?
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Correct answer:
The correct answer is . Set to get , so and . This value is the x-intercept of the line, where the function output equals zero; the value results from a sign error when isolating .
When solving the system and , what can be concluded?
- A.The system has exactly one solution
- B.The two lines are perpendicular
- C.The system has no solution because the lines are parallel
- D.The two equations represent the same line
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Correct answer: The system has no solution because the lines are parallel
The correct answer is that the system has no solution because the lines are parallel. Rearranging gives , or , which has the same slope of as but a different intercept. Equal slopes with different intercepts means the lines never meet.
Solve for : .
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- B.
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- D.
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Correct answer:
The correct answer is . Subtract from both sides to get , subtract to get , and divide by to get . Dividing by is the key step; treating it as divided by gives the incorrect value .
A plumber's total charge is modeled by a linear function. The charge is $120 for hours of work and $260 for hours of work. What is the hourly rate?
- A.$50 per hour
- B.$40 per hour
- C.$45 per hour
- D.$35 per hour
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Correct answer: $35 per hour
The correct answer is $35 per hour. The hourly rate is the slope: dollars per hour. Dividing the total charge by total hours without accounting for a fixed fee gives misleading values such as $50 or $60.
For which value of is the inequality NOT satisfied?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . Subtract to get , then multiply by and flip the inequality to get , so values above fail the inequality. Since is greater than , it does not satisfy the inequality, while , , and all do.
A line has an x-intercept at and a y-intercept at . What is the slope of the line?
- A.
- B.
- C.
- D.
Show answerHide answer
Correct answer:
The correct answer is . Using the two intercepts as points, the slope is . The negative signs in both the numerator and denominator cancel, so the slope is positive; dropping one sign produces the incorrect .
Solve for : , and identify the largest integer value of that satisfies it.
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- B.
- C.
- D.
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Correct answer:
The correct answer is . Subtract from both sides to get , subtract to get , and divide by to get . Because can equal at most , the largest integer satisfying the inequality is ; the value exceeds the limit.
A spring stretches according to the function , where is the hanging weight in pounds and is the spring length in centimeters. What does the constant represent in this context?
- A.The stretch per pound of weight
- B.The maximum length of the spring
- C.The total weight applied
- D.The natural length of the spring with no weight
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Correct answer: The natural length of the spring with no weight
The correct answer is the natural length of the spring with no weight. In , the constant is the length when , representing the unstretched spring. The coefficient is the slope, the additional length per pound of weight.
Solve for : .
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- D.
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Correct answer:
The correct answer is . Add to both sides to get , add to get , and divide by to get . Mishandling the sign when moving the variable terms leads to the incorrect value .
A line in standard form is . What is the slope of this line?
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- B.
- C.
- D.
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Correct answer:
The correct answer is . Solving for gives , so , revealing a slope of . Forgetting the negative sign that comes from dividing by leads to the incorrect .
A study app offers a free trial then a linear monthly cost. After 3 months a user has paid $24 total, and after 7 months the user has paid $56 total. What is the monthly cost?
- A.$8
- B.$6
- C.$9
- D.$10
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Correct answer: $8
The correct answer is $8. The monthly cost is the slope between the two points: dollars per month. Dividing a total by its month count without using the rate of change gives misleading values.
Solve the system and for the value of .
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- B.
- C.
- D.
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Correct answer:
The correct answer is . Substitute into the first equation: , giving , so and . Substituting back gives , which is the x-value rather than the requested y-value.
Which point lies on the line ?
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Correct answer:
The correct answer is . Substituting gives , which matches the point . The point would require from the equation, so it does not lie on the line.
Solve the inequality .
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Correct answer:
The correct answer is . Subtract 4 from both sides to get , then divide by and flip the inequality sign to get . Forgetting to reverse the inequality when dividing by a negative gives the incorrect .
A linear function passes through the points and . What is true about this function?
- A.Its slope is undefined
- B.Its slope is positive
- C.Its slope is 0 and it is a constant function
- D.Its slope is negative
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Correct answer: Its slope is 0 and it is a constant function
The correct answer is its slope is 0 and it is a constant function. The two points have the same y-value of 6, so the change in y is 0, making the slope and the function constant at . An undefined slope would require equal x-values instead.
Solve for : .
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . Distribute both sides to get , subtract to get , add 8 to get , and divide by 3 to get . The value 28 is before the final division.
Advanced Math (70)
What does the leading coefficient's sign tell you about the graph of a quadratic function written as ?
- A.The sign of determines the y-intercept of the parabola
- B.The sign of has no effect on the direction the parabola opens
- C.A positive means the parabola opens upward and has a minimum
- D.A positive means the parabola opens downward and has a maximum
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Correct answer: A positive means the parabola opens upward and has a minimum
The correct answer is that a positive means the parabola opens upward and has a minimum. In , when is positive the curve opens upward so its vertex is the lowest point, and when is negative it opens downward with a maximum. The y-intercept is set by the constant , and the sign of definitely controls the opening direction.
The number of real solutions of a quadratic equation depends on the value of its discriminant. What does a negative discriminant indicate?
- A.Exactly one repeated real solution
- B.No real solutions
- C.Infinitely many real solutions
- D.Two distinct real solutions
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Correct answer: No real solutions
The correct answer is no real solutions. When the discriminant is negative, the square root in the quadratic formula involves a negative number, so no real values satisfy the equation and the parabola never crosses the x-axis. A discriminant of zero gives one repeated solution, and a positive discriminant gives two distinct real solutions.
In the function , what does the value 5 represent?
- A.The value of the function when equals 0
- B.The growth factor applied each time increases by 1
- C.The rate at which the function decreases
- D.The value of where the graph crosses the x-axis
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Correct answer: The value of the function when equals 0
The correct answer is the value of the function when equals 0. In an exponential function of the form , the coefficient is the initial value, found by setting since , so . The growth factor here is the base 2, the function increases rather than decreases, and an exponential function of this form never crosses the x-axis.
What is the sum of the solutions to the equation ?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is 7. Factoring gives , so the solutions are and , which sum to 7. The value 10 is the product of the solutions, and confuses the sum with the middle coefficient before applying the sign relationship.
