Taking the SAT® I: Reasoning Test Introduction Sections

Taking the SAT® I: Reasoning Test Introduction Sections The materials in these files are intended for individual use by students getting ready to take an SAT Program test; permission for any other use must be sought from the SAT Program. Schools (state-approved and/or accredited diploma-granting secondary schools) may reproduce them, in whole or in part, in limited quantities, for face-to-face guidance/teaching purposes but may not mass distribute the materials, electronically or otherwise. These materials and any copies of them may not be sold, and the copyright notices must be retained as they appear here. This permission does not apply to any third-party copyrights contained herein. No commercial use or further distribution is permitted.

The College Board: Expanding College Opportunity The College Board is a national nonprofit membership association whose mission is to prepare, inspire, and connect students to college success and opportunity. Founded in 1900, the association is composed of more than 4,300 schools, colleges, universities, and other educational organizations. Each year, the College Board serves over three million students and their parents, 23,000 high schools, and 3,500 colleges through major programs and services in college admissions, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its best-known programs are the SAT®, the PSAT/NMSQT®, and the Advanced Placement Program® (AP®). The College Board is committed to the principles of excellence and equity, and that commitment is embodied in all of its programs, services, activities, and concerns. For further information, visit www.collegeboard.com. Copyright 2003 by College Entrance Examination Board. All rights reserved. College Board, Advanced Placement Program, AP, One-on-One with the SAT, SAT, and the acorn logo are registered trademarks of the College Entrance Examination Board. ELPT and English Language Proficiency Test are trademarks owned by the College Entrance Examination Board. PSAT/NMSQT is a registered trademark of the College Entrance Examination Board and the National Merit Scholarship Corporation. SAT PrepPacks is a trademark owned by collegeboard.com. Other products and services mentioned herein may be trademarks of their respective owners. This publication was prepared and produced by Educational Testing Service (ETS), which develops and administers SAT Program tests for the College Board. The College Board and Educational Testing Service are dedicated to the principle of equal opportunity, and their programs, services, and employment policies are guided by that principle.

Real Test Prep from the Test Makers!

2003 TAKING THE SAT I: ®

Reasoning Test

2004 Visit the SAT Prep Center at

www.collegeboard.com for more practice.

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Taking the SAT I: Reasoning Test

SAT Committee 2002–2003

Contents

Rebecca Zwick, University of California at Santa Barbara, Santa Barbara, California, Chair John Barnhill, Florida State University, Tallahassee, Florida Esperanza Buckhalt, Coral Gables High School, Coral Gables, Florida Suzanne T. Colligan, Georgetown Visitation Prep School, Washington, DC Joseph A. Estrada, Texas A&M University, College Station, Texas Deren Finks, Harvey Mudd College, Claremont, California Kurt F. Geisinger, University of St. Thomas, Houston, Texas Christine Gonzalez, Baltimore Public Schools, Baltimore, Maryland

SAT® I: Reasoning Test

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Frequently Asked Questions . . . . . . . . . . . . . . . . . . . . . . . . . .4 Preparing for the SAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5

Verbal Practice Questions

6

Analogies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 Sentence Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 Critical Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9

Mathematics Practice Questions 14 Calculator Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14 Mathematics Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15

Daniel Lotesto, Riverside University High School, Milwaukee, Wisconsin

Standard Multiple Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . .21

Kelly A. Walter, Boston University, Boston, Massachusetts

Student-Produced Response . . . . . . . . . . . . . . . . . . . . . . . . .26

Wilbur W. Washburn, University of Washington, Seattle, Washington

Quantitative Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . .23

Practice SAT Test

30

Susan Wilbur, University of California at Irvine, Irvine, California

About the Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30

Student Representatives

Practice SAT Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35

Joshua Bryan Henderson, Trinity University, San Antonio, Texas

Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62

Ashraf Mohammad Akhtar Hossain, Duke University, Durham, North Carolina

Score Conversion Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63

Answer Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31

Scoring the Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . .63

THE COLLEGE BOARD: EXPANDING COLLEGE OPPORTUNITY The College Board is a national nonprofit membership association whose mission is to prepare, inspire, and connect students to college success and opportunity. Founded in 1900, the association is composed of more than 4,300 schools, colleges, universities, and other educational organizations. Each year, the College Board serves over three million students and their parents, 23,000 high schools, and 3,500 colleges through major programs and services in college admissions, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its best-known programs are the SAT®, the PSAT/NMSQT®, and the Advanced Placement Program® (AP®). The College Board is committed to the principles of excellence and equity, and that commitment is embodied in all of its programs, services, activities, and concerns. For further information, visit www.collegeboard.com. This book was prepared and produced by Educational Testing Service (ETS), which develops and administers SAT Program tests for the College Board. Copies of this book are shipped to secondary schools at the beginning of each academic year for free distribution to students who plan to register for the SAT. Additional copies can be purchased for $6.00 each (50 or more, $3.00 each). To obtain a free copy, write to College Board SAT Program, P.O. Box 6200, Princeton, NJ 08541-1200. The College Board and ETS are dedicated to the principle of equal opportunity, and their programs, services, and employment policies are guided by that principle. Copyright © 2003 by College Entrance Examination Board. All rights reserved. College Board, Advanced Placement Program, AP, One-on-One with the SAT, SAT, and the acorn logo are registered trademarks of the College Entrance Examination Board. ELPT and English Language Proficiency Test are trademarks owned by the College Entrance Examination Board. PSAT/NMSQT is a registered trademark of the College Entrance Examination Board and National Merit Scholarship Corporation. SAT PrepPacks is a trademark owned by collegeboard.com. Other products and services may be trademarks of their respective owners. Taking the SAT I: Reasoning Test

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SAT I: Reasoning Test ®

questions and answers

Frequently Asked Questions

How long is the SAT? The SAT lasts three hours. It contains seven separately timed sections that may appear in different orders: ❚ Three verbal sections: two 30-minute sections and one 15-minute section (78 questions: 19 sentence completions; 19 analogies; 40 critical reading)

❚ understand and analyze what you read

❚ Three math sections: two 30-minute sections and one 15-minute section (60 questions: 35 five-choice; 15 quantitative comparisons; 10 student-produced responses)

❚ recognize relationships between parts of a sentence

❚ One 30-minute equating section*: verbal or math

❚ establish relationships between pairs of words

* The equating section does not count toward your final score. It is used to try out new questions for future editions of the SAT and to help make sure that your test scores are comparable to scores on other editions.

What do the verbal questions test? Verbal questions test your ability to:

What do the math questions test? Mathematics questions test your ability to solve problems involving:

❚ arithmetic ❚ algebra ❚ geometry A year of algebra and some geometry are sufficient to answer SAT math questions.

Are all questions worth the same? Yes. You get one point for each correct answer. You get no points for questions you omit.You lose a fraction of a point for each wrong answer, except on the studentproduced response questions in the math section. On those questions, no points are deducted for wrong answers.

Test-Taking Tips Before the Test

1.

Know the test directions for all six question types. Use the time you save to answer questions.

4

2.

Get familiar with the answer sheet. It has four pages, and you need to know what answers go in which section. Take a look at the sample answer sheet on page 31.


During the Test

3.

Answer easy questions first. You earn just as many points for easy questions as you do for hard questions. The easier questions are at the beginning of the section and the harder questions at the end—except for Critical Reading questions, which are ordered according to the logic and organization of each passage.

4.

Guess smart. If you can rule out one or more answer choices for a multiple-choice question as definitely wrong, your chances of guessing the right answer improve. For math questions without answer choices, fill in your best guess; no points are subtracted for wrong answers as they are in all other question types.

How are SAT scores reported? You get two scores—one for the verbal sections and one for the math sections. These scores range from 200 to

800, with 800 being the best score. When you get your scores, you also get information that explains how you did on each part of the test.

Preparing for the SAT What are the best ways to practice for the SAT? ❚ Take the PSAT/NMSQT®.

❚ Most programs improve scores an average of about 10 points on verbal and about 15–20 points on math. ❚ Longer programs (40 hours or so) can improve scores an average of about 25–40 points on verbal and math combined, although additional benefits get smaller as programs get longer.

Do SAT scores change without coaching programs? Yes. Students who take the SAT more than once almost always earn different scores. The average

change between the spring of the junior year and the fall of the senior year is an increase of 10–15 points on verbal and 10–15 points on math. While the scores rise or stay the same for nearly two-thirds of repeat testtakers, some do decline. The main reasons why your scores may increase are real academic growth and practice taking the SAT.

❚ Take a practice SAT test (see page 30). ❚ Visit the SAT Prep Center at www.collegeboard.com. ❚ Go over the sample questions, test-taking tips, and directions in this book.

What other test prep materials help prepare you? The best way to practice is to use real SAT questions and tests. Here are College Board materials that

use both: Do coaching programs help? There is no proven answer. Research that examined a wide range of coaching programs—from basic familiarization to extensive instruction—suggests that:

Software: One-on-One with the SAT Book: 10 Real SATs Internet: SAT Prep Center at

®

www.collegeboard.com (See page 2 for ordering and pricing information.)

Test-Taking Tips 5.

Omit questions that you really have no idea how to answer. But if you can rule out any choice, you probably should guess from among the rest of the choices.

