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ISBN 10: 1–889057–91–6
ISBN 13: 978–1–889057–91–0
SAT is a registered trademark of the College Entrance Examination Board, which was not involved in the
production of, and does not endorse, this book. P. O. Box 692023
West Hollywood, CA 90069
E-mail: [email protected]
Website: ABOUT THIS BOOK
If you don’t have a pencil in your hand, get one now! Don’t just read this book—write on it, study it,
scrutinize it! In short, for the next four weeks, this book should be a part of your life. When you have
finished the book, it should be marked-up, dog-eared, tattered and torn.
Although the SAT is a difficult test, it is a very learnable test. This is not to say that the SAT is “beatable.”
There is no bag of tricks that will show you how to master it overnight. You probably have already realized
this. Some books, nevertheless, offer "inside stuff" or "tricks" which they claim will enable you to beat the
test. These include declaring that answer-choices B, C, or D are more likely to be correct than choices A or
E. This tactic, like most of its type, does not work. It is offered to give the student the feeling that he or she
is getting the scoop on the test.
The SAT cannot be “beaten.” But it can be mastered—through hard work, analytical thought, and by
training yourself to think like a test writer.
The SAT math sections are not easy—nor is this book. To improve your SAT math score, you must be
willing to work; if you study hard and master the techniques in this book, your score will improve—
This book contains 10 full-length SAT Math Tests with detailed solutions to all the problems! These
problems and solutions will introduce you to numerous analytic techniques that will help you immensely,
not only on the SAT but in college as well. For this reason, studying for the SAT can be a rewarding and
All the mathematical properties that you need to know for the SAT math sections are carefully integrated
into the solutions. This way you don't have to spend days or even weeks studying math properties before
practicing with the tests. iii CONTENTS
ORIENTATION 7 THE TESTS 11 Test 1
Section 3 13
56 Test 2
Section 3 75
108 Test 3
Section 3 121
150 Test 4
Section 3 161
189 Test 5
Section 3 201
232 v Test 6
Section 3 247
274 Test 7
Section 3 287
315 Test 8
Section 3 327
356 Test 9
Section 3 369
396 Test 10
Section 3 409
437 SUMMARY OF MATH PROPERTIES 449 vi ORIENTATION
Format of the Math Sections
The Math sections include two types of questions: Multiple-choice and Grid-ins. They are designed to test
your ability to solve problems, not to test your mathematical knowledge. The questions in each sub-section
are listed in ascending order of difficulty. So, if a section begins with 8 multiple-choice questions followed
by 10 grid-ins, then Question 1 will be the easiest multiple-choice question and Question 8 will be the
hardest. Then Question 9 will be the easiest grid-in question and Question 18 will be the hardest. There
will be two 25-minute math sections and one 20-minute section. The sections can appear anywhere in the
54 Total Questions Time
(two 25-minute sections
and one 20-minute section) Level of Difficulty
The mathematical skills tested on the SAT are basic: only first year algebra, geometry (no proofs), and a
few basic concepts from second year algebra. However, this does not mean that the math section is easy.
The medium of basic mathematics is chosen so that everyone taking the test will be on a fairly even playing
field. This way students who are concentrating in math and science don’t have an undue advantage over
students who are concentrating in English and humanities. Although the questions require only basic
mathematics and all have simple solutions, it can require considerable ingenuity to find the simple solution.
If you have taken a course in calculus or another advanced math course, don’t assume that you will find the
math section easy. Other than increasing your mathematical maturity, little you learned in calculus will
help on the SAT.
As mentioned above, every SAT math problem has a simple solution, but finding that simple solution may
not be easy. The intent of the math section is to test how skilled you are at finding the simple solutions.
The premise is that if you spend a lot of time working out long solutions you will not finish as much of the
test as students who spot the short, simple solutions. So, if you find yourself performing long calculations
or applying advanced mathematics—stop. You’re heading in the wrong direction.
Tackle the math problems in the order given, and don’t worry if you fail to reach the last few questions.
It’s better to work accurately than quickly.