Which expression is the complete factorization of ?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . Factoring requires two numbers that multiply to and add to , which are and 3, giving . The pair 8 and adds to 5 rather than , and the pairs using 6 and 4 multiply to 24 with the wrong sign.
A bacteria population doubles every 90 minutes and starts at 200 cells. Which expression gives the population after minutes?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . Exponential growth uses ; the population starts at 200, doubles so the factor is 2, and the number of doubling periods is the minutes divided by 90, giving the exponent . The form is linear, and the other choices misplace the base or the starting value.
When the expression is written in factored form, what is the greatest common factor that can be removed?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . Both terms share a factor of 3 and a factor of , so the greatest common factor is , giving . Removing only 3 or only leaves a common factor behind, and is not a factor of the first term.
Solve the equation for all real values of .
- A. and
- B. and
- C. only
- D. only
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Correct answer: and
The correct answer is and . Factoring out gives , so either or , which yields and . Forgetting the root, or misreading the sign, produces the other choices.
A parabola is written in vertex form as . What is the axis of symmetry of its graph?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . In vertex form , the axis of symmetry is the vertical line , and here because of the term. The value 7 is the y-coordinate of the vertex, and misreads the sign inside the parentheses.
Which expression is equivalent to ?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . Squaring a binomial follows , so . Writing omits the middle term, and the other choices use the wrong coefficient.
Solve the radical equation for .
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is 13. Squaring both sides gives , then adding 1 gives and dividing by 2 gives ; checking confirms . Forgetting to add 1 before dividing or mishandling the squaring step produces the other values.
For the function , which value of makes the function undefined?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . A rational function is undefined where its denominator equals zero, and gives . The value makes only the numerator zero, which is allowed, and the other choices do not make the denominator zero.
What value completes the square for the expression ?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is 16. To complete the square, take half of the coefficient of and square it: half of 8 is 4, and 4 squared is 16, so . Stopping at 4 gives only half the coefficient, and 64 squares the full coefficient 8 instead of its half.
An investment of 800 dollars grows according to , where is in years. By what percent does the investment grow each year?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . In an exponential model , the base 1.06 equals , so the rate is 0.06, which is a annual increase. The percent change is read from the difference between the base and 1, not from the base 1.06 itself or from misplacing the decimal.
What is the product of the binomials ?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . The product of conjugate binomials equals , so the middle terms cancel and . Keeping a middle term or writing misapplies the difference-of-squares pattern.
A quadratic function has zeros at and . Where is the axis of symmetry located?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . The axis of symmetry lies exactly halfway between the two zeros, so it is the average of and 6, which is . The other choices come from adding the zeros without dividing by 2 or from sign errors.
Which expression is the complete factorization of ?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . Checking by expansion, , which matches. The choice gives a middle term of , so it does not produce the required .
What is the y-intercept of the parabola defined by ?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . The y-intercept occurs where , and substituting gives , so the constant term is the y-intercept. The values and are the coefficients of the x-terms, which vanish when .
Which equation could have no real solutions?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . Rearranging gives , and since no real number squared is negative, this equation has no real solutions. The other equations all yield real values because they set equal to a nonnegative number or factor to give real roots.
Which value of x satisfies the rational equation ?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . Multiplying both sides by gives , so , then and ; checking confirms . Forgetting to distribute the across the parentheses produces the other choices.
A model rockets's height is given by . How many seconds after launch does it return to the ground?
- A.2 seconds
- B.64 seconds
- C.4 seconds
- D.8 seconds
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Correct answer: 4 seconds
The correct answer is 4 seconds. The rocket is on the ground where , so factoring gives , with solutions (launch) and (landing). The value is when it reaches maximum height, not the ground, and is a coefficient rather than a time.
Which expression is equivalent to , where x is not equal to 3?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . The numerator is a difference of squares that factors as , and canceling the common factor with the denominator leaves . Canceling to keeps the wrong factor, and the other choices fail to factor the numerator.
Why does the graph of an exponential function such as never touch the x-axis?
- A.Because the y-intercept is at the origin
- B.Because the function has a horizontal asymptote at the x-axis that it approaches but never reaches
- C.Because the function is undefined for negative values of x
- D.Because the base 4 is greater than 1
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Correct answer: Because the function has a horizontal asymptote at the x-axis that it approaches but never reaches
The correct answer is that the function has a horizontal asymptote at the x-axis that it approaches but never reaches. As x becomes very negative, gets closer and closer to zero without ever equaling zero, so the curve approaches the x-axis as an asymptote. The function is defined for all real x, and its y-intercept is at , not the origin.
What are the solutions to the equation ?
- A. only
- B. and
- C. only
- D. and
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Correct answer: and
The correct answer is and . Adding 27 gives , dividing by 3 gives , and taking the square root gives both and . Reporting only one root ignores the negative solution, and confuses the value with .
After completing the square, is rewritten in vertex form. What is that form?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . Taking half of gives , and , so . Misreading the sign inside the parentheses or the constant produces the other choices.
Which expression results from simplifying ?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . Multiplying monomials means multiplying the coefficients and adding the exponents, so and , giving . Adding the coefficients gives , and multiplying the exponents instead of adding them gives .
The graph of is shifted to produce . How is the graph of g related to the graph of f?
- A.Shifted 4 units right and 3 units up
- B.Shifted 4 units right and 3 units down
- C.Shifted 4 units left and 3 units down
- D.Shifted 4 units left and 3 units up
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Correct answer: Shifted 4 units left and 3 units down
The correct answer is shifted 4 units left and 3 units down. In , the value (from ) shifts the graph 4 units left, and shifts it 3 units down. The plus inside the parentheses moves the graph left, not right, and the negative constant moves it down, not up.
Which statement correctly describes the end behavior of an exponential decay function as x increases without bound?
- A.The output increases without bound
- B.The output stays constant at the initial value
- C.The output approaches zero
- D.The output becomes negative
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Correct answer: The output approaches zero
The correct answer is that the output approaches zero. In exponential decay the base is between 0 and 1, so repeatedly multiplying drives the output toward zero as x grows large, approaching the x-axis as an asymptote. The output does not grow without bound, turn negative, or remain at the initial value.
Solve the radical equation for x.
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . First isolate the radical by adding 2 to get , then square both sides to get , and subtract 5 to get ; checking confirms . Squaring before isolating the radical, or forgetting to subtract 5, produces the other values.