6.

Don’t panic if you cannot answer every question. You do not have to answer every question correctly to get a good score. You can get an average score by answering about half of the questions correctly and omitting the remaining questions.

7.

Use your test book for scratch work. You can also cross off choices you know are wrong and mark questions you have omitted so you can go back to them if you have time.

8.

Keep track of time. If you finish a section before time is called, check your answers in that section only.


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Verbal

practice test questions

The verbal sections of the SAT contain three types of questions:

Think of the colon (:) and double colon (::) as shorthand for is related to and as. For example: “CRUMB is related to BREAD as ____________ is related to _________.” Follow the two outlined steps to answer each analogy.

❚ analogies (19 questions) ❚ sentence completions (19 questions) ❚ critical reading (40 questions) Note: Calculators may not be on your desk or be used on the verbal sections of the SAT.

Analogies Analogy questions measure your: ❚ knowledge of the meanings of words ❚ ability to see a relationship in a pair of words ❚ ability to recognize a similar or parallel relationship in another pair of words

DIRECTIONS Each question below consists of a related pair of words or phrases, followed by five pairs of words or phrases labeled A through E. Select the pair that best expresses a relationship similar to that expressed in the original pair.

Step 1:: Establish an exact relationship between the two capitalized words before looking at the five answer choices. Make up a short sentence or phrase that shows how the two words in capital letters relate to each other. In the example above, the relationship between CRUMB and BREAD can best be stated as “a CRUMB is a very small piece of BREAD.” Step 2:: Determine which pair of words among the five answer choices works best in the sentence that you made up. Put each answer choice in the sentence you have established and see if it fits. ❚ ounce, a very small piece of unit? No. ❚ splinter, a very small piece of wood? Yes, but continue to look at the possibilities. ❚ water, a very small piece of bucket? No. ❚ twine, a very small piece of rope? No. ❚ cream, a very small piece of butter? No. Correct answer: (B) / Difficulty level: Easy

SAMPLE QUESTIONS

Example: CRUMB:BREAD:: (A) (B) (C) (D) (E)

Answering Strategy

For each of the following:

ounce:unit splinter:wood water:bucket twine:rope cream:butter C

A

D

E

❚ State the precise relationship between the capitalized pair of words in a sentence. ❚ Decide which answer choice best fits the sentence. ❚ Read the explanation provided.

Analogy Tips 1.

Be flexible. Words can have more than one meaning, and pairs of words can have more than one relationship. If necessary, revise your sentence expressing the relationship until you can identify a single best answer.

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2.

Do not look for words that have similar meanings. The meanings of the words are less important than the relationship between each pair of words.


3.

Look for parts of speech to be used consistently. The words in the answer choices should be used in the same way as the words in CAPITAL letters.

4.

Consider each of the five choices before you select your answer. Eliminate choices that you know are wrong. Choose your answer from those remaining.

1. CLAY:POTTER:: (A) (B) (C) (D) (E)

stone:sculptor machines:mechanic hems:tailor bricks:architect chalk:teacher

Step 2:: Now decide which pair of words among the five choices works best in the sentence you made up and mark your answer here: A

B

C

D

E

Explanation Explanation One relationship between CLAY and POTTER can be stated as “CLAY is what a POTTER works with.” But this relationship allows for several possible answers and is therefore too general: a sculptor works with stone, a mechanic works with machines, a tailor works with hems, and so on. You need a more precise relationship. A sentence such as “CLAY is the material shaped by a POTTER” allows only choice (A) as the correct answer. A POTTER shapes CLAY to make something just as a sculptor shapes stone to make something. Correct answer: (A) / Difficulty level: Easy

2. DISLIKE:LOATHE:: (A) (B) (C) (D) (E)

terrorize:fear admire:despise obscure:confuse annoy:infuriate order:obey

Explanation Both DISLIKE and LOATHE have similar general meanings, but they express different degrees of feeling: “To LOATHE is to DISLIKE greatly.” Only choice (D) fits this relationship: “To infuriate is to annoy greatly.” If you don’t consider the precise meanings of the words in this analogy, you might be tempted to select (C). In (C), obscure and confuse have similar general meanings (like the capitalized pair of words), but they are also similar in degree. Thus, it is not true that “to confuse is to obscure greatly.” Correct answer: (D) / Difficulty level: Medium

YOUR TURN Apply what you have learned to find the best answer to this hard analogy: 3. CALLOW:MATURITY:: (A) (B) (C) (D) (E)

avaricious:gain sympathetic:calamity naïve:childishness tyrannical:dissent deceitful:honesty

The relationship between CALLOW and MATURITY can be stated as “Someone who is CALLOW lacks MATURITY.” If you answered (E), you are correct. Someone who is deceitful lacks honesty. CALLOW and MATURE refer to opposite human qualities; so do deceitful and honest. Here are the relationships in choices A–D: ❚ (A) Someone who is avaricious desires gain. ❚ (B) We are often sympathetic to those who experience a calamity. ❚ (C) Someone who is naïve (a word like CALLOW) may demonstrate childishness. ❚ (D) Someone who is tyrannical does not permit dissent. Correct Answer: (E) / Difficulty level: Hard

Sentence Completions Sentence Completion questions measure your: ❚ knowledge of the meanings of words ❚ ability to understand how the different parts of a sentence fit logically together

DIRECTIONS Each sentence below has one or two blanks, each blank indicating that something has been omitted. Beneath the sentence are five words or sets of words labeled A through E. Choose the word or set of words that, when inserted in the sentence, best fits the meaning of the sentence as a whole.

Example: Medieval kingdoms did not become constitutional republics overnight; on the contrary, the change was -------. (A) unpopular (B) unexpected (C) advantageous (D) sufficient (E) gradual

Step 1:: Write a clear, specific relationship between the capitalized words here (check the dictionary if you are not sure about the meaning of CALLOW):

A

B

C

D


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Answering Strategy To answer a sentence completion question, look at how the parts of the sentence relate to one another. Then pick the choice that best fits the logic of the sentence as a whole. In this example: ❚ The first part of the sentence says that the “kingdoms” did not change “overnight.” ❚ The second part begins with “on the contrary” and describes the true nature of the change. ❚ The correct answer will be a word that describes a change that is “contrary” to an “overnight” change. ❚ Change that is unpopular, unexpected, advantageous, or sufficient is not necessarily change that is slow. ❚ However, gradual change is “contrary” to “overnight” change. Correct answer: (E) / Difficulty level: Easy

SAMPLE QUESTIONS For each of the following sentence completions: ❚ Read the sentence. ❚ Decide which choice is correct. ❚ Read the explanation.

❚ You are told that the scientist is “usually reserved,” but that, at the press conference, she behaved “unexpectedly.” ❚ The word that makes sense to fill in the blank must mean the opposite of (or at least be very different from) “reserved.” ❚ Since “reserved” means restrained or uncommunicative, the correct answer must be a word like frank or open or outspoken. ❚ Only (B), forthright, is a word with this kind of meaning. ❚ Read the entire sentence with forthright filled in to see that it makes sense. Correct answer: (B) / Difficulty level: Easy 2. With foolish parsimony, the new edition of the book ------- the beautiful illustrations that so ------- the original edition. (A) (B) (C) (D) (E)

deletes..cluttered expands..popularized reproduces..graced omits..distinguished includes..blemished

Explanation

1. At a recent press conference, the usually reserved biochemist was unexpectedly ------- in addressing the ethical questions posed by her work. (A) correct (B) forthright (C) inarticulate (D) retentive (E) cautious

Explanation Some sentence completions emphasize knowledge of vocabulary. The logic of these sentences is straightforward. In this example:

One way to answer a sentence completion question with two words missing is to focus first on one of the two blanks. If one of the words in an answer choice is logically wrong, then you can eliminate the entire choice from consideration. Look at the second blank in the example above. ❚ Would it make sense to say that beautiful illustrations cluttered the original book? No, so (A) cannot be the answer. ❚ Would beautiful illustrations have blemished the original book? No, so (E) can be eliminated.

Sentence Completion Tips 1.

Read the sentence carefully. You may want to substitute the word “blank” for each blank to get an overall sense of the meaning of the sentence. Figure out how its parts relate to each other.

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2.

Pay attention to the precise meanings of the words in the sentence and answer choices. Don’t pick an answer simply because it is a popular expression or “sounds good.”


3.

Note any key transitional words in the sentence. They tell you how the parts of the sentence relate to each other. Some common transitional words are but, although, however, yet, even though, because, therefore, as a result, etc.

4.

Consider all the answer choices. Don’t overlook a choice that makes a better and more logical sentence than your choice does.

