Taking the SAT® I: Reasoning Test
Introduction Sections
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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.
www.collegeboard.com
™
2
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
4
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
3
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.
Taking the SAT I: Reasoning Test
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.
Taking the SAT I: Reasoning Test
5
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.
6
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.
Taking the SAT I: Reasoning Test
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
Taking the SAT I: Reasoning Test
7
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.
8
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.”
Taking the SAT I: Reasoning Test
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
Taking the SAT I: Reasoning Test
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.
10
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.
Taking the SAT I: Reasoning Test
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.
Taking the SAT I: Reasoning Test
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.
Taking the SAT I: Reasoning Test
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
13
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.
14
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.
Taking the SAT I: Reasoning Test
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
Taking the SAT I: Reasoning Test
15
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?
16
Taking the SAT I: Reasoning Test
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.)
Taking the SAT I: Reasoning Test
17
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.)
18
Taking the SAT I: Reasoning Test
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
Taking the SAT I: Reasoning Test
19
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:
20
Taking the SAT I: Reasoning Test
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
21
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.
Taking the SAT I: Reasoning Test
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
Taking the SAT I: Reasoning Test
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.
Taking the SAT I: Reasoning Test
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
Taking the SAT I: Reasoning Test
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.
Taking the SAT I: Reasoning Test
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
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
5
6
6
6
6
6
6
6
7
7
7
7
7
7
7
8
8
8
8
8
8
8
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
6
6
7
7
7
7
8
8
8
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
Taking the SAT I: Reasoning Test
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
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
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
Taking the SAT I: Reasoning Test
29