Which expression is the complete factorization of ?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . First factor out the common x to get , then factor the difference of squares , giving . Stopping at is not fully factored, and dropping the x loses a factor.
A square's area is modeled by , where s is the side length. If the side length triples, how does the area change?
- A.It increases by 3 square units
- B.It doubles
- C.It becomes 9 times as large
- D.It triples
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Correct answer: It becomes 9 times as large
The correct answer is that it becomes 9 times as large. Because area depends on the square of the side, replacing s with gives , so the area is multiplied by 9. The area does not simply triple, since the relationship is nonlinear, and it does not change by a fixed amount.
Using the quadratic formula, what are the solutions to ?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . With , , and , the discriminant is , so ; since , this becomes . Failing to simplify the radical or to divide both terms by 2 leads to the other choices.
Which type of function would best model a quantity that decreases by a fixed percentage during each equal time interval?
- A.A constant function
- B.A quadratic function
- C.An exponential function
- D.A linear function
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Correct answer: An exponential function
The correct answer is an exponential function. A fixed percentage change per equal interval means the quantity is multiplied by the same factor each period, which is the defining behavior of an exponential function. A linear function changes by a fixed amount rather than a fixed percentage, a quadratic changes at a varying rate tied to the square of the input, and a constant function does not change at all.
What are the solutions to the equation ?
- A. only
- B. and
- C. and
- D. and
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Correct answer: and
The correct answer is and . Taking the of both sides gives or , so or . Keeping only the positive root, or forgetting to add 4 back, produces the other choices.
Which expression is the complete factorization of ?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . Expanding gives , which matches the original. The pair produces a middle term of , so it does not give the required .
What is the minimum value of the function ?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . In vertex form , the value is the minimum when the parabola opens upward, and here . The value is the x-coordinate of the vertex, not the minimum output.
A population is modeled by , where is measured in days. By what factor does the population grow each day?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . In an exponential model , the base is the factor multiplied each period, and here means the population triples every day. The value 50 is the starting amount, not the growth factor.
What is the product ?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . This is a difference of squares where , so . The middle terms cancel, so no term appears.
Solve .
- A. and
- B. and
- C. and
- D. and
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Correct answer: and
The correct answer is and . The equation factors as , giving or . Choosing the opposite signs would not satisfy the original equation.
What value completes the square for the expression ?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . To complete the square, take half of the linear coefficient and square it: , giving . The value 6 stops at half the coefficient without squaring it.
A radioactive sample loses one-third of its mass every 5 years, starting at 600 grams. Which expression gives the mass after years?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . Losing one-third means two-thirds remains, so the decay factor is , and the exponent counts how many 5-year periods have passed. Using base would discard two-thirds instead of one-third each period.
What is the axis of symmetry of ?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . For the axis of symmetry is . The choice forgets to divide by .
Simplify the expression .
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . When multiplying monomials, the coefficients multiply () and the exponents on the same base add (). Multiplying the exponents would wrongly give .
Solve the radical equation .
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . Squaring both sides gives , so and ; checking confirms . The choice 27 stops before dividing by 3, and 23 mishandles the constant.
For , which values of make the function undefined?
- A. and
- B. and
- C. and
- D. only
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Correct answer: and
The correct answer is and . A rational function is undefined where its denominator equals zero, and gives or . The value only makes the numerator zero, which is allowed.
How many real solutions does the equation have?
- A.Two distinct real solutions
- B.Infinitely many real solutions
- C.Exactly one repeated real solution
- D.No real solutions
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Correct answer: Exactly one repeated real solution
The correct answer is exactly one repeated real solution. The discriminant is , and a discriminant of zero means a single repeated root, here . A positive discriminant would instead give two distinct solutions.
What is the y-intercept of ?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . The y-intercept occurs where , and substituting gives , so the constant term is the y-intercept. The values from the x-terms vanish when .
Factor completely: .
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . First remove the common factor 5 to get , then factor the difference of squares . Leaving is not fully factored.
A car's value is modeled by dollars after years. What does represent?
- A.The car keeps 88% of its value each year, losing 12%
- B.The starting value of the car
- C.The car loses 88% of its value each year
- D.The car gains 88% in value each year
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Correct answer: The car keeps 88% of its value each year, losing 12%
The correct answer is that the car keeps 88% of its value each year, losing 12%. In a decay model , the base means 88% remains after each year, so the annual loss is , or 12%. The starting value is the coefficient 24000, not the base.
Solve .
- A. and
- B. and
- C. and
- D. only
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Correct answer: and
The correct answer is and . Moving everything to one side gives , which factors as , so or . Dividing both sides by at the start would lose the solution .
Which expression is equivalent to , for ?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . The numerator factors as , and the common factor cancels with the denominator, leaving . Canceling the wrong factor would give a different result.
Using the quadratic formula, solve .
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . With , , and , the discriminant is , so ; since , this becomes . Failing to simplify the radical or divide both terms by 2 leads to the other choices.
The graph of is shifted to give . How does the graph move?
- A. units down
- B. units up
- C. units to the left
- D. units to the right
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Correct answer: units up
The correct answer is units up. Adding a constant outside the squared term, as in , shifts the parabola vertically by , so moves it units up. Horizontal shifts come from changes inside the parentheses, not from an added constant.
Simplify: .
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . Dividing powers with the same base subtracts the exponents: . Dividing the exponents would wrongly give , and adding them would give .
What is the extreme-value behavior of ?
- A.It has a maximum because the parabola opens downward
- B.It has a maximum because the parabola opens upward
- C.It has neither a maximum nor a minimum
- D.It has a minimum because the parabola opens upward
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Correct answer: It has a minimum because the parabola opens upward
The correct answer is that it has a minimum because the parabola opens upward. The positive leading coefficient makes the parabola open upward, so the vertex is the lowest point and the function has a minimum. A negative leading coefficient would instead create a maximum.
Solve the equation .
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . Multiplying both sides by gives , so , then and ; checking confirms . Forgetting to distribute the produces the other choices.
What is the product of the solutions to ?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . The equation factors as with roots and , and ; this also matches the constant term when the leading coefficient is . The sum of the roots, by contrast, is .
Simplify: .
