GRE Mathematics

GRE Math Set 1

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Category 1 Number Operations

1. For which of the following expressions is it true that the value of the expression is the same for

587=x as for 587-=x ?

I. x21 -

II. 221 x- III. 2)21( x-

(A) I only

(B) II only

(C) I and III only

(D) II and III only

(E) I, II, and III

2. If the numbers 24

17,

2

1,

8

3,

4

3, and

16

9 were ordered from greatest to least, the middle

number of the resulting sequence would be

(A) 24

17

(B) 2

1

(C) 8

3

(D) 4

3

(E) 16

9

GRE Math Set 1


3. The sum 8

7 +

9

1 is between

(A) 2

1 and

4

3

(B) 8

7 +

9

1 and 1

(C) 1 and 14

1

(D) 14

1 and 1

2

1

(E) 12

1 and 2

4. Of the following, which is closest to 7

1?

(A) 0.200

(B) 0.172

(C) 0.167

(D) 0.143

(E) 0.140

5. For all integers a , b , c , and d , ),,,(* dcba is defined as dcba -+- . What is the value of )5,8,3,1(* ?

(A) 1- (B) 0

(C) 1

(D) 2

(E) 3

GRE Math Set 1


6. The operation # is defined by the equation yx

yxyx

--

=2

# , where yx 2 . The value of

which of the following is 0?

(A) 2#2- (B) 2#2 - (C) 2#2

(D) 1#2

(E) 2#3

7. The number 0.756 is how much greater than 4

3?

(A) 25

6

(B) 125

3

(C) 250

3

(D) 500

3

(E) 250

1

8. What number when multiplied by 7

4 yields

7

6 as the result?

(A) 7

2 (B)

3

2 (C)

2

3 (D)

7

24 (E)

2

7

GRE Math Set 1


9. If the sum of 3 times n , 11 times n , and 111 times n is equal to 1,000, what is the value of

n ?

(A) 8

(B) 12.5

(C) 16

(D) 25

(E) 125

10. Which of the following is equal to 0.042?

(A) 10

2

10

4 +

(B) 100

2

10

4 +

(C) 100

2

100

4 +

(D) 000,1

2

100

4+

(E) 000,1

2

000,1

4+

11. In the addition problem above, if and D each represent a digit, then represents (A) 9

(B) 8

(C) 5

(D) 2

(E) 0

GRE Math Set 1


12. If represents a digit in the subtraction problem above, = (A) 1 (B) 5 (C) 6 (D) 7 (E) 9

13. The figure above shows the arrangement and code numbers of 9 fluorescent tubes. If a tube is

illuminated whenever its code number is received, which of the following shows the

arrangement of tubes illuminated when the digits in the result of 3804,3 are received?

14. Of the following, which is the closest approximation of 8.199

49.02.50 ?

(A) 10

1 (B)

8

1 (C)

4

1 (D)

4

5 (E)

2

25

GRE Math Set 1


403

)998.0(24.61 2

15. The expression above is approximately equal to

(A) 1

(B) 3

(C) 4

(D) 5

(E) 6

16. )24.0(

)15)(33.0)(2( is approximately equal to

(A) 2

(B) 2

5

(C) 15

(D) 36

(E) 40

17. Of the following, which is the best approximation for 04.012.11

98.32005.201.0

?

(A) 0.015

(B) 0.15

(C) 1.5

(D) 15

(E) 150

GRE Math Set 1


18. Of the following, the closest approximation to 202.0

)8745.0(097.8 is

(A) 40

(B) 35

(C) 30

(D) 4

(E) 3

19. If a sequence of 8 consecutive odd integers with increasing values has 9 as its 7th term, what is

the sum of the terms of the sequence?

(A) 22

(B) 32

(C) 36

(D) 40

(E) 44

20. If ,,, yxw and z are positive and z

y

x

w= , which of the following is NOT always true?

(A) xywz =

(B) y

z

w

x =

(C) w

z

y

x =

(D) z

zy

x

xw +=+

(E) y

yz

w

wx +=+

GRE Math Set 1


21. If

21 c

abba

d+

+= ,

2

ca = , and

4

3cb = , what is the value of d in terms of c?

(A) 11

10c

(B) 2

5c

(C) 3

10c

(D) c11

10

(E) c2

5

22. Tamara saves $35 each week. If she now has $100 saved, in how many weeks can she first

have enough saved to buy a lawn mower that costs $250?

(A) 2

(B) 3

(C) 4

(D) 5

(E) 6

23. If 000,3>x , then the value of 12 +x

x is closest to

(A) 6

1

(B) 3

1

(C) 21

10

(D) 2

1

(E) 2

3

GRE Math Set 1


24. A certain clock indicates 8 oclock. If the clock runs accurately for the next 11,995 hours,

what hour will it indicate at the end of that time?

(A) 1 oclock

(B) 2 oclock

(C) 3 oclock

(D) 5 oclock

(E) 10 oclock

25. If 2

1 the result obtained when 2 is subtracted from x5 is equal to the sum of 10 and x3 ,

what is the value of x ?

(A) 22- (B) 4

(C) 4

(D) 18

(E) 22

26. A light-year is approximately 12106 miles. Approximately how many miles from Earth is a galaxy that is 6102 light-years away? (A) 6100.3 (B) 18102.1 (C) 19102.1 (D) 72102.1 (E) 73102.1

GRE Math Set 1


27. If it is 6:27 in the evening on a certain day, what time in the morning was it exactly 2,880,717

minutes earlier? (Assume standard time in one location.)

(A) 6:22

(B) 6:24

(C) 6:27

(D) 6:30

(E) 6:32

28. If the sum of two positive integers is 24 and the difference of their squares is 48, what is the

product of the two integers?

(A) 108

(B) 119

(C) 128

(D) 135

(E) 143

29. On a certain scale of intensity, each i ncrement of 10 in magnitude represents a tenfold increase

in intensity. On this scale, an intensity corresponding to a magnitude of 165 is how many

times an intensity corresponding to a magnitude of 125?

(A) 40

(B) 100

(C) 400

(D) 1,000

(E) 10,000

GRE Math Set 1


30. In the formula ( )32

1

rV = , if r is halved, then V is multiplied by

(A) 64 (B) 8 (C) 1 (D) 8

1 (E)

64

1

31. In the subtraction problem above, D , , and O represent different positive digits. If is 1 greater than D , what is the value of O ?

