sat math study guide

8/17/2019 SAT Math Study Guide

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Video Aided Instruction, Inc.Roslyn Heights, New York

1

SAT MathStudy Guide

byPeter Lanzer, M.B.A.

#VAI-S1531v1.0


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2

Writing the PerfectCollege Admission Essay

1 program on 2 DVDs

This study guide should be used along with the follow-ing program published by Video Aided Instruction. Theinstructor works through the exercises found in thisguide – and much, much more – during the course of theactual program.

SAT Math6 DVDs · 11 hrs. 20 mins.item #VAI-1531 · price $159.95isbn 1-57385-153-1 · upc 600459153199

Copyright © 2005 Video Aided Instruction, Inc.

To all users of the Video Aided Instruction publication

named above, permission is hereby granted to createone (1) paper imprint of this document per person. Inaddition, teachers and librarians may reproduce paperimprints of this document in quantities not to exceedone hundred (100) imprints annually. All imprints andreproductions made must retain all copyright noticescontained herein. This authorization is granted for thisspecific document only, and only when the document isused along with the publication named above.

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other than as an aid during self-study or group instruc-tion along with the aforementioned publication.

Other Video Aided Instruction titles you might enjoy:

Complete SAT Series3 programs on 13 DVDs

SAT Math Study Guide

Introduction

Many of the questions on the Math sections of theSAT are genuine brain teasers that require sophisti-cated problem-solving skills. To succeed, you mustcombine your knowledge of number operations,algebra, geometry, data analysis, statistics, and prob-ability – with a good measure of “outside the box”thinking that you won’t learn in math class.

Luckily, this unique program can help you excel! Yourinstructor’s warm, conversational delivery and keeninsight will help you make sense of even the most

convoluted problems. You’ll learn to answer all typesof SAT Math questions – from multiple-choice togrid-ins – quickly, easily, and accurately. After view-ing this program, you’ll be able to make the most ofyour time and earn every point you can.

For easier studying and maximum success, we recom-mend that you view the program over a number ofshort sessions: don’t try to absorb too much at onetime. Review the entire program, or specific sections,as many times as you find necessary in order to mas-ter the material.

Perhaps most importantly: don’t forget to take ad-vantage of your “pause” button while viewing theprogram. This will allow you to try the problems inthis study guide before you view the solutions on thescreen. Keep plenty of scrap paper handy so you can jot down ideas, work through concepts, and more.And finally, be sure to use the myriad on-screengraphics to take notes for yourself – when you’redone, you’ll have a notebook you can refer back toagain and again.

About Your InstructorPeter Lanzer, M.B.A., is a dynamic educator with over30 years of experience teaching high school and col-lege math. His groundbreaking test prep TV showsappeared regularly on “The Learning Channel.” To-day, through his many popular video programs, Mr.Lanzer shares his expertise, energy, and enthusiasmwith grateful math students around the world.


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SAT Math Study Guid

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Copyright © 2005 Video Aided Instruction, Inc. All Rights Reserved

Notes

1. The use of a calculator is permitted. All numbers used are real numbers.

2. The figures accompanying the problems are intended to provide information useful in solving theproblems. These figures are drawn as accurately as possible EXCEPT when it is stated in the prob-lem that the figure is not drawn to scale. All figures lie in a plane unless otherwise indicated.

3. Unless otherwise specified, the domain of any function f is assumed to be the set of all real num-

bers x for which f ( x ) is a real number.

PART 1 – Format and Structure

Directions: See DVD program.

PART 2 – Number and OperationsDirections: Solve each problem and choose the best answer choice.

1. Which of the following lists the four fractions in ascending order?

(A)3

4

7

8

8

10

14

17, , , (B)

8

10

3

4

7

8

14

17, , , (C)

14

17

8

10

7

8

3

4, , ,

(D)3

4

7

8

14

17

8

10, , , (E)

3

4

8

10

14

17

7

8, , ,

2. In Sommerset High School,1

5

of the students are in the sophomore class, and1

8

of the sophomores

take Mrs. Anderson’s sophomore cooking class. If 23

of the sophomores in this class are males and

there are 10 females in the class, how many students are in Sommerset High School?

(A) 600 (B) 800 (C) 1,000 (D) 1,200 (E) 1,500

Reference Information

c 2 = a2 + b2a

bc

45°

Special Right Triangles

2 x x

60°

30° x 3 s

s

45°

s 2r h

V = r 2h

r

C = 2 r A = r 2

l

h

w

V = lwhb

h

A = bh1

2

l

w

A = lw

The number of degrees of arc in a circle is 360.The number of degrees in a straight angle is 180.The sum of the measures in degrees of the angles of a trangle is 180.


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SAT Math Study Guid

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3. A watch which originally cost D dollars was reduced by 40%. If two weeks later, it was then in-creased by 20%, what was the new price of the watch?

(A) $0.40D (B) $0.60D (C) $0.72D (D) $0.80D (E) $0.82D

4. Andre is 3 years younger than Julio was 4 years ago, and Martin is 2 years older than Andre is to-day. How many years older is Julio than Martin?

(A) 6 (B) 5 (C) 4 (D) 3 (E) 2

5. If A = − −{ }8 4 15 24 0, , , , and B = − −

3 61

31

1

2, , , , , what is the greatest possible value of

a

b

if a is a number in set A and b is a number in set B?

(A) −5 (B) 45 (C) 48 (D) 56 (E) 80

6. If m is the largest negative odd three-digit integer without repeated digits, and n is the largestpositive even three-digit integer without repeated digits, then m + n is

(A) 883 (B) 875 (C) 863 (D) 1 (E) −1

7. On the number line, w


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SAT Math Study Guid

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12. When k is divided by 9, the remainder is 5. What is the remainder when 5k is divided by 9?