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . Raising a product to a power raises each factor: and , giving . Cubing only the coefficient or adding the exponents instead of multiplying gives the wrong choices.
A diver's height is , where is in seconds. What is the diver's height at ?
- A. feet
- B. feet
- C. feet
- D. feet
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Correct answer: feet
The correct answer is feet. Substituting gives , so the constant term is the starting height. The values and are coefficients of the -terms, which vanish at .
Which equation represents exponential growth?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . Exponential growth occurs when the base is greater than , and satisfies that condition. A base between and , like , would model decay, and the linear forms do not have a variable exponent.
Solve .
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- D.
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Correct answer:
The correct answer is . Squaring both sides gives , so and ; checking confirms . The choice stops before dividing by .
What is the sum of the solutions to ?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . The equation factors as with roots and , and ; this equals for the equation. The product of the roots, , is a different quantity.
Rewrite in vertex form.
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . Completing the square adds and subtracts , giving . Misreading the sign inside the parentheses or the constant produces the other choices.
Factor completely: .
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . First remove the common factor to get , then factor the difference of squares . Stopping at is not fully factored, and dropping the loses a factor.
For , what happens to the output when is tripled?
- A.It triples
- B.It becomes times as large
- C.It stays the same
- D.It becomes times as large
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Correct answer: It becomes times as large
The correct answer is that it becomes times as large. Squaring a tripled input multiplies the result by : . The output does not merely triple because the variable is squared, making the relationship nonlinear.
Which value of satisfies the equation ?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . Since can be written as (because ), the exponents must match, giving . The choice confuses the base and the result.
Which expression is equivalent to , for ?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . Dividing each term in the numerator by gives and , so the result is . Dividing only one term or mishandling the exponents produces the other choices.
A parabola has zeros at and . What is the maximum or minimum y-value located at, in terms of position along the x-axis?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . The vertex, where the maximum or minimum occurs, lies on the axis of symmetry halfway between the two zeros: . Adding the zeros without dividing by would give the wrong value.
Which expression is equivalent to ?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . Multiplying powers with the same base adds the exponents: , so the result is . Multiplying the exponents would wrongly give .
Which expression is equivalent to ?
- A.
- B.
- C.
- D.
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Correct answer:
The correct answer is . Squaring a binomial follows , so . Writing omits the middle term, which is the difference-of-squares result instead.
Problem-Solving and Data Analysis (30)
A laptop's price is increased by 10% and then the new price is decreased by 10%. Compared with the original price, what is the net effect?
- A.The final price is 10% lower than the original price
- B.The final price is 1% higher than the original price
- C.The final price equals the original price
- D.The final price is 1% lower than the original price
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Correct answer: The final price is 1% lower than the original price
The final price is 1% lower than the original price. Increasing by 10% multiplies the price by , and decreasing that result by 10% multiplies by , so the combined factor is . Because is 1% below 1, the price ends up 1% lower. Percent changes applied in sequence are multiplied, not added, which is why a 10% increase followed by a 10% decrease does not return to the start.
A store buys a backpack for $40 and sells it at a price that gives a 35% profit on the cost. What is the selling price?
- A.$75
- B.$58
- C.$54
- D.$56
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Correct answer: $54
The selling price is $54. A 35% profit on the $40 cost adds dollars, so the selling price is dollars. Equivalently, the price is 135% of the cost: . The percent here is taken on the cost, which is the base, not on the selling price.
An investor deposits $2,000 in an account earning 5% simple interest per year. How much interest is earned after 3 years?
- A.$300
- B.$200
- C.$315
- D.$100
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Correct answer: $300
The interest earned is $300. Simple interest is found by multiplying the principal by the rate and the number of years: dollars. Each year earns $100 in simple interest, and over 3 years that totals $300. Unlike compound interest, simple interest is always calculated on the original principal only.
A blueprint is drawn so that 1 inch represents 6 feet. A wall that is actually 27 feet long is how many inches long on the blueprint?
- A.4.5 inches
- B.3.5 inches
- C.4.0 inches
- D.5.0 inches
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Correct answer: 4.5 inches
The wall is 4.5 inches long on the blueprint. Because 1 inch represents 6 feet, the number of inches equals the real length divided by 6: . Setting up the proportion 1 inch to 6 feet equals x inches to 27 feet gives the same result. Dividing the real measurement by the scale factor converts from feet to drawing inches.
A factory produces defective parts at a rate of 3 per 500 produced. At this rate, how many defective parts are expected in a batch of 8,000 parts?
- A.32
- B.40
- C.48
- D.24
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Correct answer: 48
About 48 defective parts are expected. The defect rate is 3 per 500, and groups of 500, so the expected number of defects is . Using the proportion 3 to 500 equals x to 8,000 gives the same answer. Scaling the rate up by the number of equal groups is the key step.
A faucet fills a tank at a constant rate of 9 gallons every 4 minutes. How many minutes will it take to fill a 63-gallon tank?
- A.32 minutes
- B.28 minutes
- C.24 minutes
- D.36 minutes
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Correct answer: 28 minutes
It will take 28 minutes. The unit rate is 9 gallons per 4 minutes, which is gallons per minute, so the time equals minutes. Alternatively, 63 gallons is gallons, so the time is minutes. Working with the per-minute rate or scaling the original ratio both lead to the same time.
A vehicle travels at 90 kilometers per hour. Given that 1 kilometer is approximately 0.62 miles, which expression best estimates its speed in miles per hour?
- A.
- B.
- C.
- D.
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Correct answer:
The estimate is , which is about 56 miles per hour. Each kilometer equals about 0.62 miles, so multiplying the number of kilometers by 0.62 converts the distance covered each hour into miles. Dividing would convert in the wrong direction, from miles back to kilometers, so multiplication by the conversion factor is correct here.
The seven daily high temperatures recorded one week were 71, 73, 68, 73, 75, 70, and 73 degrees. What is the median of these temperatures?
- A.73 degrees
- B.70 degrees
- C.75 degrees
- D.71 degrees
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Correct answer: 73 degrees
The median is 73 degrees. To find the median, order the values from least to greatest: 68, 70, 71, 73, 73, 73, 75. With seven values, the median is the fourth value, which is 73. The median is the middle value of an ordered list, not the most frequent value, although in this set 73 also happens to be the mode.