❚ Now you can focus on the first blank in the sentence and try to figure out whether (B), (C), or (D) is the best answer. ❚ The beginning of the sentence refers to “foolish parsimony” in the new edition of the book. Parsimony suggests stinginess, an excessive reluctance to spend money. ❚ Would it have been foolishly stingy to expand or to reproduce the beautiful illustrations? No, so (B) and (C) can be eliminated, and only (D) remains. Always check your choice, however, by reading the entire sentence with your answer filled in. ❚ Does it make sense to say “With foolish parsimony, the new edition of the book omits the beautiful illustrations that so distinguished the original edition?” Yes. Correct answer: (D) / Difficulty level: Medium

Critical Reading The critical reading questions on the SAT measure your ability to read and think carefully about several different reading passages ranging from 100 to about 850 words long. Passages are taken from a variety of fields, including the humanities, social sciences, and natural sciences. They vary in style and can include narrative, argumentative, and expository elements. Some selections consist of a pair of related passages on a shared issue or theme that you are asked to compare and contrast. The following kinds of questions may be asked about a passage: ❚ Vocabulary in Context questions ask you to determine the meanings of words from their context in the reading passage. ❚ Literal Comprehension questions assess your understanding of significant information directly stated in the passage.

YOUR TURN Now try a difficult sentence completion question on your own. Like hard analogies, hard sentence completions usually contain challenging vocabulary. 3. There is no doubt that Larry is a genuine -------: he excels at telling stories that fascinate his listeners. (A) braggart (B) dilettante (C) pilferer (D) prevaricator (E) raconteur

If you do not know the meaning of some of these words, check them in the dictionary now, and mark your answer here: A

B

C

D

E

Explanation Some sentence completions contain a colon. This is a signal that the words after the colon define or directly clarify what came before. In this case, “he excels at telling stories that fascinate his listeners” defines the word raconteur, choice (E). None of the other words is directly defined by this clause. ❚ A braggart may or may not excel at telling stories and may actually annoy listeners. ❚ A dilettante is someone who dabbles at a career or hobby and so may not excel at anything. ❚ A pilferer steals repeatedly, in small quantities; this has nothing to do with storytelling. ❚ A prevaricator tells lies, but not necessarily in an accomplished or fascinating way; and the sentence refers to stories, not lies. You should choose the word that best fits the meaning of the sentence as a whole, and only choice (E) does so. Correct answer: (E) / Difficulty level: Hard

❚ Extended Reasoning questions measure your ability to synthesize and analyze information as well as to evaluate the assumptions made and the techniques used by the author. Most of the critical reading questions fall into this category. You may be asked to identify cause and effect, make inferences, recognize implications, and follow the logic of an argument. Below are samples of the kind of reading passages and questions that may appear on your test. The first is a single long passage, the next is a pair of related passages, and the last is a short passage. For each set of sample materials: ❚ Read the text carefully. ❚ Decide on the best answer to each question. ❚ Read the explanation for the correct answer.

DIRECTIONS The passages below are followed by questions based on their content; questions following a pair of related passages may also be based on the relationship between the paired passages. Answer the questions on the basis of what is stated or implied in the passages and in any introductory material that may be provided. The following passage is from a book written by a zoologist and published in 1986.

The domestic cat is a contradiction. No other animal has developed such an intimate relationship with humanity, while at the same time demanding and Line getting such independent movement and action. The cat manages to remain a tame animal because 5 of the sequence of its upbringing. By living both with other cats (its mother and littermates) and with humans (the family that has adopted it) during its infancy and kittenhood, it becomes attached to and considers that it 10 belongs to both species. It is like a child that grows up in a foreign country and as a consequence becomes


9

15

20

25

30

35

40

45

50

bilingual. The young cat becomes bimental. It may be a cat physically but mentally it is both feline and human. Once it is fully adult, however, most of its responses are feline ones, and it has only one major reaction to its human owners. It treats them as pseudoparents. The reason is that they took over from the real mother at a sensitive stage of the kitten’s development and went on giving it milk, solid food, and comfort as it grew up. This is rather different from the kind of bond that develops between human and dog. The dog sees its human owners as pseudoparents, as does the cat. On that score the process of attachment is similar. But the dog has an additional link. Canine society is grouporganized; feline society is not. Dogs live in packs with tightly controlled status relationships among the individuals. There are top dogs, middle dogs, and bottom dogs; and under natural circumstances they move around together, keeping tabs on one another the whole time. So the adult pet dog sees its human family both as pseudoparents and as the dominant members of the pack, hence its renowned reputation for obedience and its celebrated capacity for loyalty. Cats do have a complex social organization, but they never hunt in packs. In the wild, most of their day is spent in solitary stalking. Going for a walk with a human, therefore, has no appeal for them. And as for “coming to heel” and learning to “sit” and “stay,” they are simply not interested. Such maneuvers have no meaning for them. So the moment a cat manages to persuade a human being to open a door (that most hated of human inventions), it is off and away without a backward glance. As it crosses the threshold, the cat becomes transformed. The kitten-of-human brain is switched off and the wildcat brain is clicked on. The dog, in such a situation, may look back to see if its human packmate is following to join in the fun of exploring, but not the cat. The cat’s mind has floated off into another, totally feline world, where strange bipedal* primates have no place. Because of this difference between domestic cats and domestic dogs, cat-lovers tend to be rather different

55

60

65

70

75

80

85

from dog-lovers. As a rule cat-lovers have a stronger personality bias toward working alone, independent of the larger group. Artists like cats; soldiers like dogs. The much-lauded “group loyalty” phenomenon is alien to both cats and cat-lovers. If you are a company person, a member of the gang, or a person picked for the squad, the chances are that at home there is no cat curled up in front of the fire. The ambitious Yuppie, the aspiring politician, the professional athlete, these are not typical cat-owners. It is hard to picture football players with cats in their laps—much easier to envisage them taking their dogs for walks. Those who have studied cat-owners and dog-owners as two distinct groups report that there is also a gender bias. The majority of cat-lovers are female. This bias is not surprising in view of the division of labor evident in the development of human societies. Prehistoric males became specialized as group-hunters, while the females concentrated on food-gathering and childbearing. This difference contributed to a human male “pack mentality” that is far less marked in females. Wolves, the wild ancestors of domestic dogs, also became pack-hunters, so the modern dog has much more in common with the human male than with the human female. The argument will always go on—feline selfsufficiency and individualism versus canine camaraderie and good fellowship. But it is important to stress that in making a valid point I have caricatured the two positions. In reality there are many people who enjoy equally the company of both cats and dogs. And all of us, or nearly all of us, have both feline and canine elements in our personalities. We have moods when we want to be alone and thoughtful, and other times when we wish to be in the center of a crowded, noisy room. *bipedal: having two feet

Critical Reading Tips 1.

Read each passage thoughtfully and follow the author’s reasoning. Notice attitude, tone, and general style. Details in a passage help you understand how the author wants you to feel or think. Introductions and footnotes are included to provide you with context for the passages.

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2.

You might want to read the questions before you read the passage so you get a sense of what to look for. On the other hand, you might find this approach to be a waste of time. Try both methods when you take the practice test to see which works better for you.


3.

Answer questions based on what is stated or implied in the passage(s). Remember that an answer choice can contain a true statement and still be the wrong answer.

4.

Read all of the choices and make certain the answer you choose is the best among the choices given. Don’t be misled by choices that are only partially correct.

SAMPLE QUESTIONS Following are three sample questions about this passage. In the actual test, reading passages can be followed by as few as five or as many as thirteen questions. You may be asked to interpret specific pieces of information presented in the passage. 1. The politician and athlete mentioned in lines 59–60 are presented primarily as examples of people who (A) (B) (C) (D) (E)

are powerful are intelligent are typical animal-lovers have strong personalities seek status within a group

Explanation Politicians and athletes are mentioned in the part of the passage that argues that cat-lovers, like cats, tend to feel more comfortable when alone and that dog-lovers, like dogs, tend to feel more comfortable in groups. Politicians and athletes are included as examples of dog-lovers, suggesting that they are people who prefer a society that is group-oriented and hierarchical, like that of canines. ❚ (A), (B), and (D) are not correct because the author never discusses canine or feline society in terms of how powerful or intelligent cats and dogs are or how strong their personalities are. ❚ (C) is not correct because, although politicians and athletes may indeed be “typical” dog-owners, the author’s primary point is that all “animal-lovers” are not the same: cat-lovers are generally different from dog-lovers. ❚ (E) is correct because it describes one aspect of both canine society, as mentioned in lines 24–33, and doglovers in general, as discussed in lines 55–63. Correct answer: (E) / Difficulty level: Medium

❚ (D) and (E) may seem attractive because caricature is often used to ridicule or criticize people. However, the passage does not indicate that cat-lovers or dog-lovers are being ridiculed or criticized in any way. ❚ Only (C) fits the meaning of the word as it is used in the passage. Correct answer: (C) / Difficulty level: Medium Some questions require you to analyze information in one part of the passage in terms of information presented elsewhere. 3. The last four sentences in the passage (lines 78–85) provide (A) an example of the argument that has been made earlier (B) a summary of the points made earlier (C) a reason for the generalizations made earlier (D) a modification of the position taken earlier (E) a rebuttal to opposing views referred to earlier

Explanation In lines 50–75, the author describes cat-lovers and doglovers as if they were two distinct groups with no personality traits in common. In lines 78–85, the author admits that such a clear-cut distinction is not really possible. ❚ (A), (B), and (C) are not correct because they are inaccurate descriptions of the conclusion of the passage. ❚ (E) is incorrect because the author does not completely deny the accuracy of statements made earlier in the passage but says only that they were an exaggeration of the true situation. ❚ (D) is correct because it indicates that the author has modified the statements made earlier. Correct answer: (D) / Difficulty level: Medium

Some questions ask you to recognize the meaning of a word as it is used in the context of the passage.