(A) 9

(B) 7

(C) 5

(D) 3

(E) 2

32. Which of the following procedures is always equivalent to adding 5 given numbers and then

dividing the sum by 5?

I. Multiplying the 5 numbers and then finding the 5th root of the product.

II. Adding the 5 numbers, doubling the sum, and then moving the decimal point one place to

the left.

III. Ordering the 5 numbers numerically and then selecting the middle number.

(A) None

(B) I only

(C) II only

(D) III only

(E) I and III

GRE Math Set 1


Category 1 Number Operations

1. For which of the following expressions is it true that the value of the expression is the same for

587=x as for 587-=x ? I. x21 -

II. 221 x- III. 2)21( x-

(A) I only

(B) II only

(C) I and III only

(D) II and III only

(E) I, II, and III

587=x , 587-=x . x . & (B).

2. If the numbers 24

17,

2

1,

8

3,

4

3, and

16

9 were ordered from greatest to least, the middle

number of the resulting sequence would be

(A) 24

17 (B)

2

1 (C)

8

3 (D)

4

3 (E)

16

9

GRE i) ii) . .

8

3 43

8

3 4

3

GRE Math Set 1


. 24

17 , 21 ,

8

3 , 4

3

.

=

249

83 ,

=

2412

21 ,

24

17 ,

=

24

18

4

3

. 16

9

.

### fractions( )9

11,

19

22

19

22 9

11 cross-multiplying( 22 9, 11 19)

. 22 9 < 11 19, so 9

111922 <

& (E).

3. The sum 8

7 +

9

1 is between

(A) 2

1 and

4

3 (B)

8

7 +

9

1 and 1 (C) 1 and 1

4

1

(D) 14

1 and 1

2

1 (E) 1

2

1 and 2

.

8

7 +

9

1 =

72

71 1 (C),(D),(E) .

=

72

54

4

3

72

71 (B) .

8

7 +

9

1 (C),(D),(E)

.. 8

7 1

8

1 .

9

1

9

1

8

1 .

8

7 +

9

1 1

.

GRE Math Set 1


GRE . & (B)

4. Of the following, which is closest to 7

1?

(A) 0.200

(B) 0.172

(C) 0.167

(D) 0.143

(E) 0.140

7

1 100

.

& (D).

5. For all integers a , b , c , and d , ),,,(* dcba is defined as dcba -+- . What is the value of )5,8,3,1(* ?

(A) 1- (B) 0

(C) 1

(D) 2

(E) 3

),,,(* dcba dcba -+- )5,8,3,1(* .

. .

& (C).

GRE Math Set 1


6. The operation # is defined by the equation yx

yxyx

--

=2

# , where yx 2 . The value of

which of the following is 0?

(A) 2#2- (B) 2#2 - (C) 2#2

(D) 1#2

(E) 2#3

yx

yxyx

--

=2

# 0 .

0 0 .

& (C).

7. The number 0.756 is how much greater than 4

3?

(A) 25

6

(B) 125

3

(C) 250

3

(D) 500

3

(E) 250

1

0.756 4

3 . 4

3

. 43 ,

25 100

75 . GMAT

. & (D).

GRE Math Set 1


8. What number when multiplied by 7

4 yields

7

6 as the result?

(A) 7

2 (B)

3

2 (C)

2

3 (D)

7

24 (E)

2

7

7

4

7

6 .(

7

6

7

4=x .)

GRE .

& (C) .

9. If the sum of 3 times n , 11 times n , and 111 times n is equal to 1,000, what is the value of

n ?

(A) 8 (B) 12.5 (C) 16 (D) 25 (E) 125

n 3, n 11, n 111 1,000 n . ( 000,1111113 =++ nnn )

& (A).

10. Which of the following is equal to 0.042?

(A) 10

2

10

4+

(B) 100

2

10

4+

(C) 100

2

100

4 +

(D) 000,1

2

100

4+

(E) 000,1

2

000,1

4+

1000 . & (D).

GRE Math Set 1


11. In the addition problem above, if and D each represent a digit, then represents (A) 9 (B) 8 (C) 5 (D) 2 (E) 0

,D . +2 = 2 0, 9. 10 . 10 . 0 2 D 0 . 9 . ( .)

& (A).

12. If represents a digit in the subtraction problem above, = (A) 1

(B) 5

(C) 6

(D) 7

(E) 9

.

& (E).

GRE Math Set 1


13. The figure above shows the arrangement and code numbers of 9 fluorescent tubes. If a tube

is illuminated whenever its code number is received, which of the following shows the

arrangement of tubes illuminated when the digits in the result of 3804,3 are received?

9 fluorescent tube code . Code tube 3804,3 tube .

& (D).

14. Of the following, which is the closest approximation of 8.199

49.02.50 ?

(A) 10

1 (B)

8

1 (C)

4

1 (D)

4

5 (E)

2

25

GRE . .

8

1

200

5.050 =

& (B).

GRE Math Set 1


403

)998.0(24.61 2

15. The expression above is approximately equal to

(A) 1

(B) 3

(C) 4

(D) 5

(E) 6

( )3

20

160 2=

& (B).

16. )24.0(

)15)(33.0)(2( is approximately equal to

(A) (A) 2

(B) 2

5

(C) 15

(D) 36

(E) 40

& (E).

17. Of the following, which is the best approximation for 04.012.11

98.32005.201.0

?

(A) 0.015

(B) 0.15

(C) 1.5

(D) 15

(E) 150

& (C).

GRE Math Set 1


18. Of the following, the closest approximation to 202.0

)8745.0(097.8 is

(A) 40

(B) 35

(C) 30

(D) 4

(E) 3

& (B).

19. If a sequence o f 8 consecutive odd integers with increasing values has 9 as its 7th term, what is

the sum of the terms of the sequence?

(A) 22

(B) 32

(C) 36

(D) 40

(E) 44

8 consecutive odd integers increasing value 7 9 . : -3, -1, 1, 3, 5, 7, 9, 11. 3 3 0 5 + 7 + 9 + 11 = 32.

Tips -3, -1, 1, 3, 5, 7,, 9, 11 3 ( ) 2 11 . a , d , ( n ) l n nS

2

)( lanSn

+= ,

{ }2

)1(2 dnanSn

-+=

. (B)

GRE Math Set 1


20. If ,,, yxw and z are positive and z

y

x

w= , which of the following is NOT always true?