A teacher records quiz scores and finds the mean is 82. After discovering that one student's score of 50 was entered when it should have been 90, how will correcting the error affect the mean?
- A.The mean cannot change because the number of scores is unchanged
- B.The corrected mean will be lower than 82
- C.The corrected mean will be higher than 82
- D.The corrected mean will remain exactly 82
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Correct answer: The corrected mean will be higher than 82
The corrected mean will be higher than 82. Replacing the value 50 with the larger value 90 increases the total of all scores while the number of scores stays the same, so the mean, which is the total divided by the count, must rise. The mean depends on the sum of the values, so changing any single value to a larger one raises the average.
Two factories each shipped the same number of boxes with the same mean weight. Factory P's box weights are all very close to the mean, while Factory Q's box weights are scattered far above and below the mean. Which factory's weights have the smaller standard deviation?
- A.Both are equal because the mean weights are the same
- B.Factory P, because its weights are tightly clustered around the mean
- C.It cannot be determined without knowing the exact weights
- D.Factory Q, because its weights cover a wider range
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Correct answer: Factory P, because its weights are tightly clustered around the mean
Factory P has the smaller standard deviation because its weights are tightly clustered around the mean. Standard deviation measures the typical distance of values from the mean, so tightly grouped data produce a small standard deviation while widely scattered data produce a large one. Equal means describe the center of each set, not the spread, so they do not make the standard deviations equal.
Every value in a data set is increased by adding 5 to each number. How does this change affect the mean and the standard deviation of the data set?
- A.Both the mean and the standard deviation increase by 5
- B.The mean increases by 5 and the standard deviation stays the same
- C.The mean stays the same and the standard deviation increases by 5
- D.Both the mean and the standard deviation stay the same
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Correct answer: The mean increases by 5 and the standard deviation stays the same
The mean increases by 5 and the standard deviation stays the same. Adding the same amount to every value shifts the entire data set, raising the center, the mean, by exactly that amount. However, the distances between the values do not change, so the spread, measured by the standard deviation, is unchanged. Adding a constant moves the data but does not stretch or compress it.
A two-way table classifies 150 movie-goers by age group and ticket type. Among the 90 adults, 36 bought premium tickets. What fraction of the adults bought premium tickets?
- A.
- B.
- C.
- D.
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Correct answer:
The fraction is . The question restricts attention to adults, so the denominator is the 90 adults rather than all 150 movie-goers, and the favorable group is the 36 adults who bought premium tickets. Dividing 36 by 90 gives . Using 150 as the denominator would answer a different question about the entire group rather than about adults only.
A nutrition study estimated that 42% of surveyed teenagers eat breakfast daily, with a margin of error of 3 percentage points at a 95% confidence level. What does the margin of error indicate about this estimate?
- A.Exactly 42% of all teenagers eat breakfast daily
- B.The plausible range for the true percentage of all teenagers is about 39% to 45%
- C.The survey must be repeated because 42% is not a reliable figure
- D.The true percentage for all teenagers is almost certainly above 45%
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Correct answer: The plausible range for the true percentage of all teenagers is about 39% to 45%
The plausible range for the true percentage is about 39% to 45%. A margin of error of 3 percentage points means the survey's estimate of 42% should be widened by 3 in each direction, giving an interval from 39% to 45% within which the true population percentage is likely to fall. A single sample cannot fix the true value at exactly 42%, and the margin of error expresses normal sampling uncertainty rather than a flaw requiring the survey to be redone.
A scatterplot relating the age of a used car in years to its resale value in dollars shows points that decrease from upper left to lower right and lie close to a straight line. Which best describes the relationship and what the line of best fit would estimate?
- A.A weak negative association; the line guarantees the exact value of every car
- B.A strong negative association; the line estimates a likely resale value for a given age
- C.A strong positive association; the line estimates a likely age for a given resale value
- D.No association; the line cannot be used to make any estimate
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Correct answer: A strong negative association; the line estimates a likely resale value for a given age
There is a strong negative association, and the line of best fit estimates a likely resale value for a given age. Points falling from upper left to lower right show that as age increases, resale value tends to decrease, which is a negative relationship, and points lying close to a line make the association strong. The line summarizes the trend so a chosen age yields a predicted typical value, but it cannot guarantee the exact value of any individual car because real points scatter around the line.
A cyclist rides miles in hours at a constant speed. Which expression gives the cyclist's average speed in miles per hour?
- A. divided by
- B. multiplied by
- C. plus
- D. divided by
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Correct answer: divided by
The average speed is divided by , which equals miles per hour. Speed is distance divided by time, so dividing the miles traveled by the hours gives miles per hour. Multiplying distance by time or dividing time by distance would produce units that do not represent speed.
A chemical solution is salt by mass. How many grams of salt are in grams of the solution?
- A. grams
- B. grams
- C. grams
- D. grams
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Correct answer: grams
There are grams of salt. Since the solution is salt by mass, the amount of salt is of the total mass: grams equals grams. The value grams is the mass of everything that is not salt, and the percent must be taken on the full -gram mass.
A used textbook is marked down twice: first by , then the reduced price is marked down by an additional . What single percent of the original price does a buyer ultimately pay?
- A.
- B.
- C.
- D.
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Correct answer:
The buyer pays of the original price. After a markdown, remains, which is a factor of ; a further markdown leaves , a factor of . Multiplying the factors gives , or . Successive percent changes are multiplied, so the two discounts cannot simply be added to .
A landscaping crew of workers can clear a lot in hours. Assuming all workers clear at the same constant rate, how many hours would workers take to clear the same lot?
- A. hours
- B. hours
- C. hours
- D. hours
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Correct answer: hours
Six workers would take hours. The total work is fixed at workers times hours, which equals worker-hours. Dividing worker-hours by workers gives hours. This is an inverse relationship: more workers means less time, so the time goes down rather than up.
A runner completes a race in seconds. Given that minute equals seconds, how many minutes did the race take?
- A. minutes
- B. minutes
- C. minutes
- D. minutes
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Correct answer: minutes
The race took minutes. To convert from seconds to minutes, divide by because each minute contains seconds: divided by equals . Converting from a smaller unit to a larger unit yields a smaller number, so dividing rather than multiplying is correct.