(A) 1 (B) 2 (C) 5 (D) 7 (E) 9

13. If x = 5n + 7, and n is a positive integer, then x can NEVER be divisible by which of the following?

(A) 2 (B) 3 (C) 4 (D) 5 (E) 11

14. If x is a positive even integer and y is a positive odd integer, which of the following CANNOT betrue?

I. x + y is divisible by 3

II. 3 x + 2 y is divisible by 4

III. 2 x + 3 y is divisible by 6

(A) I only (B) II only (C) III only (D) II and III only (E) I, II, and III

15. If m is the number of distinct prime divisors of 60, and n is the largest prime divisor of 24, then m − n =

(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

16. If k is the least possible integer wherek

4,

k

36, and

k

54 are all positive integers, and l is the

greatest common factor of 24 and 27, thenk

l is a multiple of

(A) 8 (B) 18 (C) 24 (D) 27 (E) 30

17. Both 15 and 24 are factors of a certain positive integer. Which of the following is NOT necessarilyalso a factor of this integer?

(A) 3 (B) 8 (C) 9 (D) 10 (E) 40

18. What is the greatest possible integer k such that 3k is a factor of 1812?

(A) 12 (B) 18 (C) 24 (D) 32 (E) 36

19. In the repeating decimal 0 142961. , how many of the first 141 digits to the right of the decimalpoint are perfect squares?

(A) 141 (B) 94 (C) 92 (D) 71 (E) 47

20. In a stadium, the first row has 50 seats. Each row after the first row has 2 more seats than the previous row. How many seats are in the 41st row?

(A) 80 (B) 128 (C) 130 (D) 132 (E) 140

21. Which term of the arithmetic sequence 11, 8, 5, … is −97?

(A) the 38th (B) the 37th (C) the 36th (D) the 35th (E) the 20th

22. If the first three terms of an arithmetic sequence are x − y , x , and x + y , then the 29th term must be

(A) x + 30 y (B) x + 29 y (C) x + 28 y (D) x + 27 y (E) x + 26 y


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23. What is the sum of the first 100 terms in the sequence whose nth term is a nnn= − ⋅( )1 2 ?

(A) −2 (B) 0 (C) 52 (D) 98 (E) 100

24. The sum of the first 50 positive even integers is 2,550. What is the sum of the next 50 positive evenintegers?

(A) 5,600 (B) 6,650 (C) 7,100 (D) 7,550 (E) 12,500

25. In a culture there are 128 bacteria. Under conditions favorable to their growth, the number of bacteria doubles every 20 minutes. What is the number of bacteria in the culture at the end of 4 hours and 20 minutes?

(A) 220 (B) 219 (C) 214 (D) 213 (E) 27

26. If −12 and 4 are the first and second terms respectively of a geometric sequence, what is the valueof the sixth term?

(A)4

81 (B)

4

9 (C) −

4

3 (D) −

4

27 (E) −

4

81

27. An auto service center handles 70 cars per day that have either tune-ups, oil changes, or both. If 60have tune-ups and 50 have oil changes, how many have tune-ups but do NOT have oil changes?

(A) 10 (B) 20 (C) 40 (D) 50 (E) 60

28. Of the 24 vice-presidents attending a corporate luncheon,3

4 have children and

2

3 are on a diet.

If at least 2 vice-presidents do not have children and are not on a diet, what is the minimum number of vice-presidents who have children and are on a diet?

(A) 9 (B) 10 (C) 11 (D) 12 (E) 13

29. If a, b, c , d , e, and f are all single, unique, non-negative digits and 573

938× abc def

, then d must be

(A) 5 (B) 4 (C) 3 (D) 2 (E) It cannot be determined from the information given.

• ERRATUM NOTICE •In his reading of problem 29, the instructor misstates that a, b, c , d , e, and f must be positive digits. Please use the correct

wording of the problem as given above instead. We sincerely apologize for any confusion that this error may cause.

30. On Friday, Joey eagerly watched his analog wall clock non-stop as it marked the time from 8:43 AM when he arrived at work until 12:02 PM when he left for lunch. Upon his return at 12:57 PM, he

resumed watching the clock non-stop until 5:12 PM when he went home. Assuming this clock included the conventional 12 hour markings, how many times in total did Joey see the clock’s minute hand cross each of the 12 hour markings on Friday?

(A) 89 (B) 90 (C) 91 (D) 92 (E) 93


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PART 3 – Algebra and Functions

Directions: Solve each problem and choose the best answer choice.

1. If2

34 8 x − = − , what is the value of

2

34 x + ?

(A) −6 (B) −4 (C) 0 (D) 4 (E) 6

2. If x 2 − 3 x − 8 = 0, what is the value of 5 x 2 − 15 x ?

(A) 8 (B) 15 (C) 24 (D) 25 (E) 40

3. If n is an integer and nn n

= −

− +24

1 1( )( ), what is the value of

n

n

−+

1

1?

(A) 2 (B) −3 (C) −9 (D) −12 (E) −15

4. Ifk

k k

−−

−

=

4

3

1

20, which of the following must be true?

I. 12

0k − =

II.k

k

−−

=4

30

III. k = 3

(A) I only (B) II only (C) II and III only (D) I and II only (E) I, II, and III

5. For what value of k isk

k

k

k − =

−−

−1

2

12?

(A) −5

7

(B)2

3

(C)3

4

(D) 1 (E) For no value of k .

6. If 3a + 4b = 16 and a ≠ 0, what is the value of12

32 8

a

b−?

(A) −2 (B) 2 (C) 4 (D) 8 (E) 16

7. Ifa b

a c

+−

= −4

3, what is the numerical value of

2a b c

a b

+ −+

?

(A) −8

3 (B) −

7

3 (C) −

1

4 (D)

1

4 (E)

3

7

8. If 3 x = 4 y + 20 and 2 x = −4 y , what is the value of xy ?

(A) −8 (B) −2 (C) 0 (D) 4 (E) 8

9. If5 3

2r s

− = − and1 1

6r s

+ = , what is the value of8

r ?

(A) 16 (B) 4 (C)1

2 (D)

1

4 (E) −16


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10. If 2 x + y − z = 6 and x + 5 y + z = 9, what is the value of 2 x + 4 y ?