(A) xywz =

(B) y

z

w

x=

(C) w

z

y

x=

(D) z

zy

x

xw +=

+

(E) y

yz

w

wx +=+

z

y

x

w = xywz = , (C)

. (C)

21. If

21 c

abba

d+

+= , 2

ca = , and

4

3cb = , what is the value of d in terms of c?

(A) 11

10c (B)

2

5c (C)

3

10c (D)

c11

10 (E)

c2

5

d c 2

ca =

4

3cb =

21

c

abba

d+

+=

.

. (A)

22. Tamara saves $35 each week. If she now has $100 saved, in how many weeks can she first

have enough saved to buy a lawn mower that costs $250?

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

GRE Math Set 1


$100 $250 lawn mower . $100 $150 . $35 4 $140 $240 5 .

. (D)

23. If 000,3>x , then the value of 12 +x

x is closest to

(A) 6

1 (B)

3

1 (C)

21

10 (D)

2

1 (E)

2

3

12 +xx =

x

12

1

+

x

1 x 3,001

x

1= 0.00033322

1/2 .

. (D)

24. A certain clock indicates 8 oclock. If the clock runs accurately for the next 11,995 hours,

what hour will it indicate at the end of that time?

(A) 1 oclock

(B) 2 oclock

(C) 3 oclock

(D) 5 oclock

(E) 10 oclock

8 11,995 . 11,995 8 12,003 hours 24 3 . .

. (C)

GRE Math Set 1


25. If 2

1 the result obtained when 2 is subtracted from x5 is equal to the sum of 10 and x3 ,

what is the value of x ?

(A) 22- (B) 4

(C) 4

(D) 18

(E) 22

xx 310)25(2

1+=-

x = 22- . (A)

26. A light-year is approximately 12106 miles. Approximately how many miles from Earth is a galaxy that is 6102 light-years away? (A) 6100.3 (B) 18102.1 (C) 19102.1 (D) 72102.1 (E) 73102.1

( light-year ) 12106 miles , 6102 . 12106 6102 = 19102.1

. (C)

27. If it is 6:27 in the evening on a certain day, what time in the morning was it exactly 2,880,717

minutes earlier? (Assume standard time in one location.)

(A) 6:22 (B) 6:24 (C) 6:27 (D) 6:30 (E) 6:32

6:27, 2,880,717 minutes . 2,880,717 minutes = (48,011 60 minutes) + 57 minutes. 57 6: 30 . . (D)

GRE Math Set 1


28. If the sum of two positive integers is 24 and the difference of their squares is 48, what is the

product of the two integers?

(A) 108 (B) 119 (C) 128 (D) 135 (E) 143

yx, 24=+ yx , 4822 =- yx xy . 4822 =- yx 2)())(24(48))(( =--==-+ yxyxyxyx

24=+ yx 2=- yx 11,13 == yx xy = 143 . (E)

29. On a certain scale of intensity, each increment of 10 in magnitude represents a tenfold increase

in intensity. On this scale, an intensity corresponding to a magnitude of 165 is how many

times an intensity corresponding to a magnitude of 125?

(A) 40

(B) 100

(C) 400

(D) 1,000

(E) 10,000

165 125 40 magnitude . magnitude 10 10 (intensity) 10101010 =10,000 . (E)

30. In the formula ( )32

1

rV = , if r is halved, then V is multiplied by

(A) 64

(B) 8

(C) 1

(D) 8

1

(E) 64

1

38

1

rV = , r

r2

1

3

1

rV = . V 8 .

. (B)

GRE Math Set 1


31. In the subtraction problem above, D , , and O represent different positive digits. If is 1 greater than D , what is the value of O ?

(A) 9 (B) 7 (C) 5 (D) 3 (E) 2

= D + 1 . D = 1 = 2 = 9 . = 9 . . (A)

32. Which of the following procedures is always equivalent to adding 5 given numbers and then

dividing the sum by 5?

I. Multiplying the 5 numbers and then finding the 5th root of the product.

II. Adding the 5 numbers, doubling the sum, and then moving the decimal point one

place to the left.

III. Ordering the 5 numbers numerically and then selecting the middle number.

(A) None

(B) I only

(C) II only

(D) III only

(E) I and III

5 5 . 5 , . 1 10 . 5 2 10 5 .

. (C)

GRE Math Set 2-1


Category 2 Number Properties

1. If x is an even integer and y is an odd integer, which of the following CANNOT be true?

(A) yx is an even integer.

(B) xy is an odd integer.

(C) x is a multiple of y .

(D) y is a multiple of x .

(E) xy is an even integer.

2. If x is an even integer, which of the following is an odd integer?

(A) 23 +x (B) x7

(C) 58 +x

(D) 2x

(E) 3x

3. If x is a positive odd integer and y is a negative even integer, which of the following could

be a negative odd integer?

(A) xy

(B) xy

(C) yx -

(D) yx +2

(E) 2yx +

GRE Math Set 2-1


4. An integer n that is greater than 1 is said to be prime-saturated if it has no prime factor

greater than or equal to n . Which of the following integers is prime-saturated?

(A) 6

(B) 35

(C) 46

(D) 66

(E) 75

5. If ba, , and c are three consecutive odd integers such that 2010

GRE Math Set 2-1


7. What is the least common multiple of 3, 4, 5, and 8?

(A) 480

(B) 240

(C) 120

(D) 105

(E) 60

8. Which of the following is NOT a factor of 252?

(A) 2

(B) 3

(C) 6

(D) 7

(E) 8

9. There are 125 chips on a table. If as many of the chips as possible are to be arranged into an

equal number of 3-chip and 4-chip stacks and the remaining chips are to be removed, how

many of the chips are to be removed?

(A) One

(B) Two

(C) Five

(D) Six

(E) Seven

GRE Math Set 2-1


9.0=x

9.0

1=y

2)9.0(=z 10. The values of ,, yx and z are shown above. Which of the following gives these numbers

in order from least to greatest?

(A) zyx ,,

(B) yzx ,,

(C) xzy ,,

(D) xyz ,,

(E) yxz ,,

11. If x and y are positive integers, which of the following is NOT necessarily an integer?

(A) yx + (B) yx -

(C) y

x

(D) xy

(E) yx

12. For any number ,x x denotes the least non-negative number y such that yx + is an

integer. What is the value of 8.4 8.4?