Related Pair of Passages 2. In line 79, “caricatured” means (A) (B) (C) (D) (E)

copied imitated distorted ridiculed criticized

Explanation A caricature is an exaggerated or distorted representation of someone or something. In the final paragraph, the author admits having exaggerated the two positions for the sake of making a point. ❚ Although (B) is one correct definition of “caricatured,” it is an incorrect answer because, like (A), it does not indicate that the author has exaggerated and distorted the characteristics of cat-lovers and dog-lovers.

In Passage 1, the author presents his view of the early years of the silent film industry. In Passage 2, the author draws on her experiences as a mime to generalize about her art. (A mime is a performer who, without speaking, entertains through gesture, facial expression, and movement.)

Passage 1 Talk to those people who first saw films when they were silent, and they will tell you the experience was magic. The silent film had extraordinary powers to Line draw members of an audience into the story, and an 5 equally potent capacity to make their imaginations work. It required the audience to become engaged—to supply voices and sound effects. The audience was the final, creative contributor to the process of making a film. The finest films of the silent era depended on two 10 elements that we can seldom provide today—a large and receptive audience and a well-orchestrated score.


11

15

20

25

30

35

40

45

50

55

60

65

70

12

For the audience, the fusion of picture and live music added up to more than the sum of the respective parts. The one word that sums up the attitude of the silent filmmakers is enthusiasm, conveyed most strongly before formulas took shape and when there was more room for experimentation. This enthusiastic uncertainty often resulted in such accidental discoveries as new camera or editing techniques. Some films experimented with players; the 1915 film Regeneration, for example, by using real gangsters and streetwalkers, provided startling local color. Other films, particularly those of Thomas Ince, provided tragic endings as often as films by other companies supplied happy ones. Unfortunately, the vast majority of silent films survive today in inferior prints that no longer reflect the care that the original technicians put into them. The modern versions of silent films may appear jerky and flickery, but the vast picture palaces did not attract four to six thousand people a night by giving them eye strain. A silent film depended on its visuals; as soon as you degrade those, you lose elements that go far beyond the image on the surface. The acting in silents was often very subtle, very restrained, despite legends to the contrary. Passage 2 Mime opens up a new world to the beholder, but it does so insidiously, not by purposely injecting points of interest in the manner of a tour guide. Audiences are not unlike visitors to a foreign land who discover that the modes, manners, and thought of its inhabitants are not meaningless oddities, but are sensible in context. I remember once when an audience seemed perplexed at what I was doing. At first, I tried to gain a more immediate response by using slight exaggerations. I soon realized that these actions had nothing to do with the audience’s understanding of the character. What I had believed to be a failure of the audience to respond in the manner I expected was, in fact, only their concentration on what I was doing; they were enjoying a gradual awakening—a slow transference of their understanding from their own time and place to one that appeared so unexpectedly before their eyes. This was evidenced by their growing response to succeeding numbers. Mime is an elusive art, as its expression is entirely dependent on the ability of the performer to imagine a character and to re-create that character for each performance. As a mime, I am a physical medium, the instrument upon which the figures of my imagination play their dance of life. The individuals in my audience also have responsibilities—they must be alert collaborators. They cannot sit back, mindlessly complacent, and wait to have their emotions titillated by mesmeric musical sounds or visual rhythms or acrobatic feats, or by words that tell them what to think. Mime is an art that, paradoxically, appeals both to those who respond instinctively to entertainment and to those whose appreciation is more analytical and complex. Between these extremes lie those audiences conditioned to resist any collaboration with what is played before them; and these the mime must seduce despite themselves. There is only one way to attack those reluctant minds—take them unaware! They will be delighted at an unexpected pleasure.


SAMPLE QUESTIONS Following are three sample questions about this pair of passages. In the actual test, paired passages are followed by more than three questions. Some of the questions will focus on one of the two passages, some on the other, and some on both. All of the sample questions below focus on the relationship between the paired passages, so you can see what these kinds of questions are like. 4. Both passages are primarily concerned with the subject of (A) (B) (C) (D) (E)

shocking special effects varied dramatic style visual elements in dramatic performances audience resistance to theatrical performances nostalgia for earlier forms of entertainment

Explanation This question asks you to think about both of the passages. You will have to figure out the main subject or focus of the pair of passages, not simply recognize that the first passage is about silent films and the second is about mime. The discussion of silent films in Passage 1 is most directly concerned with the effectiveness of silent films for audiences of that era. The discussion of a mime’s techniques in Passage 2 is most directly concerned with factors that make a mime performance effective for the audience. Thus, the main subject of both passages is the way that a visual form of entertainment affects an audience. ❚ You can eliminate (A) because shocking special effects are not a main subject of either passage. ❚ Choice (E) is wrong because a tone of mild nostalgia appears only in Passage 1; since this question asks about the pair, the correct choice must be true for both passages. ❚ Although (B), varied dramatic styles (used by film performers and by a mime), is an idea briefly touched upon in both passages, it is not their primary subject. ❚ Even though both authors are making points about the overall role of audiences in the performances, (D) is incorrect because only Passage 2 addresses audience resistance to theatrical performances. ❚ (C) is the correct answer because it refers to performances in a visual art form. Correct answer: (C) / Difficulty level: Medium 5. The incident described in lines 41–52 shows the author of Passage 2 to be similar to the silent filmmakers of Passage 1 in the way she (A) required very few props (B) used subtle technical skills to convey universal truths (C) learned through trial and error (D) combined narration with visual effects (E) earned a loyal audience of followers

Explanation This question focuses on the particular situation that the mime discusses in the second paragraph of Passage 2. The question asks you to think about how this experience of the mime is similar to the experiences of the silent filmmakers described in Passage 1. Lines 41–52 show the mime changing her performance when she realized that something was not working as she expected it to. Passage 1 mentions that silent filmmakers learned through “experimentation” (line 17) and “accidental discoveries” (line 18). So, like the mime, these people learned through trial and error, making (C) the best of the five choices. You can eliminate the other answer choices because they don’t include traits both “described in lines 41–52” and shared with “the silent filmmakers.” ❚ (A) is wrong because neither passage mentions props. ❚ (B) is wrong because Passage 1 doesn’t discuss conveying universal truths. ❚ (D) is wrong because a mime performs without narration. ❚ (E) is wrong because, while Passage 1 describes loyal audiences, lines 41–52 do not. Correct answer: (C) / Difficulty level: Hard 6. What additional information would reduce the apparent similarity between these two art forms? (A) Silent film audiences were also accustomed to vaudeville and theatrical presentations. (B) Silent film could show newsworthy events as well as dramatic entertainment. (C) Dialogue in the form of captions was integrated into silent films. (D) Theaters running silent films gave many musicians steady jobs. (E) Individual characters created for silent films became famous in their own right.

❚ (B) and (E) are wrong because it’s possible that a mime, through gesture and movement, could act out newsworthy events or create characters that become famous in their own right. ❚ (D) describes a side effect of silent films unrelated to their fundamental similarity with mime. Correct answer: (C) / Difficulty level: Hard

Short Passage “A writer’s job is to tell the truth,” said Ernest Hemingway in 1942. No other writer of our time had so fiercely asserted, so pugnaciously defended, Line or so consistently exemplified the writer’s obligation 5 to speak truly. His standard of truth telling remained, moreover, so rigorous that he was ordinarily unwilling to admit secondary evidence picked up from sources other than his own experience. This is not to say that he refused to invent freely. But he always made it a 10 sacrosanct point to invent in terms of what he knew from actually having been there.

SAMPLE QUESTION Following is a sample question about this short passage. In the actual test, short passages are followed by two to four questions. 7. According to the passage, Hemingway aimed to convey through his writing (A) (B) (C) (D) (E)

an undistorted view of reality an appreciation of the creative process a readily accessible aesthetic fundamental philosophical truths strategies for writing effectively

Explanation Explanation This question asks you to do two things. First, based on the information in the passages, figure out an “apparent similarity” between silent films and mime. Second, select an answer choice with information that isn’t found in either passage but that, if true, would make silent films and mime seem less similar. If you think about the art forms discussed in the two passages, you should realize that neither uses speech. This similarity is important. Choice (C), however, adds the information that dialogue between characters was part of silent films. Characters “spoke” to each other even though audiences read captions instead of hearing spoken words. Thus silent films indirectly used speech, making them different from mime, which (as stated in the introduction to the two passages) relies solely on “gesture, facial expression, and movement.” (A), (B), (D), and (E) are wrong because they don’t deal with a fundamental similarity between the two art forms.