(A) 10 (B) 8 (C) 6 (D) 0 (E) −6

11. If a, b, c , and d are non-zero numbers, and 3b = 5c , 10c = 9d, and a = 2b, what is the value of a in terms of d ?

(A) 6d − 4 (B) 3d (C) d (D)2

3

d (E)d

2

12. One of the factors of ay 2 − 3a + y 2 − 3 is

(A) a − 3 (B) a − 2 (C) a − 1 (D) a + 1 (E) a + 2

13. If a ≠ x and a ≠ 2 x , the expression xy ay

a ax x

−+ −2 22

is equivalent to

(A) y

a (B)

y

a2 (C)

y

a x − 2 (D)

y

a x + 2 (E) −

+ y

a x 2

14. If a2 + b2 = 8 and (a + b)2 = 9, then (a − b)2 = (A) 5 (B) 6 (C) 7 (D) 8 (E) 9

15. If x 2 − y 2 = 135 and x − y = 9, what is the value of x ?

(A) 24 (B) 15 (C) 12 (D) 3 (E) −12

16. Ifc c

c

c

c

c

c

3

2

2 2

1

1

1

1

14

++

+ −

+ +

−−

=( )

, what is the value of c ?

(A)2

3 (B)

1

2 (C) 1 (D) 2 (E) 12

17 If x = −.125, what is the value of1 1

1

8

71 x x

x

−

−?

(A) −8.75 (B) −8 (C) −2 (D) −.125 (E) .125

18. If 2a + 3 is a factor of 6a2 + a − k , what is the value of k ?

(A) −12 (B) −8 (C) 4 (D) 12 (E) 18

19. If 3 is one solution of the equation x 2 + 5 x + k = 29, where k is a constant, what is the other solution?

(A) 8 (B) 5 (C) −8 (D) −12 (E) −24

20. If (3t − a)(3t − b) = 9t 2 + 36t − 28, then b could be

(A) −14 (B) −7 (C) −2 (D) 7 (E) 14

21. For which value of k will (−3)3k be least?

(A) −3 (B) −2 (C) 0 (D) 1 (E) 2


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22. If x = 8, what is the value of x x −( )( )[ ]2 02

327 ?

(A)27

64 (B)

9

16 (C) 2 (D) 8 (E) 16

23. If1

0086 x − =

.. , what is the value of x ?

(A) .075 (B) .0032 (C) .0016 (D) .00048 (E) .00032

24. If 6m · 36m + 3 = 1, what is m?

(A) 3 (B) 0 (C) −1 (D) −2 (E) −3

25. If 41

802 1

1

− +−

−

= x x

, what is the value of x ?

(A) −1 (B) 0 (C) 1 (D) 2 (E) 4

26. If 2 · 2k 2

− 6 = (−2)4k

, then k could be (A) −5 (B) −4 (C) 1 (D) 5 (E) 6

27. If 5− x = 10, what is the value of 53 x ?

(A) 10000 (B) 1000 (C) 100 (D)1

100 (E)

1

1000

28. If 1252

3a

y −

= , in terms of y, what is 125a?

(A) 125 y (B) 25 y (C) 5 y (D) 2 y (E) y

3

29. If mn =1

64, where m and n are integers, which of the following could be a value for m?

I. m = 8

II. m = −2

III. m = −4

(A) I only (B) I and II only (C) II and III only (D) I and III only (E) I , II, and III

30. If 2100 + 814k = 1625k + 316, what is the value of k ?

(A) 1 (B) 2 (C) 3 (D) 4 (E) 1000

31. If m = 2999 − 2997 and n = 2998, then

(A) m n=2

3 (B) m = n (C) m n=

3

2 (D) m = 2n (E) m = 4n

32. If y y − =3 10, what is the value of y y + 3 ?

(A) 10 (B) 16 (C) 40 (D) either 10 or 40 (E) 100


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33.5 3 2 1

2 6

2 6

5 3 2 1

−( ) +( )−

÷ +

+( ) −( ) =

(A) 1 (B) 0 (C) −1 (D) −7

2 (E) −

−−

10 3

4 6

34. If x x

3 23332

72

4= , then x =

(A) 9 (B) 3 2 (C) 3 23 (D) 2 2 (E) 2 23

35. What is the smallest integral value of x which satisfies the inequality x x

4

2

31− < ?

(A) −3 (B) −2 25 (C) −2 (D) 2

(E) It cannot be determined from the information given.

36. If a and b are integers, a + b > −3, and a 0 and abc 0

II. b4 > b5

III. a4 > a2

(A) I only (B) II only (C) III only (D) II and III only (E) I, II, and III

38. If pq = −2 and sq = 1, which of the following must be true?

(A) p k

. ?

(A)0 7−5

(B)0 5−7

(C)0 7−5

(D)0 6−6

(E)0 5−7


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42. In a medical research lab, a certain experiment requires that the temperature, t , can deviate from–15°C by no more than 1°C. Which of the following expresses the possible range in temperature?

(A) t + ≥15 1 (B) t + ≤15 1 (C) t − ≥15 1 (D) t − ≤15 1 (E) t − ≥ −1 15

43. A telecommunications company is hiring 10 new engineers. The annual salary the company is offering, s, ranges between $75,000 and $82,000, and is based on experience. Which of the following inequalities describes all possible values of s?

(A) s − ≤75 000 3 500, , (B) s − ≤3 500 82 000, ,

(C) s − ≤82 000 3 500, , (D) s − ≥3 500 75 000, ,

(E) s − ≤78 500 3 500, ,

44. If y varies directly as x and y = 8 when x =1

3, what is the value of y when x = 2?

(A) 12 (B) 16 (C) 24 (D) 36 (E) 48

45. If y varies directly as the square root of x and y = 10 when x = 25, what is the value of y when x = 100?

(A) 2.5 (B) 10 (C) 12.5 (D) 20 (E) 40

46. If y varies directly as x , which of the following must be true?

I. y

x k = , where k is the constant of variation

II. when x is multiplied by a number, y is multiplied by the same number

III. when x increases, y increases

(A) I only (B) II only (C) I and II only (D) II and III only (E) I, II, and III

47. If y varies inversely as x and y = 8 when x = 6, what is the value of y when x = 12? (A) 2 (B) 4 (C) 6 (D) 14 (E) 16

48. Mr. Hooper drove his car for x hours and 12 minutes. If he averaged 50 mph, in terms of x , howmany miles did he drive?

(A) 50 x + 10 (B) x +12

50 (C) 50 x + 600 (D)

50

12 x + (E)

50100

x +

49. Mrs. Cobb drove her car through a snowstorm, traveling 240 miles at a certain rate of speed. If she

had traveled 15 mph faster, she would have been able to travel 30 miles further in3

4 of the time

that she spent on her trip. In miles per hour, what was Mrs. Cobb’s rate for the trip?