(A) -0.4

(B) 0

(C) 0.6

(D) 7.8

(E) 8.0

GRE Math Set 2-1


13. What is the least odd integer, greater than 1, that is both the square of an integer and the cube

of an integer?

(A) 9 (B) 27 (C) 81 (D) 243 (E) 729

14. What is the least possible product of 4 different integers, each of which has a value between 5

and 10, inclusive?

(A) 5040- (B) 3600- (C) 720- (D) 600- (E) 120-

15. The number line above shows the position of a point that has coordinate x . Which of the

following statements about x must be true?

I. 42

GRE Math Set 2-1


16. Which of the following must be true?

I. If caba +

GRE Math Set 2-1


19. When the integer n is divided by 6, the remainder is 3. Which of the following is NOT a

multiple of 6?

(A) 3-n (B) 3+n (C) n2

(D) n3

(E) n4

20. If the remainder is 7 when positive integer n is divided by 18, what is the remainder when

n is divided by 6?

(A) 0

(B) 1

(C) 2

(D) 3

(E) 4

21. If the two-digit integers M and N are positive and have the same digits, but in reverse

order, which of the following CANNOT be the sum of M and N ?

(A) 181

(B) 165

(C) 121

(D) 99

(E) 44

GRE Math Set 2-1


22. If the product of two positive integers is 630, which of the following must be true?

I. Both integers are even numbers.

II. At least one of the integers is a multiple of 3.

III. One of the integers is 10.

(A) I only

(B) II only

(C) III only

(D) I and II

(E) II and III

23. For any numbers a and b , abbaba -+= . If 0=ba , which of the following CANNOT be a value of b ?

(A) 2

(B) 1

(C) 0

(D) 1-

(E) 2

3-

24. When the integer k is divided by 12, the remainder is 3. Which of the following, when

divided by 12, will have a remainder of 6?

I. 2k

II. 6k

III. 4 k + 6

(A) I only

(B) II only

(C) III only

(D) I and II only

(E) I, II, and III

GRE Math Set 2-1


25. What is the least number of digits (including repetitions) needed to express 10100 in decimal

notation?

(A) 4

(B) 100

(C) 101

(D) 1,000

(E) 1,001

26. What is the smallest positive integer n for which 324 is a factor of n6 ? (A) 2

(B) 3

(C) 4

(D) 5

(E) 6

27. If n is an integer, which of the following CANNOT be a factor of 43 +n ? (A) 4

(B) 5

(C) 6

(D) 7

(E) 8

28. If n and k are integers whose product is 400, which of the following statements must be

true?

(A) 0>+ kn (B) kn (C) Either n or k is a multiple of 10.

(D) If n is even, then k is odd.

(E) If n is odd, then k is even.

GRE Math Set 2-1


29. If a is a positive integer, and if the units digit of 2a is 9 and the units digit of ( )21+a is

4, what is the units digit of ( )22+a ?

(A) 1 (B) 3 (C) 5 (D) 7 (E) 9

30. An Armstrong number is an n -digit number that i s equal to the sum of the n th powers of

its individual digits. For example, 153 is an Armstrong number because it has 3 digits and

153351 333 =++ . What is the digit k in the Armstrong number 1, 46k ? (A) 2

(B) 3

(C) 4

(D) 5

(E) 6

31. If the sum of the first n positive integers is S , what is the sum of the first n positive even

integers, in terms of S ?

(A) 2

S

(B) S

(C) S2

(D) 22 +S (E) S4

GRE Math Set 2-1


32. The positive integers ,,, cba and d are such that cba >> . If dbca +=+ , which of the following CANNOT be true?

(A) ad > (B) bd = (C) bd > (D) cd > (E) db >

33. If [ ]x is the greatest integer less than or equal to x , what is the value of ]7.2[]4.3[]6.1[ ++- ?

(A) 3

(B) 4

(C) 5

(D) 6

(E) 7

STOP

GRE Math Set 2-1



1. If x is an even integer and y is an odd integer, which of the following CANNOT be true?

(A) yx is an even integer.

(B) xy is an odd integer.

(C) x is a multiple of y .

(D) y is a multiple of x .

(E) xy is an even integer.

(A) yx , . .

(B) xy , .

(C) . (D) , . . ## summary of arithmetic . & (D).

2. If x is an even integer, which of the following is an odd integer?

(A) 23 +x (B) x7

(C) 58 +x

(D) 2x

(E) 3x

+ , . x 58 +x (+). & (C).

GRE Math Set 2-1


3. If x is a positive odd integer and y is a negative even integer, which of the following could

be a negative odd integer?

(A) xy

(B) xy

(C) yx -

(D) yx +2

(E) 2yx +

x , y

. (A) . (B) . (C) .

(D) yx

GRE Math Set 2-1


5. If ba, , and c are three consecutive odd integers such that 2010

GRE Math Set 2-1


8. Which of the following is NOT a factor of 252?

(A) 2 (B) 3 (C) 6 (D) 7 (E) 8

252 . !!!

252732 22 = & (E).

9. There are 125 chips on a table. If as many of the chips as possible are to be arranged into an

equal number of 3-chip and 4-chip stacks and the remaining chips are to be removed, how

many of the chips are to be removed?

(A) One (B) Two (C) Five (D) Six (E) Seven

125 3, 4 . 3 4 125 . 3 4 7 . 7125 . & (D).

9.0=x

9.0

1=y

2)9.0(=z 10. The values of ,, yx and z are shown above. Which of the following gives these numbers

in order from least to greatest?

(A) zyx ,,

(B) yzx ,,

(C) xzy ,,

(D) xyz ,,

(E) yxz ,,

10

99.0 ==x , 9

9.01

==y , 2

2

10

9)9.0(

==z

& (E).

GRE Math Set 2-1


11. If x and y are positive integers, which of the following is NOT necessarily an integer?

(A) yx + (B) yx -

(C) y

x

(D) xy

(E) yx

. & (C).

12. For any number ,x x denotes the least non-negative number y such that yx + is an

integer. What is the value of 8.4 8.4?

(A) -0.4

(B) 0

(C) 0.6

(D) 7.8

(E) 8.0

x x . 8.4 8.4 . 8.4 0.6,1.6, . 0.6. & (D).

13. What i s the least odd integer, greater than 1, that is both the square of an integer and the cube

of an integer?

(A) 9 (B) 27 (C) 81 (D) 243 (E) 729

(the square of an integer) (the cube of an integer) .