You may bring a calculator to the test, but all questions can be answered without using a calculator. Be
careful not to overuse the calculator; it can slow you down. 7 SAT Math Tests Scoring the SAT
The three parts of the test are scored independently. You will receive a reading score, a writing score, and
a math score. Each score ranges from 200 to 800, with a total test score of 600–2400. The average score of
each section is about 500. Thus, the total average score is about 1500.
In addition to the scaled score, you will be assigned a percentile ranking, which gives the percentage of
students with scores below yours. For instance, if you score in the 80th percentile, then you will have
scored better than 80 out of every 100 test takers. The PSAT
The only difference between the SAT and the PSAT is the format and the number of questions (fewer),
except for Algebra II questions, which do not appear. Hence, all the techniques developed in this book
apply just as well to the PSAT. Questions and Answers
When is the SAT given?
The test is administered seven times a year—in October, November, December, January, March, May, and
June—on Saturday mornings. Special arrangements for schedule changes are available.
If I didn’t mail in a registration form, may I still take the test?
On the day of the test, walk-in registration is available, but you must call ETS in advance. You will be
accommodated only if space is available—it usually is.
How important is the SAT and how is it used?
It is crucial! Although colleges may consider other factors, the majority of admission decisions are based
on only two criteria: your SAT score and your GPA.
How many times should I take the SAT?
Most people are better off preparing thoroughly for the test, taking it one time and getting their top score.
You can take the test as often as you like, but some schools will average your scores. You should call the
schools to which you are applying to find out their policy. Then plan your strategy accordingly.
Can I cancel my score?
Yes. To do so, you must notify ETS within 5 days after taking the test.
Where can I get the registration forms?
Most high schools have the forms. You can also get them directly from ETS by writing to:
Scholastic Assessment Test
Educational Testing Service
P.O. Box 6200
Princeton, NJ 08541
Or through the Internet:
8 Orientation Directions and Reference Material
Be sure you understand the directions below so that you do not need to read or interpret them during the
Solve each problem and decide which one of the choices given is best. Fill in the corresponding circle on
your answer sheet. You can use any available space for scratchwork.
3. All numbers used are real numbers.
Figures are drawn as accurately as possible EXCEPT when it is stated that the figure is not drawn to
scale. All figures lie in a plane unless otherwise indicated.
Unless otherwise stated, the domain of a function f should be assumed to be the set of all real
numbers x for which f(x) is real number. Note 1 indicates that complex numbers, i = 1 , do not appear on the test.
Note 2 indicates that figures are drawn accurately. Hence, you can check your work and in some cases even
solve a problem by “eyeballing” the figure. If a drawing is labeled “Figure not drawn to scale,” then the
congruent to another object may not be. The statement “All figures lie in a plane unless otherwise
indicated” indicates that two-dimensional figures do not represent three-dimensional objects. That is, the
drawing of a circle is not representing a sphere, and the drawing of a square is not representing a cube.
Note 3 indicates that both the domain and range of a function consist of real numbers, not complex
numbers. It also indicates that a function is defined only on its domain. This allows us to avoid stating the
, we do not need to
domain each time a function is presented. For example, in the function f (x) =
state that the 4 is not part of the domain since f (4) =
is not a
= is undefined. The expression
real number; it does not even exist.
r l h
h w b
2 A = r
C = 2r A = lw A= 1
2 l r
2x c h b w a
2 V = lwh V = r h 2 2 c = a +b 2 x s
Special Right Triangles The number degrees of arc in a circle is 360.
The sum of the measures in degrees of the angles of a triangle is 180.
Although this reference material can be handy, be sure you know it well so that you do not waste time
looking it up during the test. 9 s 2 THE TESTS Test 1 SAT Math Tests Section 1
Time: 25 minutes
1. If n is an odd integer, which one of the following is an even integer?
(E) 2. Define xy by the equation xy = xy – y. Then 23 =
(E) 3. n3
2n + 3
n(n + 3)
18 2 he graph of x = –y + 2 and the graph of the line k intersect at (0, p) and (1, q). Which one of the
following is the smallest possible slope of line k ?