(A) 30 (B) 35 (C) 45 (D) 50 (E) 55

50. Derrick bicycled up a hill at an average speed of 4 mph and returned immediately along the sameroute at an average speed of 12 mph. What is his average speed for the entire trip ?

(A) 4 23

mph (B) 6 mph (C) 6 12 mph (D) 8 mph (E) 9 mph


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51. If 3 gardeners, working together at the same rate, can cut the grass at a golf course in 5 13

hours,what fraction of that job can 2 of the gardeners accomplish in one hour?

(A)1

16 (B)

1

8 (C)

3

16 (D)

1

4 (E)

5

16

52. Amanda, Boyd, and Carlos are bricklayers. Working alone, they would take 3, 5, and 6 days, respec-tively, to finish a wall. If one calls in sick, what is the minimum fractional part of the wall that canbe completed in one day?

(A)2

3 (B)

8

15 (C)

1

2 (D)

11

30 (E)

1

3

53. Two robots can paint a car in 3 minutes and three men can paint the same car in 4 minutes. At thisrate, how many minutes would it take one man and one robot working together to paint the car?

(A) 2 13

(B) 2 56

(C) 4 (D) 5 12

(E) 7

54. Jessica can type a manuscript twice as quickly as Alyson. After working alone for two hours, Jessicagot Alyson to help her and they completed the job two hours later. How many hours would it have

taken Jessica to do the entire job by herself?

(A) 6 (B) 5 (C) 4 (D) 2.5 (E) .5

55. If 3 quarts of milk cost c cents, how many dollars do 10 gallons cost?

(A)c

10 (B)

2

15

c (C) 4c (D) 10c (E) 40c

56. The army provided supplies for 400 soldiers for a 40-day-long exercise in the desert. If only 125soldiers participate, how many days would the supplies last?

(A) 12.5 (B) 68 (C) 125 (D) 128 (E) 165

57. If 150 pencils last 60 students 10 days, at this rate, how many days will 200 pencils last 50 students?

(A) 6 (B) 16 (C) 20 (D) 24 (E) 30

58. In hospital H, on the average, x boxes of diapers are used by each baby weekly. At this rate, howmany babies per week can be supplied if a total of k boxes must last w weeks?

(A)kw

x (B)

kx

w (C)

k

xw (D)

x

kw (E)

w

kx

59. The ratio of Emily’s salary to Jake’s salary is 2 : 3, while the ratio of Jake’s salary to Laura’s is 4 : 5. What is the ratio of Emily’s salary to Laura’s salary?

(A) 1 : 3 (B) 2 : 5 (C) 1 : 2 (D) 8 : 15 (E) 15 : 8

60. A box containing 36 jars of pickles falls, breaking some of the jars. Which of the following CANNOTbe the ratio of broken jars to the ones that were not broken?

(A) 1 : 1 (B) 1 : 2 (C) 1 : 3 (D) 1 : 4 (E) 1 : 5


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61. Eight people decided to rent a fishing boat for an afternoon for a total of x dollars. If three peoplecancelled, how many extra dollars did each person have to pay?

(A) x

84− (B)

3

40

x (C)

x

5 (D)

x

83− (E) 8 x − 3

62. The retail price of a refrigerator represents an 80% mark-up over the wholesale cost. For ColumbusDay, the refrigerator went on sale at 33 1

3

% off the retail price. The profit is what percent of thesale price?

(A) 15% (B) 16 23% (C) 20% (D) 33 1

3% (E) 46 2

3%

63. The buyer for a chain of 1,000 office supply stores purchased 12,000 fax machines at a cost of $240each. She plans to run a 20% off President’s Day sale. During the sale, she needs to make a profit of40% on the cost of each fax machine sold. At what price should she mark each fax machine beforethe sale?

(A) $384 (B) $400 (C) $420 (D) $438.60 (E) $450

64. Mr. Hitchcock bought 20 exotic birds, all at the same price, for a total cost of N dollars. Two of the

birds died, and he sold the rest at 50% more per bird than its cost. His total profit in terms of N was

(A)7

20

N (B)

3

40

N (C)

27

20

N (D)

N +103

(E)20

3N

65. If x , y , and z are positive numbers, 70 percent of x is 120 percent of y , and 5 percent of y is3 1

2 percent of z , what percent of z is 2 x ?

(A) 480 (B) 360 (C) 350 (D) 280 (E) 240

66. A drawer contains white, green, and black socks. The number of white socks exceeds the numberof green socks by 3, and the number of black socks exceeds twice the number of green socks by 10.

If the drawer contains w white socks, then what percent of all the socks are white?

(A)100

3 1w +% (B)

100

4 1

w

w +% (C)

100

3 1

w

w +% (D)

100

4 1w +% (E)

25

w %

67. Jack and Jill are standing on line to buy tickets to a rock concert. The ratio of the number of peopleahead of them to the number of people behind them is 2 : 3. After five people ahead of them de-cide to leave the line and nine new people join the line behind them, the ratio becomes 3 :5. Whatis the total number of people that are on line now?

(A) 268 (B) 266 (C) 264 (D) 262 (E) 260

68. A plumber and his assistant are working together on a job. The plumber’s daily wage exceeds twice the assistant’s daily wage by $80. If they complete the job in 3 days and earn at least $1,680in total, which of the following could be the assistant’s daily wage?

(A) 120 (B) 130 (C) 140 (D) 150 (E) 170

69. If f x x x

x x

( ) = −

+

2

2 14, then which of the following is closest to the value of f (1000)?

(A) 0 (B) 0.04 (C) 0.25 (D) 0.9 (E) 250


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70. If f ( x ) = − x 3 + 3 x + h and f (0) = −2, then what is the value of f (−2)?

(A) −16 (B) −7 (C) 0 (D) 2 (E) 7

71. If f ( x ) = 2 x and g( x ) = x + 3, then which of the following is equal to f (g( x )) − f ( x )?

(A) 7f ( x ) (B) 6f ( x ) (C) 5f ( x ) (D) 4f ( x ) (E) 2f ( x )

72. If g x x

x ( ) = −

−1

4, h( x ) = x − 1 , and k ( x ) = x, which of the following is NOT in the domain

of g(h(k ( x )))?