729333333 332226 ===

& (E).

GRE Math Set 2-1


14. What is the least possible product of 4 different integers, each of which has a value between 5

and 10, inclusive?

(A) 5040- (B) 3600- (C) 720- (D) 600- (E) 120-

5 10 4 . 600,310985 -=-

& (B).

15. The number line above shows the position of a point that has coordinate x . Which of the

following statements about x must be true?

I. 42

GRE Math Set 2-1


16. Which of the following must be true?

I. If caba +

GRE Math Set 2-1


19. When the integer n is divided by 6, the remainder is 3. Which of the following is NOT a

multiple of 6?

(A) 3-n (B) 3+n (C) n2

(D) n3

(E) n4

n 6 3 . 36 += an . 36 += an 6 3 6 .

& (D).

20. If the remainder is 7 when positive integer n is divided by 18, what is the remainder when

n is divided by 6?

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

n 18 7 . 718 += an . n

6 6

7

6

18+=

an . 1 .

& (B).

21. If the two-digit integers M and N are positive and have the same digits, but in reverse

order, which of the following CANNOT be the sum of M and N ?

(A) 181 (B) 165 (C) 121 (D) 99 (E) 44

M N 19 91 . : M = 10a + b, N = 10b + a M + N = 11( a + b )

M + N 11 . (A) (E) 11 . (A)

GRE Math Set 2-1


22. If the product of two positive integers is 630, which of the following must be true?

I. Both integers are even numbers.

II. At least one of the integers is a mul tiple of 3.

III. One of the integers is 10.

(A) I only

(B) II only

(C) III only

(D) I and II

(E) II and III

630 . 630 = 7532 2 2 I III 3 210 . II . (B)

23. For any numbers a and b , abbaba -+= . If 0=ba , which of the following CANNOT be a value of b ?

(A) 2 (B) 1 (C) 0 (D) 1- (E) 2

3-

abbaba -+= = 0 b b

aababa

--

=-=-1

01 - b 1=b . (B)

24. When the integer k is divided by 12, the remainder is 3. Which of the following, when

divided by 12, will have a remainder of 6?

I. 2 k

II. 6 k

III. 4 k + 6

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

312 += Qk I. 2 k = 2( 312 +Q ) = Q24 + 6, 6. ! II. 6 k = 18126 + Q , 18 12 6. ! III. 4 k + 6 = 4( 312 +Q ) + 6 = 18124 + Q , 18 12 6,

! (E)

GRE Math Set 2-1


25. What is the least number of digits (including repetitions) needed to express 10100 in decimal

notation?

(A) 4 (B) 100 (C) 101 (D) 1,000 (E) 1,001

10100 . 110 = 2 210 =3 10 3 = 4. 10010 100 1 101 .

(C)

26. What is the smallest positive integer n for which 324 is a factor of n6 ? (A) 2

(B) 3

(C) 4

(D) 5

(E) 6

324 n6 factor . 324 = 22 36 n6 = 324Q n6 324. 4=n . 6 4 1296 324 4 .

(C)

27. If n is an integer, which of the following CANNOT be a factor of 43 +n ? (A) 4

(B) 5

(C) 6

(D) 7

(E) 8

43 +n = 1)1(3 ++n , n 3 . 3 6 43 +n factor .

(C)

GRE Math Set 2-1


28. If n and k are integers whose product is 400, which of the following statements must be

true?

(A) 0>+ kn (B) kn (C) Either n or k is a multiple of 10.

(D) If n is even, then k is odd.

(E) If n is odd, then k is even.

400=nk = 2452 . (A) n = -2, k = -5 400=nk 0

GRE Math Set 2-1


30. An Armstrong number is an n -digit number that is equal to the sum of the n th powers of

its individual digits. For example, 153 is an Armstrong number because it has 3 digits and

153351 333 =++ . What is the digit k in the Armstrong number 1, 46k ? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6

n n Armstrong number .

1, 46k = 4444 461 +++ k = 1553 + 4k , 4k 1553 3 4 . 4k 1. (A) (E) 4 1 3.

(B)

31. If the sum of the first n positive integers is S , what is the sum of the first n positive even

integers, in terms of S ?

(A) 2

S (B) S (c) S2 (D) 22 +S (E) S4

. n S ,

1 n 2

)1( +nn= S )1( +nn = 2 S .

(C)

32. The positive integers ,,, cba and d are such that cba >> . If dbca +=+ , which of the following CANNOT be true?

(A) ad > (B) bd = (C) bd > (D) cd > (E) db >

cba >> dbca +=+ , ba > dbca +=+ ad > . ad > cb > dbca +=+

. (A)

GRE Math Set 2-1


33. If [ ]x is the greatest integer less than or equal to x , what is the value of ]7.2[]4.3[]6.1[ ++- ?

(A) 3

(B) 4

(C) 5

(D) 6

(E) 7

[ ]x x , [ ]6.1- = -2, [ ]4.3 = 3, [ ]7.2 = 2 . 3.

(A)

< Summary of Arithmetic >

Divisibility Tests for 2, 3, 5, and 10

A number is divisible by

2, if its last digit is even 0, 2, 4, 6, or 8;

3, if the sum of its digits is a number divisible by 3

5, if its last digit is 0 or 5; and

10, if its last digit is 0

Rules for Odds and Evens

odd + odd = even odd odd = odd

even + even = even even even = even

odd + even = odd odd even = even

Factors() The factors of a number are the positive integers that evenly divide that number.

Ex: 36 9 factors : 1, 2, 3, 4, 6, 9, 12, 18, 36

36 = 22 32 2 (exponent) 2 1 , 3 2 1 (2 + 1 )(2 + 1) = 9( factors )

GRE Math Set 2-1


(the greatest common divisor ) : , , .

(the lowest common divisor): , , .

Ex. : 180, 420 (GCD) (LCD)? ) 180 = 2 2 3 3 5, 420= 2 2 3 5 7 GCD = 2 2 3 5= 60 , LCD = 2 2 3 3 5 7 = 1260

Decimals()

1) : 0 (5

3 =0.6)

2) : (0.123123123 =0. 321 )

3) : 0 ex) 1/6 = 0.1666

4) : 2 5

ex.) 15.052

3

20

3

80

122

=

==

GRE Math Set 2-2



1. If k is a positive integer and if the numeral in the decimal system that represents the product

k60 contains only the digits 0 and 2, what is the least possible value of k60 ?