The passage discusses Ernest Hemingway’s conviction that writers should only write about what is true or real and should base their writing on their own experience of the world. ❚ The correct answer is (A) because it indicates that Hemingway sought to communicate a truthful view of the world. ❚ (B) is not correct because the passage does not indicate that Hemingway tried to convince readers of the value of the creative process. ❚ (C) and (D) are incorrect because the passage states that Hemingway wanted to convey reality in his writing, not an artistic vision or underlying philosophical principles. ❚ (E) is incorrect because the passage does not state that Hemingway tried to provide particular strategies for good writing. Correct answer: (A) / Difficulty level: Medium

❚ If true, (A) would not “reduce the apparent similarity” between silent films and mime. Taking the SAT I: Reasoning Test

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Mathematics practice test questions

The mathematics sections of the SAT contain three types of questions: ❚ Standard five-choice multiple-choice (35 questions) ❚ Four-choice quantitative comparisons that emphasize the concepts of equalities, inequalities, and estimation (15 questions) ❚ Student-produced response questions that provide no answer choices (10 questions) Some questions are like the questions in your math textbooks. Others ask you to do original thinking and may not be as familiar to you. Become familiar with the directions and the reference formulas on pages 21, 23–24, and 27, which are also in the test book.

Acceptable Calculators Calculators permitted during testing are: ❚ graphing calculators ❚ scientific calculators ❚ four-function calculators If you use a calculator with characters that are 1 inch or higher, or if your calculator has a raised display that might be visible to other test takers, you will be seated at the discretion of the test supervisor. You will not be allowed to share calculators. You will be dismissed and your scores canceled if you use your calculator to share information during the test or to remove test questions or answers from the test room.

Unacceptable Calculators Unacceptable calculators are those that:

CALCULATOR POLICY We recommend that you bring a calculator to use on the math sections of the SAT. Calculator use tends to ensure that you will not miss a question because of computation errors. However, no questions on the test require a calculator.

❚ ❚ ❚ ❚ ❚

use QWERTY (typewriter-like) keypads require an electrical outlet “talk” or make unusual noises use paper tape are electronic writing pads, pen input devices, pocket organizers, cell phones, or hand-held or laptop computers

Calculator Tips 1.

Bring a calculator with you, even if you’re not sure if you will use it. Calculators will not be available at the test center.

2.

If you don’t use a calculator regularly, practice before the test.

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3.

All questions can be answered without a calculator. No questions require complicated or tedious calculations.

4.

Don’t buy an expensive, sophisticated calculator just to take the test. Although you can use them for the test, a basic function calculator is sufficient.


5.

Don’t try to use a calculator on every question. First, decide how you will solve the problem, and then decide whether to use the calculator. The calculator is meant to aid you in problem-solving, not to get in the way.

6.

It may help to do scratchwork in the test book. Get your thoughts down before using your calculator.

7.

Make sure your calculator is in good working order and that batteries are fresh. If your calculator fails during testing, you’ll have to complete the test without it.

8.

Take the practice test in this booklet with a calculator at hand. This will help you determine how much you will probably use a calculator the day of the test.

Addition and Multiplication of Odd and Even Numbers

Arithmetic ❚ ❚ ❚ ❚ ❚

applications involving computation data interpretation (including mean, median, and mode) percent prime numbers ❚ odd and even numbers ratio and proportion ❚ divisibility

Algebra ❚ ❚ ❚ ❚ ❚

❚ ❚ ❚ ❚ ❚ ❚

area and perimeter of a polygon area and circumference of a circle volume of a box, cube, and cylinder Pythagorean Theorem and special properties of isosceles, equilateral, and right triangles 30°-60°-90° and 45°-45°-90° triangles properties of parallel and perpendicular lines coordinate geometry geometric visualization slope similarity

Other ❚ logical reasoning ❚ newly defined symbols based on commonly used operations ❚ probability and counting

ARITHMETIC

Multiplication

even even even

even even even

odd odd even

even odd even

even odd odd

odd odd odd

Percent

factoring ❚ linear equations and positive integer exponents inequalities sequences ❚ quadratic equations algebraic word problems ❚ simplifying algebraic substitution expressions

Geometry ❚ ❚ ❚ ❚

Addition

AND

ALGEBRAIC CONCEPTS

Integers, Odd and Even Numbers, Prime Numbers, Digits ❚ Integers: . . . , 4, 3, 2, 1, 0, 1, 2, 3, 4, . . . ❚ Consecutive Integers: Integers that follow in sequence; for example, 22, 23, 24, 25. Consecutive integers can be more generally represented by n, n 1, n 2, n 3, . . . ❚ Odd Numbers: . . . , 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, . . . ❚ Even Numbers: . . . , 8, 6, 4, 2, 0, 2, 4, 6, 8, . . . (Note: zero is an even number.)

Percent means hundredths or number out of 100. For example, 40 percent means 40 or 0.40 or 2 . 100 5 Percent Less than 100 Problem 1: If the sales tax on a $30 item is $1.80, what is the sales tax rate? Solution: $1.80 n $30 100 n 6, so 6% is the sales tax rate. Percent Greater than 100 Problem 2: What number is 250 percent of 2?

Mathematics Review

GENERAL MATHEMATICS CONCEPTS

Solution: n 250 2 100 n 5, so 5 is the number. Percent Less than 1 Problem 3: 3 is 0.2 percent of what number? Solution: 3 0.2 n 100 n 1,500, so 1,500 is the number. Percent Increase/Decrease Problem 4: If the price of a computer was decreased from $1,000 to $750, by what percent was the price decreased? Solution: The price decrease is $250. The percent decrease is the value of n in the equation 250 n . The value of 1, 000 100 n is 25, so the price was decreased by 25%. Note: n% increase means increase n ; original 100 n decrease n% decrease means 100 . original

❚ Prime Numbers: 2, 3, 5, 7, 11, 13, 17, 19, . . .

(Note: 1 is not a prime and 2 is the only even prime.)

❚ Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9


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Mathematics Review

Average

Solution: In this situation, the average speed is:

An average is a statistic that is used to summarize data. The most common type of average is the arithmetic mean. The average (arithmetic mean) of a list of n numbers is equal to the sum of the numbers divided by n. For example, the mean of 2, 3, 5, 7, and 13 is equal to 2 3 5 7 13 6 5

When the average of a list of n numbers is given, the sum of the numbers can be found. For example, if the average of six numbers is 12, the sum of these six numbers is 12 6, or 72. The median of a list of numbers is the number in the middle when the numbers are ordered from greatest to least or from least to greatest. For example, the median of 3, 8, 2, 6, and 9 is 6 because when the numbers are ordered, 2, 3, 6, 8, 9, the number in the middle is 6. When there is an even number of values, the median is the same as the mean of the two middle numbers. For example, the median of 6, 8, 9, 13, 14, and 16 is 9 13 11 2

The mode of a list of numbers is the number that occurs most often in the list. For example, 7 is the mode of 2, 7, 5, 8, 7, and 12. The numbers 10, 12, 14, 16, and 18 have no mode and the numbers 2, 4, 2, 8, 2, 4, 7, 4, 9, and 11 have two modes, 2 and 4. Note: The mean, median, and mode can each be considered an average. On the SAT, the use of the word average refers to the arithmetic mean and is indicated by “average (arithmetic mean).” The exception is when a question involves average speed (see problem 2 below). Questions involving the median and mode will have those terms stated as part of the question’s text. Weighted Average Problem 1: In a group of 10 students, 7 are 13 years old and 3 are 17 years old. What is the average (arithmetic mean) age of these 10 students? Solution: The solution is not the average of 13 and 17, which is 15. In this case the average is 7(13) 3(17) 91 51 10 14.2 years 10

Total distance Total time

The total distance is 2(70) 5(60) 440 km. The total time is 7 hours. Thus, the average speed was 440 6 62 7 kilometers per hour. 7 Note: In this example the average speed is not the average of the two separate speeds, which would be 65 kilometers per hour.

Properties of Signed Numbers positive positive positive negative negative positive negative positive negative Factoring You may need to apply these types of factoring: x 2 2x x(x 2) x 2 1 (x 1)(x 1) x 2 2x 1 (x 1)(x 1) (x 1)2 x 2 3x 4 (x 4)(x 1) Probability Probability refers to the chance that a specific outcome can occur. It can be found by using the following definition when outcomes are equally likely. Number of ways that a specific outcome can occur Total number of possible outcomes

For example, if a jar contains 13 red marbles and 7 green marbles, the probability that a marble selected from the jar at random will be green is 7 7 or 0.35 7 13 20

If a particular outcome can never occur, its probability is 0. If an outcome is certain to occur, its probability is 1. In general, if p is the probability that a specific outcome will occur, values of p fall in the range 0 p 1. Probability may be expressed as either a decimal or a fraction.

The expression “weighted average” comes from the fact

Geometric Figures

that 13 gets a weight factor of 7, whereas 17 gets a

Figures that accompany problems are intended to provide information useful in solving the problems. They are drawn as accurately as possible EXCEPT when it is stated in a particular problem that the figure is not drawn to scale. In general, even when figures are not drawn to scale, the relative positions of points and angles may be assumed to be in the order shown. Also, line segments that extend through points and appear to lie on the same line may be assumed to be on the same line. The text “Note: Figure not drawn to scale.” is

weight factor of 3. Average Speed Problem 2: Jane traveled for 2 hours at a rate of 70 kilometers per hour and for 5 hours at a rate of 60 kilometers per hour. What was her average speed for the 7-hour period?