(A) 2 (B) 3 (C) 4 (D) 5 (E) 22

73. Which of the following represents the domain of the function p x x x ( ) = − + +2 20 ?

(A) x x x or> < −{ }5 4 (B) x x x or≥ ≤{ }5 4

(C) x x x or≥ ≤ −{ }5 4 (D) x x − ≤ ≤ −{ }5 4

(E) x x − ≤ ≤{ }4 5

74. If y = f ( x ), then what is the range of f x x x ( ) = − ?

(A) −1


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y

x

(−12, −4) (6, −4)

y = f ( x )

O

77. Given the graph y = f ( x ) as shown above, which of the following represents the graph of y = f ( x + 5) + 7?

y

x (−5, 1) (13, 1)O

y

x

(−7, −11) (11, −11)

O

y

x (−19, 1) (−1, 1)O

(A) (B) (C)

y

x (−17, 3) (1, 3)

O

y

x

(−5, −9) (13, −9)

O (D) (E)

78. An object is projected straight up with an initial velocity of k ft /sec. The height of the object, h, atany time, t , is given by the function h(t ) = −16t 2 + kt . If the object returns to the ground at t = 8, infeet, what was the height of the object at t = 7?

(A) 96 (B) 108 (C) 112 (D) 128 (E) 132

79. The operation is defined by the equation a b = ab + 1. If (a b) c = a (b c ), then c =

(A) a (B) b (C) 2a (D) 2b (E) ab

80. For all real numbers x and y , x y = x 2 + 2 y . What is the value of x 2 2 y ?

(A) x 2 + 8 y (B) x 4 + 8 y (C) x 4 + 4 y (D) 2 x 2 + 8 y 2 (E) x 4 + 4 y 2


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PART 4 – Geometry and Measurement


A

C

30°

y °

B

F

E

DG

1. In the figure above, AB and CD intersect at E . If CD bisects AEF , what is the value of y ?

(A) 10 (B) 15 (C) 25 (D) 75 (E) 120

40° p° q°

2. In the figure above, if p > q > 0, which of the following CANNOT be the value of 2 p?

(A) 138 (B) 142 (C) 144 (D) 146 (E) 147

110°

x °

m

l

R Q

P

3. In the figure above, l || m and PR = RQ. What is the value of x ?

(A) 140 (B) 120 (C) 110 (D) 70 (E) 40

150°

z °20°

m

l

Note: Figure not drawn to scale.

4. In the figure above, if l || m, what is the value of z ?

(A) 100 (B) 90 (C) 50 (D) 30 (E) 20


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6. In the figure above, what is the value of a + g + h + f ?


a° b°

c °g°

d °h°

e° f °

a°

b° A C

B

D

E


7. If AD ⊥ BC and BE ⊥ AC , which of the following statements about a and b in the figure above must be true?

I. a and b are supplementary II. a = b III. a = 120°

(A) I only (B) II only (C) I and II only (D) II and III only (E) I and III only


8. In the figure above, l1 || l2 and l3 || l4. If 4 x + 5 y = 180, what is the value of y ?

(A) 90 (B) 22.5 (C) 18 (D) 16 (E) It cannot be determined from the information given.

y

x �

��

� �

a° b°

c ° d °


5. In the figure above, a + b = 130. What is the value of c + d ? (A) 50 (B) 130 (C) 180 (D) 230 (E) 310


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10. In parallelogram ABCD, AB = 11 and BC = 13. All of the following are possible lengths of BD EXCEPT

(A) 3 (B) 21 (C) 22 (D) 23 (E) 24

11. The lengths of two sides of a triangle are x + 1 and x − 1 where x > 1. Which of the following rangesincludes all possible values of y , the third side?

(A) 0


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17. In the figure above, if a > 0, what is the value of b?

(A)2

2 (B) 2 (C) 4 2 (D) 8

(E) It cannot be determined from the information given.

2 x ° x °

ba2 x °

2a

18. In the figure above, what is the value of x ?

(A) 2 (B) 2 3 (C) 3 2 (D) 6 (E) 6 2

60° 45°

x 2 3


19. In the figure above, what is the value of x ?


6 3

x

4 3


16. In the figure above, what is the value of y ?

(A) 3 (B)3 5

2 (C) 2 3 (D) 3 2 (E) 6

3 x ° x °

y 2 x °

3


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22. In the figure above, a portion of a rectangle whose width is 6 and length is 8 has been shaded. The vertical sides of the shaded region are parallel to the vertical sides of the rectangle, and the

horizontal sides of the shaded region are parallel to the horizontal sides of the rectangle. What is the perimeter of the shaded region?


8

6

23. While doing construction work on his house, Mr. Abel increased the length of his rectangular driveway by 50% and decreased the width of his driveway by 50%. Then the area of his driveway

(A) decreased by 50% (B) increased by 50% (C) decreased by 25% (D) increased by 25% (E) remained the same


20. In the figure above, the perimeter of ACD is closest to which of the following?

(A) 43 (B) 37 (C) 35 (D) 33 (E) 29

17

5

5

B C

A D


21. In the figure above, if the perimeter of DEF is 36, what is the value of q?


y °

y °

x ° x °

123

q

4

B

E

C A F D


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26. In the figure above, ABCD is a parallelogram. What is the value of h?

(A) 4.6 (B) 4.8 (C) 5.0 (D) 5.2 (E) 5.5

6 8

E

h

A D

B C

24. In the figure above, ABDE is a square with perimeter 68, BCDF is a rectangle, and BF = 8. What isthe area of pentagon ABCDE ?


8

A

B C

D

E

F


25. In the figure above, a large rectangle ABCD with area 117 is divided into a square with area a2 andthree smaller rectangles. If the areas of two of the smaller rectangles are 5 a and 9a as shown, whatis the value of a?

(A) 2.7 (B) 2.5 (C) 3.0 (D) 4.0 (E) 5.9

5a

9a

A B

D C

a2

4a

b

z °

27. In the figure above, 90


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28. In the figure above, if ADEF is a square, AB = 20, and AC = 5, what is the area of the entire shadedportion?