(A) 220

(B) 2,020

(C) 2,220

(D) 20,020

(E) 20,220

2. There are between 100 and 110 cards in a collection of cards. If they are counted out 3 at a

time, there are 2 left over, but if they are counted out 4 at a time, there is 1 left over. How

many cards are in the collection?

(A) 101

(B) 103

(C) 106

(D) 107

(E) 109

3. A necklace is made by stringing N individual beads together in the repeating pattern red

bead, green bead, white bead, blue bead, and yellow bead. If the necklace design begins with a

red bead and ends with a white bead, then N could equal

(A) 16

(B) 32

(C) 41

(D) 54

(E) 68

GRE Math Set 2-2


4. Mary Lou has 969 dimes in a piggy bank. What is the least number of dimes that she can

remove from the bank so that she could divide the remaining dimes equally among 7 people?

(A) 2

(B) 3

(C) 4

(D) 5

(E) 6

5. Which of the following integers equals the product of two prime numbers?

(A) 19

(B) 27

(C) 30

(D) 41

(E) 58

6. What is the smallest positive integer by which 126 can be multiplied so that the product is the

square of an integer?

(A) 4

(B) 7

(C) 9

(D) 14

(E) 126

GRE Math Set 2-2


7. The sum of three consecutive odd integers, ,, yx and z , in ascending order, is 39. What is

the sum of the three consecutive odd integers that immediately follow z ?

(A) 78

(B) 57

(C) 48

(D) 45

(E) 42

8. If the average (arithmetic mean) of ,, yx and z is an even integer, then zyx ++ could

be

(A) 15

(B) 16

(C) 18

(D) 20

(E) 21

9. The number 4

3 is what fraction of its reciprocal?

(A) 16

9

(B) 4

3

(C) 1

(D) 3

4

(E) 9

16

GRE Math Set 2-2


10. If 10

GRE Math Set 2-2



1. If k is a positive integer and if the numeral in the decimal system that represents the product

k60 contains only the digits 0 and 2, what is the least possible value of k60 ?

(A) 220

(B) 2,020

(C) 2,220

(D) 20,020

(E) 20,220

k k60 ( k60 ) 0 2 , k60 . . 60 .

: (C). :

2. There are between 100 and 110 cards in a collection of cards. If they are counted out 3 at a

time, there are 2 left over, but if they are counted out 4 at a time, there is 1 left over. How

many cards are in the collection?

(A) 101 (B) 103 (C) 106 (D) 107 (E) 109

100 ~ 110 3 2 , 4 1 . 100 110 3 2 101,104,107,110. 4 1 101.

: (A). :

GRE Math Set 2-2


3. A necklace is made by stringing N individual beads together in the repeating pattern red

bead, green bead, white bead, blue bead, and yellow bead. If the necklace design begins with

a red bead and ends with a white bead, then N could equal

(A) 16

(B) 32

(C) 41

(D) 54

(E) 68

N . , , , , N . 5 3 . 5 3 .

: (E). :

4. Mary Lou has 969 dimes in a piggy bank. What is the least number of dimes that she can

remove from the bank so that she could divide the remaining dimes equally among 7 people?

(A) 2

(B) 3

(C) 4

(D) 5

(E) 6

969 10cent , 7 . 969 7 . 969 7 966. 3 966 7 .

: (B). :

GRE Math Set 2-2


5. Which of the following integers equals the product of two prime numbers?

(A) 19 (B) 27 (C) 30 (D) 41 (E) 58

. . 2 . 30, 58 2 .. 2 !

: (E). :

6. What is the smallest positive integer by which 126 can be multiplied so that the product is the

square of an integer?

(A) 4

(B) 7

(C) 9

(D) 14

(E) 126

126

. 126 732 2 . 14 )732(42 2222 = .

: (D). :

7. The sum of three consecutive odd integers, ,, yx and z , in ascending order, is 39. What is

the sum of the three consecutive odd integers that immediately follow z ?

(A) 78 (B) 57 (C) 48 (D) 45 (E) 42

,, yx z 39 , z 3

. 2,,2 +- aaa . 393 =a

15=z . z 3 17, 19, 21 .

: (B). :

GRE Math Set 2-2


8. If the average (arithmetic mean) of ,, yx and z is an even integer, then zyx ++ could

be ?

(A) 15

(B) 16

(C) 18

(D) 20

(E) 21

,, yx z , evenzyx

=++

3 ,, yx z 3

. 18.

: (C). :

9. The number 4

3 is what fraction of its reciprocal?

(A) 16

9

(B) 4

3

(C) 1

(D) 3

4

(E) 9

16

4

3 .

4

3

3

4.

16

9

3

4

4

3=

: (A). :

GRE Math Set 2-2


10. If 10 n

11

1 +n

1 . (5) n 1 )2( +n )3( . 1 .

: (E). :

11. For how many of the integers from 10 to 99 is at least one of the two digits a 4?

(A) 9

(B) 10

(C) 18

(D) 19

(E) 20

10 99 4 . 40~49 10, 14,24,34,.94 8(44) 18 .

: (C). :

GRE Math Set 2-2


12. The series of numbers 3, 4, and 5 has the property that the square of the greatest number is

equal to the sum of the squares of the other two numbers. Which of the following series of

numbers does NOT have this property?

(A) 0.6, 0.8, 1.0

(B) 8

3,

2

1,

8

5

(C) 23 , 24 , 25

(D) 3 , 4 , 5

(E) 1.5, 2, 2.5

3, 4, 5 (5) (3, 4) . . 3, 4, 5 . (A) 0.2 3, 4, 5 . (B) 8 3, 4, 5 . (D) .

: (D). :

GRE . . .

GRE Math Set 3

, http://www.vstudy.co.kr, [email protected], 02-538-5999. Page1

Category 3 Averages

1. What is the average (arithmetic mean) of the numbers 15, 16, 17, 17, 18, and 19?

(A) 14.2

(B) 16.5

(C) 17

(D) 17.5

(E) 18

2. If n red pencils cost 10 cents each and m blue pencils cost 9 cents each, what is the average (arithmetic mean) cost, in cents, per pencil?