A set of seven test scores is , , , , , , and . If the lowest score of is removed, which statement best describes the effect on the mean?
- A.The mean will stay the same because only one value changed
- B.The mean cannot be determined without the exact new total
- C.The mean will increase because a value below the mean was removed
- D.The mean will decrease because the data set is now smaller
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Correct answer: The mean will increase because a value below the mean was removed
The mean will increase because the removed score of is below the original mean, so taking it out raises the average of the remaining values. Removing a value smaller than the mean always pulls the mean upward, while removing a value larger than the mean would pull it down. The change in count alone does not determine the direction.
The dot plot of daily customer counts for a small shop shows the value appearing more often than any other value. Which measure of center does represent?
- A.The range
- B.The median
- C.The mode
- D.The mean
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Correct answer: The mode
The value represents the mode, which is the value that occurs most frequently in a data set. The mean is the arithmetic average of all values, the median is the middle value when the data are ordered, and the range measures the spread between the largest and smallest values rather than identifying a center.
Two data sets have the same range, but Set A has most of its values bunched near the center with two far-apart endpoints, while Set B has its values spread evenly across the whole range. Which set most likely has the smaller standard deviation?
- A.Both must be equal because the ranges are equal
- B.Set B, because its values are spread evenly
- C.It is impossible to compare without the exact values
- D.Set A, because most of its values lie close to the center
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Correct answer: Set A, because most of its values lie close to the center
Set A most likely has the smaller standard deviation because most of its values lie close to the center, keeping their typical distance from the mean small despite the two distant endpoints. Standard deviation measures the typical distance of values from the mean, which depends on where most data sit, not on the range. Equal ranges describe only the extreme spread, so they do not force equal standard deviations.
A two-way table records 200 commuters by transportation mode and whether they were late. Of the 75 commuters who took the train, 15 were late. What is the probability that a randomly chosen train commuter was late?
- A.
- B.
- C.
- D.
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Correct answer:
The probability is . Because the question restricts attention to train commuters, the denominator is the 75 train riders rather than all 200 commuters, and the favorable group is the 15 train riders who were late. Dividing 15 by 75 gives . Using 200 as the denominator would answer a different, unconditional question.
A bag holds 5 yellow, 8 white, and 7 black tiles. Two tiles will be picked, but the first tile picked is white and is not put back. What is the probability the next tile picked is yellow?
- A.
- B.
- C.
- D.
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Correct answer:
The probability is . After the white tile is removed and not replaced, 19 tiles remain instead of the original 20, while all 5 yellow tiles are still in the bag. The probability of drawing yellow is therefore 5 favorable outcomes out of 19 remaining tiles. Forgetting to reduce the total after removing the first tile would give the incorrect .
A polling firm reports that 56% of likely voters favor a proposal, with a margin of error of 5 percentage points. Which conclusion is best supported by these results?
- A.The true level of support is plausibly between 51% and 61%
- B.The poll is unreliable because no estimate can have error
- C.Exactly 56% of all likely voters favor the proposal
- D.Support is guaranteed to be above 50% for all voters
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Correct answer: The true level of support is plausibly between 51% and 61%
The best conclusion is that the true level of support is plausibly between 51% and 61%. A margin of error of 5 percentage points is applied in both directions around the 56% estimate, producing an interval from 51% to 61% likely to contain the true population value. A single poll cannot fix the value at exactly 56%, and the margin of error reflects normal sampling uncertainty rather than a defect in the poll.
A scientist wants to estimate the average weight of fish in a large lake and plans to compare two sampling plans. Which change to the sampling plan would most likely reduce the margin of error of the estimate?
- A.Reporting the result without a confidence level
- B.Selecting only the fish caught near one dock
- C.Rounding each weight to the nearest whole pound
- D.Selecting a much larger random sample of fish
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Correct answer: Selecting a much larger random sample of fish
Selecting a much larger random sample would most likely reduce the margin of error, because a larger sample provides more information about the population and produces a more precise estimate. Sampling only near one dock introduces bias rather than improving precision, omitting the confidence level does not change the underlying uncertainty, and rounding the weights does not address sampling variability.
A line of best fit for data relating outdoor temperature in degrees to daily ice cream sales in dollars is . Based on the model, what is the predicted change in sales for each one-degree rise in temperature?
- A.Sales increase by $150
- B.Sales decrease by $150
- C.Sales increase by $12
- D.Sales decrease by $12
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Correct answer: Sales increase by $12
Sales are predicted to increase by $12 for each one-degree rise in temperature. In the line of best fit, the slope is the coefficient of , here , and it gives the change in the predicted output for each one-unit increase in the input. The constant is the predicted value when temperature is zero, so it does not describe the rate of change.
A scatterplot of the number of hours a phone battery is used per day versus the number of days until the battery needs replacement shows points scattered widely with no clear upward or downward trend. Which description is most accurate?
- A.Little or no association between the variables
- B.A perfect linear association
- C.A strong negative linear association
- D.A strong positive linear association
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Correct answer: Little or no association between the variables
There is little or no association between the variables, because points scattered widely with no clear upward or downward trend show that knowing one value gives little information about the other. A positive association would rise from lower left to upper right and a negative one would fall in the opposite direction, while a perfect association would place every point exactly on a single line.
A photo measuring 4 inches wide and 6 inches tall is enlarged proportionally so that its new width is 10 inches. What is the new height?
- A.15 inches
- B.16 inches
- C.8 inches
- D.12 inches
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Correct answer: 15 inches
The new height is 15 inches. Because the enlargement is proportional, the ratio of width to height stays constant, so to equals to the new height. The width grew by a factor of , or , and applying that same factor to the height gives , which is inches.
A data set of monthly rainfall amounts has every value converted from inches to centimeters by multiplying each value by 2.54. How does this conversion affect the standard deviation of the data?
- A.The standard deviation increases by
- B.The standard deviation stays exactly the same
- C.The standard deviation is multiplied by
- D.The standard deviation becomes zero
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Correct answer: The standard deviation is multiplied by
The standard deviation is multiplied by . Multiplying every value by a constant scales the distances between values by that same constant, so the typical distance from the mean, which the standard deviation measures, is also multiplied by . Adding a constant would leave the standard deviation unchanged, but multiplying stretches the spread.