(A) 4 (B) 16 (C) 32 (D) 34 (E) 84

C F A

B

D E


29. In the figure above, chord AB equals the radius of the circle with center O. What is the measure ofarc ADB?

(A) 30 (B) 60 (C) 80 (D) 90 (E) 120

O

BD

A

C

30. A jogger is running on a circular track with a radius of 280 feet at the rate of 880 ft/min. To thenearest minute, how many minutes will it take her to go around the track 3 times?

(A) 2 (B) 3 (C) 6 (D) 7 (E) 12

31. A sporting goods store sells two sizes of circular exercise mats – one with a circumference of 12 ftand another with a circumference of 18 ft. To the nearest inch, by how much does the radius of thelarger exercise mat exceed the radius of the smaller exercise mat?

(A) 1 (B) 9 (C) 10 (D) 11 (E) 12


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R

15

P

Q

O


32. In the figure above, QR is tangent to a circle with center O. If the circumference of the circle is 16π , what is the length of QP ?

(A) 8 (B) 9 (C) 10 (D) 11 (E) 12

33. In the figure above, RQ is an arc of a circle whose center is P and RP is an arc of a circle whose cen-ter is Q. If PQ = 12, in terms of π , what is the length of arc RQ?

(A) 12π (B) 8π (C) 6π (D) 4π (E) π

Q

R

12P


34. The figure above consists of two circles that have the same center. The area of the outer circle is 9times the area of the inner circle. If the area of the shaded region is 32 π , what is x , the width (orthickness) of the shaded region?

(A) 7 (B) 6 (C) 5 (D) 4 (E) 3

x


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35. In the figure above, the large semicircle has center O and diameter AD. Three smaller circles withequal radii have diameters AB, BC , and CD respectively. If AD = 12, in terms of π , what is the area ofthe shaded region?

(A) 4π (B) 9π (C) 12π (D) 16π (E) 18π

O C DB A


36. In the figure above, point O is the center of the circle. If tangents AC and AB are perpendicular toeach other, in terms of π , what is the area of the shaded region?

(A) 64 − 18π (B) 64 − 16π (C) 64 − 8π (D) 64 − 4π (E) 64 − π

O

B

8

A

C

37. In the figure above, square ABCD is circumscribed by a circle with center O and diameter 16 2. Points W , X , Y , and Z are the midpoints of the sides of square ABCD. In terms of π , what is the area

of the shaded region?

(A) 64 (π − 1) (B) 72 (π − 1) (C) 96 (π − 1) (D) 108 (π − 1) (E) 128 (π − 1)

O

B X

W Y

C

A Z D


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40. All of the following points lie on the same circle with center (4, 1) EXCEPT

(A) (9, 1) (B) (0, 5) (C) (7, 5) (D) (−1, 1) (E) (4, −4)

38. In the figure above, square ABCD is circumscribed by a circle with center O. If the area of the circleis 36π , In terms of π , what is the area of the shaded region?

(A) 9π (B) 12π (C) 18π (D) 27π (E) 30π

O

B C

A D

(−4, 2)

y

x

S

T

O

−3

39. In the figure above, what are the coordinates of the point on segment ST that is equally distantfrom S and T ?

(A) (−2 12,−1) (B) (−2,−11

2) (C) (−2,−1) (D) (−2, − 1

2) (E) (−11

2,−2)

Note: Figure not drawn to scale. 41. In the figure above, EG and FG are parallel to the x -axis and y -axis respectively. EG is twice as long

as FG. What is the length of EF ?

(A) 5 2 (B) 3 3 (C) 3 2 (D) 2 5 (E) It cannot be determined from the information given.

y

x E (−1, b) G

F (3, d )

O


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46. In right triangle ABC , C is a right angle. If the coordinates are A (0, 2), B (1, 5), and C (2,t ), then t could be

(A) 1 (B) 2 (C) 2.5 (D) 4 (E) 6

45. In the xy plane, what is the area of the triangle bounded by the lines y = 2 x − 6, y = 2, and x = 0?

(A) 14 (B) 15 (C) 16 (D) 17 (E) 18

42. In the figure above, points A, B, and C are collinear and are located at the coordinates shown.What is the equation of the line that passes through point B and is parallel to the y -axis?

(A) x = 3 (B) y = 3 (C) x = 1 (D) y = 4 (E) x = 4

y

x (0, 1)

(6, 4)(k , 3)

O

A

BC

47. An open box is constructed by starting with a rectangular piece of cardboard 8" by 7", cutting asquare with side 2" from each corner and folding up the sides. How many 1" cubes will it take to fillthe box?

(A) 12 (B) 18 (C) 24 (D) 28 (E) 112


43. The figure above shows line segment in the xy coordinate plane. If h > 0 and ’ (not shown) is theimage of under a reflection in the y -axis, then what is the equation of ’?

(A) y x h= +1

2 (B) y x h= −

1

2 (C) y x h= − +

1

2 (D) y x h= − −

1

2 (E) y = 2 x

x

y

(0, h)

(2h, 0)O

44 If p and q are constants, and the solution set of the system of equations px + y − 4 = 0 and 6 x − 2 y − q = 0 contains an infinite set of ordered pairs, then which of the following must be true?

(A) p q= =13

8and (B) p = −3 and q = −8 (C) p = −3 and q = 8

(D) p q= = −1

38and (E) p q= − =

1

38and


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50. A glass company packages each of its drinking glasses in a rectangular box that measures 4" long,4" wide, and 6" high. A number of these boxes are then stacked, as many as will fit, into a cubethat measures 1' on each edge. In square inches, what is the difference between the sum of thesurface areas of this number of boxes before they are stacked and the surface area of the 1' cube?

(Include the bottom surfaces in your calculations.)

(A) 2304 (B) 2208 (C) 1728 (D) 1568 (E) 1440

49. After a blizzard, Tamika made a snowman whose head was a perfect sphere of radius R. After aday in the sun, the head has melted into a sphere whose volume is now 18 of the original head’svolume. In terms of R, what is the radius of the snowman’s head now? [V sphere =

43 π r

3]

(A)3

4

R (B)

R

2 (C)

R

3 (D)

R

4 (E)

R

8

48. In the figure above, a water cooler bottle (shown at left) is shaped like a right circular cylinder witha diameter of 10" and a height of 15" and is full of water. The paper cups that are used with thiscooler (one of which is shown at right) are shaped like right circular cones with diameters of 4" andheights of 4". What is the maximum number of paper cups that the bottle can fill completely?