(A) 2

)(19 mn +

(B) 19

910 mn +

(C) mn

mn

++ 910

(D) 2

mn +

(E) mn +

19

3. If the average (arithmetic mean) of ,,ba and c is 40, what is the average (arithmetic

mean) of )103(),103( ++ ba , and )103( +c ? (A) 50

(B) 70

(C) 130

(D) 150

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

GRE Math Set 3


4. The cost of item A is 2

y each and the cost of item B is

3

t each. What is the average

(arithmetic mean) cost per unit of a collection consisting of x units of A and k units of B ?

(A) 12

23 ty +

(B) 12

)23()( tykx +++

(C) kx

ktxy

++ 23

(D) 6

23 ktxy +

(E) )(6

23

kx

ktxy

++

5. If each of 4 subsidiaries of Corporation R has been granted a line of credit of $700,000 and each of the other 3 subsidiaries of Corporation R has been granted a line of credit of $112,000, what is the average (arithmetic mean) line of credit granted to a subsidiary of

Corporation R ? (A) $1,568,000

(B) $448,000

(C) $406,000

(D) $313,600

(E) $116,000

6. What is the average (arithmetic mean) of 8 numbers if the average of 5 of the numbers is 24

and the sum of the remaining 3 numbers is 40?

(A) 8

(B) 20

(C) 29

(D) 30

(E) 37.5

GRE Math Set 3


7. A managers annual salary is $8,400 greater than the annual salary of each of two assistant

managers, who earn x dollars each per year. The average (arithmetic mean) of the 3 annual salaries, in dollars, is

(A) 800,2+x

(B) 200,4+x

(C) 800,22 +x

(D) 3

400,82 +x

(E) 2

400,83 +x

8. If the average (arithmetic mean) of the four numbers K , 32 +K , 53 -K , and 15 +K is 63, what is the value of K ?

(A) 11

(B) 4

315

(C) 22

(D) 23

(E) 10

325

9. The average (arithmetic mean) of 10, 30, and 50 is 5 more than the average of 20, 40, and

(A) 15

(B) 25

(C) 35

(D) 45

(E) 55

GRE Math Set 3


10. A total of $2,000 was spent for a 30-day vacation. If transportation expenses were $230,

lodging was $750, and exactly $20 was spent per day for meals and tips, what was the average

(arithmetic mean) amount per day that was spent on other expenses?

(A) $66

(B) $42

(C) $33 (D) $20

(E) $14

11. In a certain company, the total monthly payroll for the 12 production workers is $18,000 and

the total monthly payroll for the 36 office workers is $63,000. By how much does the average

(arithmetic mean) monthly salary of an office worker exceed that of a production worker in

this company?

(A) $62.50

(B) $187.50

(C) $250.00

(D) $375.00

(E) $500.00

12. On a certain test, 3 students each had a score of 90, 9 students each had a score of 80, 4

students each had a score of 70, and 4 students each had a score of 60. What was the average

(arithmetic mean) score for the 20 students?

(A) 70.5 (B) 75.0

(C) 75.5

(D) 80.0

(E) 80.5

GRE Math Set 3


13. On 3 sales John has received commissions of $240, $80, and $110, and he has 1 additional sale

pending. If John is to receive an average (arithmetic mean) commission of exactly $150 on

the 4 sales, then the 4th commission must be.

(A) $164

(B) $170

(C) $175

(D) $182

(E) $185

14. The 10 households on a certain street have household incomes that range from $34,000 to

$150,000 and an average (arithmetic mean) household income of $60,000. If the household

with the highest income and the one with the lowest income are excluded, what is the average

household income for the remaining 8 households?

(A) $41,600

(B) $47,000

(C) $52,000

(D) $61,000

(E) $75,000

15. The graph above shows the distribution of the 100 scores in a certain competition. What is

the median score for the competition?

(A) 7.5

(B) 7.75

(C) 8.0

(D) 8.25

(E) 8.5

GRE Math Set 3


16. A certain elevator has a safe weight limit of 2,000 pounds. What is the greatest possible

number of people who can safely ride on the elevator at one time with the average (arithmetic

mean) weight of half the riders being 180 pounds and the average weight of the others being

215 pounds?

(A) 7

(B) 8 (C) 9

(D) 10

(E) 11

17. For the month of April, the average (arithmetic mean) of the daily high temperatures

recorded at a certain weather station was x degrees. If the average for the first 13 days of

the month was )10( -x degrees, what was the average, in degrees, for the remaining 17 days of the month?

(A) 10+x (B) 13017 +x

(C) x

x 17130 +

(D) 17

1017 +x

(E) 17

13017 +x

18. A statistician calculated the average (arithmetic mean) for 20 measurements to be 50. On

rechecking the calculations, the statistician found that a measurement of 78 had been treated

as 18 and a measurement of 40 had been treated as 50. If no other errors were made,

what is the correct average for the 20 measurements?

(A) 47.5

(B) 50.0

(C) 51.0

(D) 52.5

(E) 53.5

GRE Math Set 3


19. If k is an integer and 2515

GRE Math Set 3


22. The average (arithmetic mean) of ten numbers on a list is m . If the numbers 10 and 24 are added to the list, the average (arithmetic mean) of these twelve numbers on the list is also m . What is the value of m ?

(A) 3

111

(B) 14

(C) 17 (D) 20


23. The average (arithmetic mean) price of the 5 houses on a certain street is $60,000. If the

price of the least expensive house is $55,000 which of the following could NOT be the price of

the most expensive house?

(A) $82,000

(B) $78,000

(C) $72,000

(D) $65,000

(E) $62,500

24. If the average (arithmetic mean) of x and y is 60 and the average (arithmetic mean) of y

and z is 80, what is the value of xz - ? (A) 70

(B) 40

(C) 20

(D) 10


GRE Math Set 3


Mon Tue Wed Thu Fri

2

12+

4

11-

8

1-

0 8

31+

25. The table above shows the net change, in dollars, in the price of a share of a certain stock each

day last week. What was the average (arithmetic mean) daily net change, in dollars, for the 5-

day week?

(A) 2

1- (B)

4

1+ (C)

2

1+ (D)

4

5+ (E)

2

5+

26. If the average (arithmetic mean) of 5 positive temperatures is x degrees Fahrenheit, then the sum of the 3 greatest of these temperatures, in degree Fahrenheit, could be

(A) x6 (B) x4

(C) 3

5x

(D) 2

3x

(E) 5

3x

Score Number of

Students

83 5

70 6

92 3

5

64 1

27. The incomplete table above shows a distribution of scores for a class of 20 students. If the

average (arithmetic mean) score for the class is 78, what score is missing from the table?