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Mathematics Review

included with the figure when degree measures may not be accurately shown and specific lengths may not be drawn proportionally. The following examples illustrate what information can and cannot be assumed from figures.

Geometric Skills and Concepts Properties of Parallel Lines 1.

Example 1:

If two parallel lines are cut by a third line, the alternate interior angles are congruent.

Example:

Since AD and BE are line segments, angles ACB and DCE are vertical angles. Therefore, you can conclude that x y. Even though the figure is drawn to scale, you should NOT make any other assumptions without additional information. For example, you should NOT assume that AC CD or that the angle at vertex E is a right angle even though they might look that way in the figure.

2.

If two parallel lines are cut by a third line, the corresponding angles are congruent.

Example:

Example 2:

Note: Words like “alternate interior” or “corresponding” are generally not used on the test, but you do need to know which angles involving parallel lines are congruent.

3. A question may refer to a triangle such as ABC above. Although the note indicates that the figure is not drawn to scale, you may assume that: ❚ ABD and DBC are triangles. ❚ D is between A and C. ❚ A, D, and C are points on a line. ❚ The length of AD is less than the length of AC. ❚ The measure of angle ABD is less than the measure of angle ABC. You may not assume the following: ❚ The length of AD is less than the length of DC. ❚ The measures of angles BAD and BDA are equal. ❚ The measure of angle ABD is greater than the measure of angle DBC. ❚ Angle ABC is a right angle.

If two parallel lines are cut by a transversal, the sum of the measures of the interior angles on the same side of the transversal is 180°.

If AB is parallel to CD, then x y 180. (Because x z 180 and y z.)


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Mathematics Review

Angle Relationships 1.

5.

The sum of the measures of the interior angles of a triangle is 180°.

Example:

The sum of the measures of the interior angles of a polygon can be found by drawing all diagonals of the polygon from one vertex and multiplying the number of triangles formed by 180°.

Example:

x 70 (Because 60 50 x 180.) 2.

Since the polygon is divided into 3 triangles, the sum of the angles is 3 180° or 540°.

When two lines intersect, vertical angles are congruent.

In the SAT, the term “polygon” will be used to mean a convex polygon, that is, a polygon in which each interior angle has a measure of less than 180°.

Example:

Side Relationships 1.

xy 3.

A straight angle measures 180°.

Pythagorean Theorem: In any right triangle, a2 b2 c2, where c is the length of the longest side and a and b are the lengths of the two shorter sides.

Example:

x5 (By the Pythagorean Theorem, x 2 32 42 x 2 9 16 x 2 25 25 5) x

Example:

y 30 (Because y 150 180.) 4.

The sum of the two acute angles in a right triangle is 90°.

Example:

2.

In any equilateral triangle, all sides are congruent and all angles are congruent. Example:

x y 10 (Because the measure of the unmarked angle is 60°, the measure of all angles of the triangle are equal, and, therefore, the lengths of all sides of the triangle are equal.) x 10 (Because 4x 5x 90.)

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A polygon is “regular” if all sides are congruent and all angles are congruent.

In an isosceles triangle, the angles opposite congruent sides are congruent. Also, the sides opposite congruent angles are congruent.

Example:

Area and Perimeter Rectangles Area of a rectangle length width l w Perimeter of a rectangle 2(l w) 2l 2w Examples:

ab 4.

xy

In any triangle, the longest side is opposite the largest angle (and the shortest side is opposite the smallest angle).

Example:

Area 12 Perimeter 14

Area (x 3)(x 3) x 2 9 Perimeter 2(x 3) 2(x 3) 2x 6 2x 6 4x

Circles Area of a circle πr 2 (where r is the radius) Circumference of a circle 2πr πd (where d is the diameter) abc

5.

Mathematics Review

3.

Examples:

Two polygons are similar if the lengths of their corresponding sides are in the same ratio and the measures of their corresponding angles are equal.

Example:

Area π(32) 9π Area π(82) 64π Circumference 2π(3) Circumference π(16) 16π 6π

If polygons ABCDEF and GHIJKL are similar, and if AF and GL are corresponding sides, then 10 2 AF 5 1 GL

and

2 BC 1 . Therefore, HI 9 HI


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Mathematics Review

Triangles Area of a triangle 1 (base altitude) 2 Examples:

Area 1 8 6 24 2

Area 1 10 6 30 2

Area 1 5 12 30 2 Perimeter 12 5 13 30

Coordinate Geometry 1. In questions that involve the xand y-axes, x-values to the right of the y-axis are positive and x-values to the left of the y-axis are negative. Similarly, y-values above the x-axis are positive and y-values below the x-axis are negative. In an ordered pair (x, y), the x-coordinate is written first. For example, in the pair (2, 3), the x-coordinate is 2 and the y-coordinate is 3. 2. Slope of a line vertical change horizontal change

or

Examples:

Volume Volume of a rectangular solid (or cube) length width height l w h Examples:

Slope of PQ 4 2 2 1 − ( −2) 3 Slope of LM 4 −2 − 2

Volume of a cylinder πr 2h (where r is the radius of the base and h is the height) Example:

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A line that slopes upward as you go from left to right has a positive slope. A line that slopes downward as you go from left to right has a negative slope. A horizontal line has a slope of zero. The slope of a vertical line is undefined. Parallel lines have the same slope. The product of the slopes of two perpendicular lines is 1, provided the slope of each of the lines is defined.

Standard Multiple Choice The questions that follow will give you an idea of the type of mathematical thinking required to solve problems on the SAT. First, try to answer each question yourself and then read the solutions that follow. These solutions may give you new insights into solving the problems or

point out techniques you’ll be able to use again. Most problems can be solved in a variety of ways, so don’t be concerned if your method is different from the one given. Note that the directions indicate that you are to select the best of the choices given.

Directions: In this section solve each problem, using any available space on the page for scratchwork. Then decide which is the best of the choices given and fill in the corresponding oval on the answer sheet. Notes: 1. The use of a calculator is permitted. All numbers used are real numbers. 2. Figures that accompany problems in this test are intended to provide information useful in solving the problems. They are drawn as accurately as possible EXCEPT when it is stated in a specific problem that the figure is not drawn to scale. All figures lie in a plane unless otherwise indicated.

1. Al, Sonja, and Carol are to divide n coins among themselves. If Sonja receives twice as many coins as Carol and if Al receives twice as many coins as Sonja, then how many coins does Carol receive, in terms of n? n n n n n (A) (B) (C) (D) (E) 2 3 4 6 7

Solution ❚ Suppose Carol received x coins. Then Sonja received 2x coins and Al received 2(2x), or 4x, coins. ❚ Since the three values add up to n, then x 2x 4x 7x n. ❚ Therefore, x n 7 Correct answer: (E) / Difficulty level: Easy

2. If the perimeter of ∆PQR above is 12, what is the perimeter of ∆PQS ? (A) 15

(B) 19

(C) 20

(D) 21

(E) 22

Solution ❚ Although the figure is not drawn to scale, it does convey information: PQ 4, PS 7, and ∠RPS has the same measure as ∠RSP . ❚ From this information you can conclude that PR RS, and this fact can be recorded on the figure. ❚ One way to do this is by assigning the same variable to both lengths. Assign a different variable for the length of QR. Taking the SAT I: Reasoning Test

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Questions 3–4 refer to the following chart, which shows the professional preferences of 1,000 college students at the beginning and end of their freshman year. In this chart, the number 10 in the 3rd row, 4th column, gives the number of freshmen who had at the beginning of the year preferred medicine and at the end of the year had changed their preference to physical science. Professional Preferences of 1,000 College Students Preference at End of Freshman Year EngiPhysical Social neering Law Medicine Science Science

Correct answer: (B) / Difficulty level: Medium

Engineering

190

10

15

25

10

250

Law

0

80

5

0

15

100

Medicine

5

20

100

10

15

150

Physical Science

20

5

20

250

5

300

Social Science

10

25

20

15

130

200

Totals at End of Year

225

140

160

300

175

1,000

Preference at Beginning of Freshman Year

❚ Since the perimeter of ∆PQR is 12, then 4 k m 12, therefore k m 8. ❚ The perimeter of ∆PQS is 4 (k m) 7, which is equal to 4 8 7 19.

Totals at Beginning

3. How many of the students who preferred law at the beginning of the year changed their preference by the end of the year? (A) None

(B) 20

(C) 40

(D) 60

(E) 80

4. How many students had the same preference at the end of the year as at the beginning of the year? (A) 750

(B) 775

(C) 875

(D) 900

(E) 920

General Math Question Tips 1.

Read the problem carefully. Note key words that tell you what the problem calls for.

2.

Determine what you must do first: compute the answer and then select the answer choice, or look at the answers before you compute.

22

3.

Decide when to use a calculator.

4.

Make sure your answer is a reasonable answer to the question asked.


5.

With some problems, it may be useful to draw a sketch or diagram of the given information.

6.

The test does not require you to memorize formulas. Commonly used formulas are provided in the test book at the beginning of each mathematics section.

7.

Use the test book for scratchwork. You are not expected to do all the reasoning and figuring in your head.

8.

Geometric figures are drawn to scale unless otherwise indicated.