[V cylinder = π r 2h and V cone =

13 π r

2h]

(A) 60 (B) 63 (C) 69 (D) 70 (E) 71

15" 4"

4"

10"


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PART 5 – Probability, Statistics, and Data Analysis


1. The Monday dinner menu at the Majestic Diner offers 4 soup selections, numerous entrée selec-tions, and several dessert selections. If there are more entrée selections than dessert selections,and there are 140 possible meals that consist of one soup, one entrée, and one dessert, how many

dessert selections are offered on Monday? (A) 1 (B) 5 (C) 7 (D) 9 (E) It cannot be determined from the information given.

2. The postal service in a certain English-speaking country is considering a new scheme for generatingpostal codes that will include five characters with no repeating letters or digits. The codes will con-sist of one letter, followed by the number 9, followed by one letter, followed by two digits. Howmany different postal codes would be possible using this scheme?

(A) 468,000 (B) 421,200 (C) 67,600 (D) 46,800 (E) 44,600

3. How many 3-digit numbers that are multiples of 5 and greater than 500 can be formed from thedigits 0, 3, 5, and 7 if no digit appears more than once in the number?

(A) 1 (B) 2 (C) 4 (D) 6 (E) 8

Questions 4–6 refer to the following information:

The Hendersons – Mom, Dad, and their three children: Amy, Billy, and Cindy – arrive at the dentist’soffice and sit down in the waiting room in a row of five chairs

4. In how many ways could the family sit if Mom and Dad must sit next to each other and the threechildren must sit next to each other?

(A) 48 (B) 32 (C) 24 (D) 16 (E) 12

5. In how many ways could the family sit if Cindy must sit next to Amy?

(A) 24 (B) 32 (C) 36 (D) 48 (E) 54

6. Compared to the total number of different ways the family could sit in the row, in how manyFEWER ways could the family sit if no two children could sit next to each other?

(A) 112 (B) 108 (C) 96 (D) 64 (E) 12

7. A 3-person committee is being chosen from a group of 3 men and 3 women. In how many ways can

the committee be chosen? (A) 120 (B) 20 (C) 9 (D) 6 (E) 3

8. Luís must answer 5 out of 7 essay questions on his U.S. History final exam. If he must answer thefirst 2 questions, how many different sets of 5 questions could Luís answer?

(A) 10 (B) 9 (C) 7 (D) 5 (E) 4


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9. An urn contains a total of 18 red, white, and blue marbles. There are twice as many white marbles

as red marbles. One marble is chosen at random. If the probability that it is blue is1

6, what is the

probability that it is red?

(A)1

6 (B)

2

9 (C)

5

18 (D)

1

3 (E)

7

18

10. Three bicyclists – Lionel, Kim, and Jake – are in a three-person race. Lionel is twice as likely to winas Kim, and Kim is three times as likely to win as Jake. The probability that Kim loses the race is

(A)1

10 (B)

1

5 (C)

3

10 (D)

3

5 (E)

7

10

11. A fair coin is tossed 10 times. The probability of getting exactly 3 heads exceeds the probability ofgetting exactly 7 heads by

(A) 0 (B)1

5 (C)

2

5 (D)

1

2 (E)

45

64

12. A circle with area π 2 is inscribed in a square. If a dart hits the square, what is the probability that itdoesn’t hit inside the circle?

(A) 4 − �

(B)�

(C)3

4 (D)

1

4 (E) 1 −

�

13. At Jefferson Elementary School, there are four Sixth Grade classes that include 50 students in total.One of these classes has 14 students, whose average height is 5' 2". The average height of all thestudents in the other three Sixth Grade classes is 4' 8". To the nearest inch, what is the averageheight of a Sixth Grade student at Jefferson Elementary School?

(A) 5' 1" (B) 5' (C) 4' 10" (D) 4' 9" (E) 4' 8"

14. Mark’s final grade in Computer Science this semester will be the average of his scores on 5 tests. Oneach of the tests, he receives a score that is a whole number ranging from 0 to 100 inclusive. Theaverage of his scores on the first 2 tests was 89. If the difference in his scores on the next 2 tests is78, what is the highest possible final grade Mark could achieve in Computer Science?

(A) 84 (B) 80 (C) 78 (D) 66 (E) 53

15. If a + b = 1

8, c + b =

3

8, and a + c =

1

2, what is the average of a, b, and c ?

(A)1

6 (B)

1

4 (C)

1

3 (D)

3

8 (E)

1

2


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16. The table above shows Kenny’s scores on 8 of the 10 math quizzes he took last semester. Possiblescores on theses quizzes were integers that ranged from 0 to 10. If Kenny’s mean score on the 10quizzes was 7, what was his mode score?

(A) 6 (B) 7 (C) 9 (D) 9.5 (E) It cannot be determined from the information given.

17. Miss Palumbo wanted to rent an apartment in Cambridge. She looked at 4 apartments in total, thefirst 3 of which had the following monthly rents: $1,680, $1,750, and $1,640. The median of themonthly rents of all 4 apartments was $1,715. What was the monthly rent of the fourth apartment?

(A) $1,705 (B) $1,740 (C) $1,750 (D) $1,790 (E) It cannot be determined from the information given.

Quiz Score

1 7

2 9

3 4

4 7

5 9

6 3

7 6

8 6


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Questions 18–21 refer to the following information:

Where the Government of Country XCollected Money From (Receipts) and

How It Spent the Money (Outlays) in 1994

18. Income from which of the following sources was most nearly equal to Cost of Thrift Industry Bail-out?

(A) Social Security (B) Borrowing (C) Corporate Income Tax (D) Excise Tax (E) Other

19. To the nearest percent, by what percent was the amount received from Borrowing greater than theamount spent on Net Interest?

(A) 0% (B) 5% (C) 21% (D) 25% (E) 32%

20. Which of the following statements can be inferred from the information given in the graphs?

I. Excluding spending on National Defense, more than half of the remaining outlays went for Direct Benefit Payments to Individuals.