(E) 2 1
2+2 14 Test I Section 1—Questions 4. What is the area of the triangle shown? 5 h
(E) 5. 6
15 When the integer n is divided by 2, the quotient is u and the remainder is 1. When the integer n is
divided by 5, the quotient is v and the remainder is 3. Which one of the following must be true?
(E) 2u + 5v = 4
2u – 5v = 2
4u + 5v = 2
4u – 5v = 2
3u – 5v = 2 15 SAT Math Tests 6. 2 If xy z < 0 , then which one of the following statements must also be true?
(E) 7. None
I and II
II and III Which of the following fractions is the largest in the group?
(E) 8. xz < 0
xyz < 0 10/11
6/7 If a + 3a is 4 less than b + 3b, then a – b =
2 16 Test I Section 1—Questions 9. What is the average of x, 2x, and 6?
(E) 10. What is the ratio of 2 feet to 4 yards?
(E) 11. x/2
(x + 2)/6
(x + 2)/3 1:9
1:5 If x 0,
(E) x (x5 )
x4 2 = x5
x9 17 SAT Math Tests 12.
If x – y = 9, then x y =
(E) 13. ( 3 ) 2 5 3 [ 4 ÷ 2 + 1] =
(E) 14. –4
78 What percent of 25 is 5?
40% 18 Test I Section 1—Questions Questions 15-18 refer to the following graphs.
SALES AND EARNINGS OF CONSOLIDATED CONGLOMERATE
(in millions of dollars) Earnings
(in millions of dollars)
85 86 87 88 89 85 90 86 87 Note: Figure drawn to scale.
15. During which year was the company’s earnings 10 percent of its sales?
(E) 16. 85
90 During the years 1986 through 1988, what were the average earnings per year?
(E) 6 million
27 million 19 88 89 90 SAT Math Tests 17. In which year did sales increase by the greatest percentage over the previous year?
(E) 18. If Consolidated Conglomerate’s earnings are less than or equal to 10 percent of sales during a year,
then the stockholders must take a dividend cut at the end of the year. In how many years did the
stockholders of Consolidated Conglomerate suffer a dividend cut?
(E) 19. 86
Four Scott starts jogging from point X to point Y. A half-hour later his friend Garrett who jogs 1 mile per
hour slower than twice Scott’s rate starts from the same point and follows the same path. If Garrett
overtakes Scott in 2 hours, how many miles will Garrett have covered?
(E) 2 1/5
6 2/3 20 Test I Section 1—Questions 20. In sequence S, the 3rd term is 4, the 2nd term is three times the 1st, and the 3rd term is four times the
2nd. What is the 1st term in sequence S?
4 21 SAT Math Tests Answers and Solutions Setion 1:
B 1. We will use the Substitution Method to solve this problem. Substitution is a very useful technique for
solving SAT math problems. It often reduces hard problems to routine ones. In the substitution method,
we choose numbers that have the properties given in the problem and plug them into the answer-choices.
Now, we are told that n is an odd integer. So choose an odd integer for n, say, 1 and substitute it into each
Now, n3 becomes 13 = 1, which is not an even integer. So eliminate (A).
Next, n/4 = 1/4 is not an even integer—eliminate (B).
Next, 2n + 3 = 2 1 + 3 = 5 is not an even integer—eliminate (C).
Next, n(n + 3) = 1(1 + 3) = 4 is even and hence the answer is possibly (D).
Finally, = 1, which is not even—eliminate (E). The answer is (D). 2. We call this type of problem a Defined Function. Defined functions are very common on the SAT, and
at first most students struggle with them. Yet once you get used to them, defined functions can be some of
the easiest problems on the test. In this type of problem, you will be given a symbol (in this case, ) and a
property that defines the symbol.