Geometry and Trigonometry (30)
In a right triangle, the two legs measure 8 and 15. What is the length of the hypotenuse?
- A.
- B.
- C.
- D.
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Correct answer:
The hypotenuse is . By the Pythagorean theorem, the square of the hypotenuse equals the sum of the squares of the legs, so the hypotenuse squared equals , which is , or . The is . The value wrongly adds the legs, is too small, and does not satisfy the theorem.
On the Digital SAT Math section, which complementary-angle relationship is always true for an acute angle of measure x degrees?
- A.The sine of equals the cosine of .
- B.The sine of equals the sine of .
- C.The tangent of equals the cosine of .
- D.The sine of equals the cosine of .
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Correct answer: The sine of equals the cosine of .
The correct relationship is that the sine of equals the cosine of . For complementary acute angles, the sine of one angle equals the cosine of its complement because the side opposite one acute angle is adjacent to the other. Sine and cosine of the same angle are equal only at , tangent and cosine are not generally equal, and sine of equals sine of is false in general.
A circle in the xy-plane has the equation . What is the radius of the circle?
- A.
- B.
- C.
- D.
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Correct answer:
The radius is . In the standard form of a circle's equation, the constant on the right side equals the radius squared, so the radius squared is and the radius is the , which is . The value mistakes the radius squared for the radius, wrongly doubles the radius, and incorrectly halves the constant.
A circle has a radius of 9. A sector of this circle is bounded by a central angle of 80 degrees. What is the area of the sector?
- A.
- B.
- C.
- D.
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Correct answer:
The sector area is . The full circle has area times the radius squared, which is , and the sector represents 80 out of 360 degrees, or two-ninths, of the circle. Two-ninths of equals . The choice uses one-ninth, uses one-third, and is the whole circle's area, not the sector.
In right triangle PQR, the right angle is at Q. The leg adjacent to angle P measures 7 and the leg opposite angle P measures 24. What is the value of the tangent of angle P?
- A.
- B.
- C.
- D.
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Correct answer:
The tangent of angle P is . The tangent of an angle equals the length of the opposite side divided by the length of the adjacent side, which is 24 divided by 7. The fraction reverses opposite and adjacent, while and use the hypotenuse of 25 and would represent cosine and sine, not tangent.
Two angles are supplementary. One angle measures 47 degrees. What is the measure of the other angle?
- A.
- B.
- C.
- D.
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Correct answer:
The other angle measures . Supplementary angles sum to , so the missing angle is , which equals . The value comes from subtracting from (complementary, not supplementary), is unrelated, and comes from subtracting from .
An angle measures 270 degrees. What is the equivalent measure of this angle in radians?
- A.
- B.
- C.
- D.
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Correct answer:
The angle equals radians. To convert degrees to radians, multiply by , so equals , which simplifies to . The choice equals , equals , and equals .
A cone has a radius of 3 inches and a height of 7 inches. Using the volume formula , what is the volume of the cone in cubic inches?
- A.
- B.
- C.
- D.
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Correct answer:
The volume is cubic inches. Substituting the radius of 3 and height of 7 gives , which is , or , equaling . The choice omits the one-third factor, forgets to square the radius, and uses an incorrect combination.
Triangle ABC is similar to triangle DEF. Side AB corresponds to side DE, and AB measures 6 while DE measures 15. If side BC measures 8, what is the length of the corresponding side EF?
- A.
- B.
- C.
- D.
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Correct answer:
Side EF measures . In similar triangles, corresponding sides are proportional, so the ratio of DE to AB is 15 to 6, or 5 to 2. Multiplying BC's length of 8 by gives . The value adds the difference of 9, adds 9 to 8 incorrectly, and uses the wrong ratio direction.
On the Digital SAT, a question gives the equation of a circle as and asks for its center. The most efficient way to identify the center is to
- A.Take the to find the coordinate values
- B.Read the values that make each squared term zero, giving the point
- C.Add the constants inside the parentheses to get the center coordinates
- D.Use the point directly from the equation
Show answerHide answer
Correct answer: Read the values that make each squared term zero, giving the point
The best approach is to read the values that make each squared term zero, giving the center . In standard form, a circle's equation is , where the center is ; setting and to zero yields and . Taking the gives the radius, not the center; adding the constants and using negated values both misread the form.
In a 30-60-90 right triangle, the side opposite the 30-degree angle measures 5. What is the length of the side opposite the 60-degree angle?
- A.
- B.
- C.
- D.
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Correct answer:
The side opposite the 60-degree angle is . In a 30-60-90 triangle, the sides are in the ratio , where the shortest side faces the 30-degree angle; the side facing 60 degrees is the short side times , so . The value is the hypotenuse, belongs to a 45-45-90 triangle, and halves the correct length.
In right triangle XYZ, the right angle is at Z. The hypotenuse measures 10 and the side adjacent to angle X measures 6. What is the value of the cosine of angle X?
- A.
- B.
- C.
- D.
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Correct answer:
The cosine of angle X is . The cosine of an angle equals the length of the adjacent side divided by the hypotenuse, which is 6 divided by 10, or . The value would be the sine using the opposite leg of 8, inverts the ratio, and divides the legs rather than using the hypotenuse.
Two parallel lines are cut by a transversal. A pair of alternate interior angles is formed, and one of these angles measures 72 degrees. What is the measure of the other alternate interior angle?
- A.
- B.
- C.
- D.
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Correct answer:
The other alternate interior angle measures . When a transversal crosses two parallel lines, alternate interior angles are congruent, so the second angle equals the first at . The value treats them as supplementary, treats them as complementary, and doubles the angle, none of which applies to alternate interior angles.
A circle has a radius of 12. An arc is intercepted by a central angle of 30 degrees. What is the length of this arc? (The circumference of a circle is .)
- A.
- B.
- C.
- D.
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Correct answer:
The arc length is . The full circumference is , which is , and the arc represents 30 out of 360 degrees, or one-twelfth, of the circle. One-twelfth of equals . The choice uses one-sixth, uses one-fourth, and uses one-twenty-fourth, all incorrect fractions of the circumference.
In right triangle JKL, the right angle is at K. The leg opposite angle J measures 9 and the hypotenuse measures 41. What is the value of ?
- A.
- B.
- C.
- D.