II. The total of the three smallest categories comprises less than 1/6 of the receipts.

III. If, for 1995, the outlays for Net Interest went up 10% and the outlays for National Defense went up 20%, while the other outlays remained the same, then the total outlays exceeded $950 billion.

(A) I only (B) II only (C) III only (D) I and II only (E) I, II, and III

21. What is the angle at the center of the sector for Social Security?

(A) 120° (B) 108° (C) 100° (D) 90° (E) 30°

36¢

7¢

3¢5¢

IndividualIncome Tax

30¢

Social

Security

19¢Borrowing

CorporateIncome Tax

Excise TaxOther

43¢

12¢

6¢5¢

DirectBenefit

Payments toIndividuals

20¢

NationalDefense

14¢Net Interest

Grants toStates andLocalities

Cost of ThriftIndustry Bailout

Other

Where the Tax DollarCame FromTotal Receipts: $800 Billion

Where the Tax DollarWas SpentTotal Outlays: $900 Billion


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Questions 22–26 refer to the following graphs:

22. For which nation is the number of retirees per 100 projected to have the greatest increase over 30years?

(A) A (B) B (C) C (D) D (E) E

23. What is the median number of retirees per 100 workers for the five nations in 2005?

(A) 25 (B) 26 (C) 28 (D) 37 (E) 38

24. For the nation with the least number of projected retirees per 100 workers, what is the difference

in years between male and female life expectancies? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8

25. For the nation with the least difference in male-female life expectancy, what number most closelyapproximates the ratio of projected retirees per 100 to present retirees per 100 workers?

(A) 2 15

(B) 145

(C) 112 (D) 1 (E) 2

5

26. In 2005, if there are 1.5 million workers in nation B, which of the following expressions gives thenumber of retired people in nation B?

(A) 2.7 × 103 (B) 2.7 × 104 (C) 2.7 × 105 (D) 2.7 × 106 (E) 2.7 × 107

60

50

40

30

20

10

A

2005

2035 (projected)

B C

Nation

N u m b e r

D E

Retired People Per 100 Workers

90

80

70

60

50

40

A

Men

Women

B C

Nation

Y e a r s

D E

Life Expectancy


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1,500

1,000

500

01988 1989 1990 1991 1992 1993 1994 1995

Year

E n r o l l m e n t

Business (B)Natural Science (N)

Engineering (E)Social Science (S)

Fall Enrollment by Majorat Trescott University for 1988–1995

S

N

B

E

Questions 27–29 refer to the following graph:

27. What was the approximate percent decrease in Engineering enrollment from 1988 to 1995?

(A) 25% (B) 35% (C) 45% (D) 55% (E) 65%

28. Between which two consecutive years was the change in Business enrollment the greatest?

(A) 1992–1993 (B) 1991–1992 (C) 1990–1991 (D) 1989–1990 (E) 1988–1989

29. Between 1988 and 1995, what was the difference in ranges between the discipline whose enroll-ment had the largest range and the discipline whose enrollment had the smallest range?

(A) 900 (B) 860 (C) 800 (D) 720 (E) 600


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30. The table above shows Fragrant Flower Company’s bouquet sales (in thousands) for four differenttypes of flower bouquets during the first 6 months of 2005. In February, Irresistible Irises and CoolCarnations sold for the same price, while Dynamite Daisies sold for half of that price. If the pricesof all four types of flower bouquets were integers and did not fluctuate , which of the followingcould be the total revenues (in thousands of dollars) for February?

(A) 8,900 (B) 9,100 (C) 9,200 (D) 9,300 (E) It cannot be determined from the information given.

Jan Feb March April May June

Irresistible Irises 10 12 22 24 30 32

Really Red Roses 40 120 62 68 100 34

Dynamite Daisies 32 40 40 35 20 38

Cool Carnations 48 28 20 36 30 36

Fragrant Flower CompanyOnline Flower ShopBouquet Sales, 2005

(in thousands)


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PART 6 – Grid-ins

Directions: Solve each problem and enter your answer by darkening in the appropriate circles in each grid.

1. Today, the 5 coins in Mrs. Patel’scoin collection – a half-dollar,

a quarter, a dime, a nickel, and

a penny – would have a totalmarket value that is $21.17

more than the total value ontheir faces. If the nickel alonehas a market value of $12.24,what is the difference betweenthe total market value and thetotal face value of the otherfour coins?

.

1

2

3

4

5

6

7

8

9

.

0

1

2

3

4

5

6

7

8

9

.

0

1

2

3

4

5

6

7

8

9

.

0

1

2

3

4

5

6

7

8

9

2. On a certain number line,

A


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8. In the regular square pyramidabove, the length of altitudeTU is 12 and the length of slantheight TV is 13. What is thelength of lateral edge PT ?

.

1

2

3

4

5

6

7

8

9

.

0

1

2

3

4

5

6

7

8

9

.

0

1

2

3

4

5

6

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9. The data point for when therabbit population is 21 rabbitsper 10 acres (omitted in thegraph above) falls on the line ofbest fit for the data presented,a line which passes through thepoints (3, 56) and (30, 38). What

is the size of the crop harvest,to the nearest ton per 100 acres,for this data point?

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Rabbit Population(in rabbits per 10 acres)

C r o p H a r v e s t

( i n t o n s p e r 1 0 0 a c r e s )

Relationship Between

Crop Harvest and Rabbit Populationat Jacobsen’s Farms

10. The only horses at Silver MapleFarm are the race horses ownedby Mr. Davis. If 8 of these horsesare taken out for a race, therewould be at most 47 horses re-maining. If Mr. Davis purchased12 additional horses, he would

then own more than 65 horses.What is one possible value forthe total number of race horsesMr. Davis currently owns?

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7. The town of Hicksville plansto plant a new garden that isshaped like a triangle and mea-sures 48 feet on each side. Thetown plans to build a sidewalkaround the outside of the gar-den such that the distance fromthe outer edge of the sidewalkto the outer edge of the gardenwill always measure exactly 3feet. To the nearest foot, whatwill be the perimeter of theouter edge of the sidewalk?

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sat math study guide

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