From the given definition, we know that xy = xy – y. So, all we have to do is replace x with 2 and y with
3 in the definition:
23 = 2 3 – 3 = 3
Hence, the answer is (B). 22 Test 1 Section 1—Solutions 2 3. Let’s make a rough sketch of the graphs. Expressing x = –y + 2 in standard form yields
2 x = –1y + 0 y + 2
Since a = –1, b = 0, and c = 2, the graph opens to the left and its vertex is at (2, 0). y x Since p and q can be positive or negative, there are four possible positions for line k (the y-coordinates in
the graphs below can be calculated by plugging x = 0 and x =1 into the function x = –y + 2): y y (0, 2) (1, 1) (0, 2)
x x (1, –1) y y (1, 1)
(0, 2 ) (0, 2 ) (1, –1) Since the line in the first graph has the steepest negative slope, it is the smallest possible slope. Calculating
the slope yields m= 2 (1)
0 1 2 +1
1 ( The answer is (A). 23 ) 2 +1 = 2 1 SAT Math Tests 4. Since the triangle is a right triangle, the Pythagorean Theorem applies: h2 + 32 = 52, where h is the height
of the triangle (see summary of the Pythagoran Theorem below). Solving for h yields h = 4. Hence, the
area of the triangle is
( base )( height ) = (3)(4 ) = 6
2 The answer is (A).
> Pythagorean Theorem (For right triangles only): c a c2 = a2 + b 2 b 5. Before we begin solving this problem, let’s review the definition of division:
> “The remainder is r when p is divided by k” means p = kq + r; the integer q is called the quotient.
For instance, “The remainder is 1 when 7 is divided by 3” means 7 = 3 2 + 1. Solution: Translating “When the integer n is divided by 2, the quotient is u and the remainder is 1” into an
n = 2u + 1
Translating “When the integer n is divided by 5, the quotient is v and the remainder is 3” into an equation
n = 5v + 3
Since both expressions equal n, we can set them equal to each other:
2u + 1 = 5v + 3
Rearranging and then combining like terms yields
2u – 5v = 2
The answer is (B). 24 Test 1 Section 1—Solutions 6. Since a number raised to an even exponent is greater than or equal to zero, we know that y2 is positive (it
cannot be zero because the product xy z would then be zero). Hence, we can divide both sides of the
2 inequality xy z < 0 by y2: x y2 z 0
xz < 0
Therefore, I is true, which eliminates (A), (C), and (E). Now, the following illustrates that z < 0 is not
–1 22 3 = –12 < 0
This eliminates (D). Hence, the answer is (B). 7. To solve this problem, note the following strategy:
> To compare two fractions, cross-multiply. The larger number will be on the same side as the
larger fraction. Solution: Cross-multiplying the fractions 9/10 and 10/11 gives 9 11 versus 10 10, which reduces to 99
versus 100. Now, 100 is greater than 99. Hence, 10/11 is greater than 9/10. Continuing in this manner
shows that 10/11 is the largest fraction in the group. Hence, the answer is (A). 8.
> In Algebra, you solve an equation for, say, y by isolating y on one side of the equality symbol. On
the SAT, however, you are often asked to solve for an entire term, say, 3 – y by isolating it on one
side. Solution: Translating the sentence into an equation gives
a + 3a = b + 3b – 4
Combining like terms gives
4a = 4b – 4
Subtracting 4b from both sides gives
4a – 4b = –4
Finally, dividing by 4 gives
a – b = –1
Hence, the answer is (B). 25 SAT Math Tests 9. First, let's review the definition of an average:
> The average of N numbers is their sum divided by N, that is, average = sum
N Solution: By the definition of an average, we get x + 2 x + 6 3 x + 6 3( x + 2 )
Hence, the answer is (D).
10. Before presenting the solution, let's review the concept of a ratio.
A ratio is simply a fraction. The following notations all express the ratio of x to y:
x : y, x ÷ y, or x/y
Writing two numbers as a ratio provides a convenient way to compare their sizes. For example, since 3/ <
1, we know that 3 is less than . A ratio compares two numbers. Just as you cannot compare apples and...
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- Elementary arithmetic, Negative and non-negative numbers, Parity