Question 3 Solution ❚ At the beginning of the year, 100 students preferred law. ❚ Of these, 5 students changed their preference to medicine and 15 changed their preference to social science. ❚ A total of 20 students changed their preference from law to another profession. Correct answer: (B) / Difficulty level: Easy

Question 4 Solution ❚ The students who had the same preference at the beginning and end of the year are those who preferred engineering at the beginning and end of the year, those who preferred law at the beginning and end of the year, etc. ❚ These numbers can be found in the cells along one diagonal of the chart. ❚ Their sum is 190 80 100 250 130 750. Correct answer: (A) / Difficulty level: Medium 5. In a group of 5 students, scores on a certain test ranged from a low of 60 to a high of 100. If the median score was 75 and the average (arithmetic mean) of the scores was 80, which of the following could be the other two scores? I. 66 and 99 II. 70 and 90 III. 76 and 89 (A) (B) (C) (D) (E)

None I only II only I and III only I, II, and III

6. For positive integer values of n, define n▲ to be the sum of the integers from 1 to n, inclusive. For example, 3▲ 1 2 3 6. If 10▲ 9▲ k▲, and k is a positive integer, what is the value of k? (A) 1

(B) 4

(C) 10

(D) 46

(E) 55

Solution ❚ You are not expected to be familiar with the symbol n▲. The phrase “define n▲ to be. . .” indicates that the definition was made up for this problem. The SAT often includes questions with newly defined operations. ❚ Using the definition, 10▲ 9▲ (1 2 3 . . . 10) (1 2 3 . . . 9) 10. ❚ This tells you that k▲ must equal 10, so you need to find the value of k for which 1 2 . . . k 10. ❚ Since 1 2 3 4 10, you know that 4▲ 10. ❚ Therefore, k 4. Correct answer: (B) / Difficulty level: Hard

Quantitative Comparison

Solution

Quantitative comparison questions emphasize the concepts of equalities, inequalities, and estimation. They generally involve less reading, take less time to answer, and require less computation than regular multiple-choice questions. Be careful not to mark an (E) response when answering the four-choice quantitative comparison questions.

In questions of this type, statements I, II, and III should each be considered independently of the others. You must determine which of those statements could be true.

DIRECTIONS

❚ Since the mean is 80, the sum of all 5 scores is 5 80 400. ❚ Three of the scores are 60, 75, and 100. Therefore, the sum of the other two scores must be 400 235 165. ❚ The numbers in statements I and III have a sum of 165, but those in statement II have a sum of 160. Therefore, the values in II are not possible. You are also given that the median score is 75. Recall that when an odd number of scores (or values) are arranged in order, the median is the middle value. ❚ In statement I, the ordered scores would be 60, 66, 75, 99, and 100. The median is 75, so statement I is possible. ❚ In statement III, the ordered scores would be 60, 75, 76, 89, and 100. In this case, the median is 76, not 75, so the values in statement III are not possible. ❚ Therefore, of the three statements, only statement I could be true.

Each of the following questions consists of two quantities in boxes, one in Column A and one in Column B. You are to compare the two quantities and on the answer sheet fill in oval A if the quantity in Column A is greater; B if the quantity in Column B is greater; C if the two quantities are equal; D if the relationship cannot be determined from the information given. AN E RESPONSE WILL NOT BE SCORED. Notes: 1. In some questions, information is given about one or both of the quantities to be compared. In such cases, the given information is centered above the two columns and is not boxed. 2. In a given question, a symbol that appears in both columns represents the same thing in Column A as it does in Column B. 3. Letters such as x, n, and k stand for real numbers.

Correct answer: (B) / Difficulty level: Hard Taking the SAT I: Reasoning Test

23

EXAMPLES Column A

Column B

52

20

E1

150˚ E2

Answers A

B

C

D

Explanations E

The answer is (C) because 150 x 180, thereby making x 30.

x˚

x

30

A

B

C

D

E

r and s are integers r1

E3

s1

A

B

C

D

E

To solve a quantitative comparison problem, you compare the quantities in the two columns and decide whether one quantity is greater than the other, whether the two quantities are equal, or whether the relationship cannot be determined from the information given. Remember that your answer should be: A B C D

if the quantity in Column A is greater; if the quantity in Column B is greater; if the two quantities are equal; if the relationship cannot be determined from the information given.

Problems are clearly separated. The quantities to be compared are boxed in and always appear on the same line as the number of the problem. Figures and additional information provided for some problems appear above the quantities to be compared.

SAMPLE QUESTIONS

1.

The answer is (A) because 25 is greater than 20.

Column A

Column B

The number of positive divisors of 21

The number of positive divisors of 12

The answer is (D) because not enough information is known about r and s.

you could answer the question without listing all the positive divisors of 12. Correct answer: (B) / Difficulty level: Easy Column A 2.

Column B

3m

5m

Solution ❚ Your first impulse may be to say that 5m is greater, but you should realize that there are no restrictions put on m, so it could be negative, zero, or positive. ❚ If m 0, then 3m 5m. ❚ If m 0, then 3m 5m. (Recall, for instance, that 3 is greater than 5.) ❚ If m 0, then 3m 5m. ❚ If m had been restricted to positive values, the correct answer would have been (B), but since m can be any value, the correct answer is (D). Correct answer: (D) / Difficulty level: Medium Column A

Solution ❚ The positive divisors of 21 are 1, 3, 7, and 21, and the positive divisors of 12 are 1, 2, 3, 4, 6, and 12. ❚ Since 12 has the greater number of positive divisors, the correct answer is (B). Note: The list of divisors for 21 is complete because 3 7 is 21 factored as a product of prime numbers. If you factor 12 as 2 6 and realize that 6 is not prime, and therefore can be broken down into other factors, you will see that 12 has more divisors than 21. In this way

24


3.

4 2 x

Column B x0

4x

Question 3 Solution ❚ Since x is greater than zero, both 4 2x and 4 x are positive, and therefore both square roots are real numbers. 4 x , you should notice that, ❚ To compare 4 2 x with no matter what positive value x has, 4 2x must be greater than 4 x. The square root of a larger number is greater than the square root of a smaller number. 4 x . The correct answer is (A). ❚ Therefore 4 2 x

Note: It is important to recall that, for positive values of m means the positive number which, when m, the symbol 25 5 since 5 is a positive squared, gives m. For example, number and 52 25. Even though (5)2 also equals 25, 5 is 25 . not a positive number and therefore 5 is not equal to 25 ). (5 is equal to

5. P15 R5

2075 2490 26975

Correct answer: (A) / Difficulty level: Easy

P and R represent digits in the correctly solved multiplication problem. Column A

Column B

P

R

Solution

Line m (not shown) goes through the point (5, 6) and is parallel to line l.

4.

Column A

Column B

The slope of m

3 5

Solution ❚ To compare the slope of m with the quantity 3 , you 5 need to find the slope of m. ❚ The fact that m goes through (5,6) is not helpful in finding the slope, but the fact that m is parallel to l is helpful because parallel lines have equal slopes. ❚ Line l goes through (0,0) and (5,3), so its slope is 3 . 5 That means that the slope of m is also 3 , so the 5 correct answer is (C).

❚ To compare the digits P and R, you need to either find the possible values of P and R or find out enough about them to determine which is greater. In this problem it is fairly easy to find the values of P and R. ❚ The multiplication problem includes not only the multipliers and their product, but also the partial products 2075 and 2490. ❚ Recalling the method used to multiply a three-digit number by a two-digit number, you can get two relationships. The first is 5 P15 2075, which is true when P 4. You may find a calculator useful in computing 2075 5. ❚ The second equation is R P15 2490. Since P 4, R 415 2490. Again, the calculator may be useful for dividing 2490 by 415 to get R 6. ❚ Once we know the values of P and R, it is easy to compare them. Since 4 6, the correct answer is (B). Correct answer: (B) / Difficulty level: Medium

Correct answer: (C) / Difficulty level: Medium

Quantitative Comparison Tips 1.

Become familiar with the directions so you do not have to refer to them.

2.

When comparing quantities that involve variables, don’t forget that the variables may represent fractions, negative numbers, and zero.

3.

Sometimes it is not necessary to compute the quantities in the columns in order to compare them.

4.

Do not spend too much time on the quantitative comparison questions (more than 15 minutes) and not have enough time for the student-produced response questions.


25

Column A 6.

The area of the largest circle that could be cut from a rectangular piece of paper 1.5 cm by 1.8 cm

Column B The area of the largest circle that could be cut from a square piece of paper with side 1.5 cm

Solution It is helpful to visualize the largest circle that could be cut from each piece of paper.

❚ There are many ways to sketch this triangle, but no matter how you draw it, angle A will be opposite side BC and angle B will be opposite side AC. Also, since angle A measures 70°, the measures of angles B and C must add up to 110°. ❚ Could BC 8? Yes, if angle B measures 70°, tABC would be isosceles and AC BC 8. But angle B could also have other measures such as 65° or 90°. Recall that in any triangle the longest side is opposite the greatest angle and the shortest side is opposite the smallest angle. ❚ Since angle A could be less than, equal to, or greater than angle B, side BC could be less than, equal to, or greater than side AC, which is 8. Since there is not enough information to tell whether BC is more or less than 8, the correct answer is (D). Correct answer: (D) / Difficulty level: Hard

❚ The largest circle in the Column A rectangle would have diameter 1.5 cm. ❚ The largest circle in the Column B square would touch all four sides and would also have diameter 1.5 cm. ❚ Therefore, the area of the largest circle that can be cut from a 1.5 cm by 1.8 cm rectangle is the same as that cut from a 1.5 cm by 1.5 cm square. The correct answer is (C). Note: It is not necessary to compute the areas in order to compare them. Correct answer: (C) / Difficulty level: Medium Column A

Column B

In tABC, angle A measures 70˚ and AC 8. 7.