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Correct answer:
The value of is because sine of an angle equals the side opposite the angle divided by the hypotenuse, and here the opposite leg is while the hypotenuse is . The fraction would be using the other leg of from the -- right triangle, inverts the ratio, and is using the two legs.
A square has a diagonal of length 10. What is the length of each side of the square?
- A.
- B.
- C.
- D.
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Correct answer:
Each side measures because a square's diagonal splits it into two -- triangles where the diagonal is the side times ; setting the side times equal to and dividing gives , which rationalizes to . The value multiplies instead of divides, ignores the radical factor, and repeats the diagonal.
A regular hexagon has interior angles that are all equal. What is the measure of each interior angle of a regular hexagon?
- A.
- B.
- C.
- D.
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Correct answer:
Each interior angle measures because the sum of interior angles of a polygon equals times the quantity of sides minus , which for six sides is , or , and dividing by the equal angles gives . The value is the total of all angles rather than one, is the interior angle of a regular pentagon, and is the central angle, not the interior angle.
In a circle of radius 10, a sector has an area of . What is the measure of the sector's central angle in degrees?
- A.
- B.
- C.
- D.
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Correct answer:
The central angle measures because the sector area equals the fraction of the circle times the full area , so equals the fraction times , giving a fraction of one-fifth; one-fifth of is . The value halves the correct angle, confuses the area coefficient with the angle, and doubles the correct fraction.
A rectangular box has a length of 5, a width of 4, and a height of 3. What is the surface area of the box?
- A.
- B.
- C.
- D.
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Correct answer:
The surface area is because a rectangular box has three pairs of identical faces with areas equal to length times width, length times height, and width times height; that is , , and , and doubling their sum of gives . The value is the volume, forgets to double the three face areas, and doubles the volume.
In right triangle MNP, the right angle is at N. If , what is the value of ?
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- B.
- C.
- D.
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Correct answer:
The value of is because equal to means the opposite leg is and the adjacent leg is , so the hypotenuse is , which is ; sine is opposite over hypotenuse, or . The value would be , repeats the tangent, and inverts a ratio.
An angle measures 225 degrees. What is the equivalent measure of this angle in radians?
- A.
- B.
- C.
- D.
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Correct answer:
The angle equals radians because converting degrees to radians means multiplying by , so equals , which reduces to . The choice equals , equals , and equals .
A circle in the xy-plane has the equation . What is the radius of this circle?
- A.
- B.
- C.
- D.
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Correct answer:
The radius is because the equation is the standard form of a circle centered at the origin where the right side equals the radius squared, so the radius is , which is . The value mistakes the radius squared for the radius, halves the constant, and doubles the radius.
Triangle RST is similar to triangle UVW with a scale factor of 2 from RST to UVW. If the perimeter of triangle RST is 18, what is the perimeter of triangle UVW?
- A.
- B.
- C.
- D.
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Correct answer:
The perimeter of triangle UVW is because perimeter is a linear measurement that scales by the same factor as the side lengths, so multiplying the perimeter of by the scale factor of gives . The value divides instead of multiplies, adds rather than scaling, and incorrectly applies the square of the scale factor, which would apply to area, not perimeter.
Two complementary angles have measures in the ratio 2 to 3. What is the measure of the larger angle?
- A.
- B.
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- D.
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Correct answer:
The larger angle measures because complementary angles sum to and the ratio to divides into equal parts of each, so the larger angle is parts, or , which is . The value is the smaller angle of parts, wrongly uses a total, and does not match the to ratio.
A right circular cylinder has a radius of 5 and a height of 8. What is the lateral surface area of the cylinder, using ?
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- B.
- C.
- D.
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Correct answer:
The lateral surface area is because the lateral surface of a cylinder unrolls into a rectangle whose area is the circumference multiplied by the height, giving , which is . The value omits the factor of , mistakenly squares the radius, and adds the base areas to a wrong lateral value.
In a circle, a central angle and an inscribed angle both intercept the same arc. If the inscribed angle measures 35 degrees, what is the measure of the central angle?
- A.
- B.
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- D.
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Correct answer:
The central angle measures because a central angle equals the measure of its intercepted arc while an inscribed angle equals half that arc, so the central angle is twice the inscribed angle, or , which is . The value wrongly sets them equal, halves the inscribed angle, and is the complement, which is unrelated.
In right triangle ABC, the right angle is at C, angle A measures 60 degrees, and the side opposite angle A measures . What is the length of the hypotenuse?
- A.
- B.
- C.
- D.
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Correct answer:
The hypotenuse is because with a angle and a angle this is a -- triangle whose sides follow the ratio to to ; the side opposite equals the short side times , so the short side is and the hypotenuse is twice the short side, which is . The value is the short side, triples the short side, and repeats the given side.
A triangle has vertices at the points , , and . What is the length of the longest side of this triangle?
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- B.
- C.
- D.
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Correct answer:
The longest side has length because the points form a right triangle with horizontal leg and vertical leg , and the hypotenuse connecting to is , which is , or . The value adds the legs, and are the two legs, each shorter than the hypotenuse.
In a circle of radius 6, an arc has a length of . What is the measure of the central angle that intercepts this arc, in degrees?
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- B.
- C.
- D.
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Correct answer:
The central angle measures because arc length equals the fraction of the circle times the circumference , so equals the fraction times , giving a fraction of five-twelfths; five-twelfths of is . The value halves the correct angle, uses one-third of the circle, and does not match the five-twelfths fraction.
In right triangle DEF, the right angle is at F. If , what is the value of ?
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- D.
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Correct answer:
The value of is because angles D and E are the two acute angles of a right triangle and are therefore complementary, and the sine of one acute angle equals the cosine of the other, so equals , which is . The value subtracts from incorrectly, would be in a specific triangle, and is unrelated.
References
- 1.College Board. “Digital SAT: What's on the Test — Structure.” satsuite.collegeboard.org, 2026. ↑
- 2.College Board. “SAT Reading and Writing Section.” satsuite.collegeboard.org. ↑
- 3.College Board. “SAT Math Section.” satsuite.collegeboard.org. ↑
- 4.College Board. “SAT Scores — Understanding Scores.” satsuite.collegeboard.org. ↑
- 5.College Board. “Register for the SAT.” satsuite.collegeboard.org. ↑
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