BC

Solution It may be helpful to sketch tABC.

26


8

Student-Produced Response Questions of this type have no answer choices provided. Instead, you must solve the problem and fill in your answer on a special grid. Ten questions on the test will be of this type. It is very important for you to understand the directions for entering answers on the grid! You will lose valuable testing time if you read the directions for the first time when you take the test. The directions are fairly simple and the gridding technique is similar to the way other machine-readable information is entered on forms. A primary advantage of this format is that it allows you to enter the form of the answer that you obtain, whether whole number, decimal, or fraction. For example, if you obtain 2 , 5 you can grid 2/5. If you obtain .4, you can grid .4. Generally, you should grid the form of the answer that you obtain naturally in solving the problem. The grid will only hold numbers that range from 0 to 9999. Decimals and fractions can also be gridded. On the next page are the actual directions that you will find on the test—read them carefully.

Answer Grid Tips 1.

Decide in which column you want to begin your answers before the test starts. This strategy saves time. We recommend that you grid in the first (left-hand) column of the grid or that you right-justify your answers (see below).

Left Justify

Right Justify

2.

If the answer is 0, grid it in column 2, 3, or 4. Zero has been omitted from column 1 to encourage you to grid the most accurate values for rounded answers. For 1 example,an answer of 8 could also be gridded as .125 but not as 0.12, which is less accurate.

3.

A fraction does not have to be reduced unless it will not fit the grid.

15

For example, will not 25 fit. You would need to grid

3 or .6 instead. 5

5.

Check your work if your answer does not fit on the grid. If you obtain a negative value, a value greater than 9999, or an irrational number, you have made an error.

4.

Do your best to be certain of your answer before you grid it. If you erase your answer, do so completely. Incomplete erasures may be picked up by the scoring machine as intended answers.


27

EXAMPLES Below are six examples of student-produced response questions. Following each question, you will find a solution and one way to enter the correct answer.

1. In a certain group of 1,000 students, 20 percent have reached the voting age of 18. If 60 percent of those 18 or older are registered to vote, how many students in the entire group are registered to vote?

❚ The numbers 4.5, 13 and 4.95 are all 3 possible answers. The . mixed number equiv1 13 1 alent to is 4 3 3 2 which cannot be grid- 3 ded. If 41/3 is gridded, 4 5 this will be read 6 41 as . 7 3 8 Difficulty level: Easy 9

1 3 / 3 4 . 9 5 .

.

.

0

0

0

1

1

1

2

2

2

3

3

4

.

.

.

.

0

0

0

1

1

1

1

2

2

2

2

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3

3

3

3

4

4

4

4

4

4

5

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5

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9

9

9

9

9

9

9

Solution ❚ Of the entire group, 20 percent have reached the voting age of 18. ❚ Since 20% of 1,000 200, we know that 200 students have reached the voting age of 18. ❚ The question goes on to say that 60 percent of those 18 or older are registered to vote. ❚ We then know that 60% of 200 120 students who are registered to vote. Difficulty level: Easy

120 .

.

.

.

0

0

0

1

1

1

1

2

2

2

2

3

3

3

3

4

4

4

4

5

5

5

5

6

6

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6

7

7

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8

8

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8

9

9

9

9

10 3x 2 13 2. What is one possible value of x that satisfies the inequality above?

3. Points P, Q, R, and T lie on a line, in that order. If PQ 1, QR 3, and QT 8, what is the value PR ?

of

PT

Solution ❚ It is often helpful to sketch a figure of the situation described. In this question, we know the order of the points on the line, so the figure would be something like this:

❚ We are also told three lengths, so that information can be added to the figure, giving:

Solution ❚ One approach to this problem is to simplify the inequality. For any inequality, adding or subtracting the same amount from each side or multiplying or dividing each side by the same positive number yields an equivalent inequality. Adding 2 gives 12 3x 15, and then dividing by 3 gives 4 x 5. ❚ The question asks for one possible value of x. Any number greater than 4 and less than 5 satisfies the inequality. Choose only one correct answer to enter in the grid. The number you choose can be entered in decimal form or in fraction form. ❚ When there is a range of possible correct answers, your gridded response must lie within the range. For this question, although 4.002 is within the range (4 x 5), its rounded value 4.00 is not within the range and would not therefore be considered a correct answer to the problem.

(It is not necessary to redraw the figure to scale.)

❚ From the figure it is clear that PR 1 3 4 and PT 1 8 9. . . . . 4 PR ❚ Therefore, 9. 0 0 0 PT 1 1 1 1 The answer can be correctly 2 2 2 2 entered in the grid as 4/9. 3 3 3 3 4 4 4 4 ❚ It is also possible to convert to 9 5 5 5 5 a decimal by computing (either by 6 6 6 6 7 7 7 7 calculator or hand) 4 9. The 4 decimal equivalent of 9 is .4444 . . . 8 8 8 8 9 9 9 The most accurate form of the decimal that fits in the grid is .444, which would also be counted as correct.

4 / 9

In some styles of writing decimals less than 1, a 4 lead zero is preferred. (In this style, would be 9 written 0.4444. . . .) In recording answers in the grid, however, this style can lead to difficulty because of the limitation of the grid. Using a lead

28


zero, .444 would be entered as 0.4, which is less accurate than .444 as 4 an approximation to and would 9 be counted as an incorrect answer. Note that the zero has been eliminated from the left-most column of the grid to discourage the use of lead zeros. (For some decimals, such as .2, lead zeros are acceptable because they fit into the grid and do not result in less accurate answers.)

. 444 .

.

.

.

0

0

0

1

1

1

1

2

2

2

2

3

3

3

3

4

4

4

4

5

5

5

5

6

6

6

6

7

7

7

7

8

8

8

8

9

9

9

9

Difficulty level: Easy

4. In a certain dormitory, there are 30 rooms available to house 42 students. If each room is occupied by either one or two students, how many rooms are occupied by just one student?

Solution ❚ Each of the 30 rooms is occupied by either one or two students. ❚ After 30 of the 42 students have been assigned one to a room, the remaining 12 students would be assigned to 12 of the rooms, each of which would then be occupied by two students. ❚ This would leave 18 rooms occupied by only one student. Difficulty level: Medium

18 .

Solution ❚ Based on the information given, the probability that a car will 7 turn toward B is . . . . . 12 0 0 0 ❚ If a car arrives at B, the probability 1 1 1 1 3 that it will turn toward D is . 2 2 2 2 7 3 3 3 3 Therefore, the probability 4 4 4 4 that a car will follow the path 5 5 5 5 6 6 6 6 A B D is 7 3 12 7. 7 7 7 7 ❚ This product simplifies to 8 8 8 8 1 3 9 9 9 9 or . You may grid either 4 12 , 3/12, 1/4, or .25 (note that there is not enough room to grid 21/84).

. 25

Note: You cannot grid your answer as a percent because there is no oval for the percent sign. If you grid 25, it will be interpreted as 25, not 25 percent. Instead, grid .25 or 1/4. Another way to solve this problem is to observe that of the 12 cars that arrive at A, 3 eventually turn 3 toward D. Therefore, the required probability is . 12 Difficulty level: Hard

.

.

.

0

0

0

1

1

1

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9

A, B, C, D, E, F, G 6. How many four-letter arrangements, such as DCAF, can be formed using the letters above, if the first letter must be D, one of the other letters in the arrangement must be A, and no letter can be used more than once in an arrangement?

Solution ❚ In a question like this you need to develop a systematic way to account for all possibilities. . . . . ❚ The letters D and A must appear 0 0 in one of the following three ways: 1 1 1 1 DA_ _ , D_A_ , or D_ _ A. 2 2 2 2 3 3 3 3 ❚ In the first case, DA_ _ , there are 4 4 4 4 5 possibilities for the 3rd letter 5 5 5 5 (B, C, E, F, or G) and once that 6 6 6 letter is determined, 4 possibili7 7 7 7 ties remain for the 4th letter. 8 8 8 8 Therefore, there are a total of 9 9 9 9 5 4 20 possibilities when the first two letters are DA_ _. ❚ When A occurs in the 3rd position (D_A_) or the 4th position (D_ _A), there are also 20 possible arrangements in each case. Therefore, there are a total of 3 20 60 different four-letter arrangements that satisfy the given conditions.

60

5. A traffic survey shows that of every 12 cars that arrive at point A above, 7 turn toward point B and 5 turn toward point C. Of every 7 cars that arrive at point B, 3 turn toward point D and 4 turn toward point E. Based on this information, what is the probability that a car will follow the path A B D?

Difficulty level: Hard


29

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