(A) 73

(B) 75

(C) 77

(D) 79

(E) 81

GRE Math Set 3


28. If x is the average (arithmetic mean) of 5 consecutive even integers, which of the following must be true?

I. x is an even integer. II. x is a nonzero integer. III. x is a multiple of 5.

(A) I only

(B) III only

(C) I and II only

(D) I and III only

(E) I, II, and III

29. The average (arithmetic mean) of 4 positive integers is 50. If the average of 2 of these

integers i s 45, what is the greatest possible value that one of the other 2 integers can have?

(A) 55

(B) 65

(C) 100 (D) 109

(E) 115

30. The trade name for a product has 6 letters , and the length of the package for the product is

l centimeters. The trade name of the product is printed lengthwise on one side of the package with a margin of m centimeters on each end of the side. If the space between each letter is considered negligible, what is the average (arithmetic mean) width, in centimeters, of each letter in the trade name?

(A) m2-l (B) m-l (C) 8l (D) 6)( - ml

(E) 6)2( - ml

GRE Math Set 3


Category 3 Average

1. What is the average (arithmetic mean) of the numbers 15, 16, 17, 17, 18, and 19? (A) 14.2

(B) 16.5

(C) 17

(D) 17.5

(E) 18

176

191817171615=

+++++ ,

: (C) :

2. If n red pencils cost 10 cents each and m blue pencils cost 9 cents each, what is the average (arithmetic mean) cost, in cents, per pencil?

(A) 2

)(19 mn +

(B) 19

910 mn +

(C) mn

mn

++ 910

(D) 2

mn +

(E) mn +

19

N1() A1, N2() A2

Combined Average(N1 N2 ) = 21

)22()11(

NN

ANAN

++

n 10cents, m 9 cents , .

: (C) :

Point! . N1 , N2 . (Combined Average) .

GRE Math Set 3


3. If the average (arithmetic mean) of ,,ba and c is 40, what is the average (arithmetic

mean) of )103(),103( ++ ba , and )103( +c ? (A) 50

(B) 70

(C) 130

(D) 150 (E) It cannot be determined from the information given.

403

=++ cba , then

3

)103()103()103( +++++ cba =?

120=++ cba . : (C) :

4. The cost of item A is 2

y each and the cost of item B is

3

t each. What is the average

(arithmetic mean) cost per unit of a collection consisting of x units of A and k units of B ?

(A) 12

23 ty +

(B) 12

)23()( tykx +++

(C) kx

ktxy

++ 23

(D) 6

23 ktxy +

(E) )(6

23

kx

ktxy

++

N1( x ) A1(2

y), N2( k ) A2(

3

t)

Combined Average(N1 N2 )

= 21

)22()11(

NN

ANAN

++

=kx

tk

yx

+

+ )3

()2

(

: (E) :

5. If each of 4 subsidiaries of Corporation R has been granted a line of credit of $700,000 and each of the other 3 subsidiaries of Corporation R has been granted a line of credit of $112,000, what is the average (arithmetic mean) line of credit granted to a subsidiary of

Corporation R ? (A) $1,568,000

(B) $448,000

(C) $406,000

(D) $313,600

(E) $116,000

GRE Math Set 3


Combined Average !!

21)22()11(

NNANAN

++

= 000,448$7

)000,112$3()000,700$4(=

+

: ( B ) :

6. What is the average (arithmetic mean) of 8 numbers if the average of 5 of the numbers is 24

and the sum of the remaining 3 numbers is 40? (A) 8

(B) 20

(C) 29

(D) 30

(E) 37.5

Combined Average ! 8 5 24, 3 40 3 40/3 . :

)(208

)3

403()245(

Average=+

,

: (B) :

7. A managers annual salary is $8,400 greater than the annual salary of each of two assistant

managers, who earn x dollars each per year. The average (arithmetic mean) of the 3 annual salaries, in dollars, is

(A) 800,2+x

(B) 200,4+x

(C) 800,22 +x

(D) 3

400,82 +x

(E) 2

400,83 +x

A managers annual salary = $ x (salary of each of two assistant managers) + $8,400

Average= 800,2$3

)400,8$(+=

+++x

xxx ,

: (A) :

GRE Math Set 3


8. If the average (arithmetic mean) of the four numbers K , 32 +K , 53 -K , and 15 +K is 63, what is the value of K ? (A) 11

(B) 4

315

(C) 22

(D) 23

(E) 10

325

63

4)15()53()32(

=++-+++ KKKK

K=23, : (D) :

9. The average (arithmetic mean) of 10, 30, and 50 is 5 more than the average of 20, 40, and

(A) 15 (B) 25 (C) 35 (D) 45 (E) 55

3

4020530

3

503010 x+++==

++ X .

: (A) :

10. A total of $2,000 was spent for a 30-day vacation. If transportation expenses were $230,

lodging was $750, and exactly $20 was spent per day for meals and tips, what was the average

(arithmetic mean) amount per day that was spent on other expenses?

(A) $66

(B) $42

(C) $33

(D) $20

(E) $14

($2,000) - ($230+$750+($2030days) = $420(other expenses) $420 30 = $14,

: (E) :

Tip :exactly $20 was spent per day for meals and tips $20 30 days .

GRE Math Set 3


11. In a certain company, the total monthly payroll for the 12 production workers is $18,000 and

the total monthly payroll for the 36 office workers is $63,000. By how much does the average

(arithmetic mean) monthly sal ary of an office worker exceed that of a production worker in

this company?

(A) $62.50 (B) $187.50

(C) $250.00

(D) $375.00

(E) $500.00

12 production workers = 500,1$12

000,18$=

36 office workers = 750,1$36

000,63$=

$1,750 -- $1,500 = $250, : (C) :

12. On a certain test, 3 students each had a score of 90, 9 students each had a score of 80, 4

students each had a score of 70, and 4 students each had a score of 60. What was the average (arithmetic mean) score for the 20 students?

(A) 70.5

(B) 75.0

(C) 75.5

(D) 80.0

(E) 80.5

Combined average !

Combined Average = 5.754493

)604()704()809()903(=

++++++ ,

: (C) :

13. On 3 sales John has received commissions of $240, $80, and $110, and he has 1 additional sale

pending. If John is to receive an average (arithmetic mean) commission of exactly $150 on

the 4 sales, then the 4th commission must be. (A) $164 (B) $170 (C) $175 (D) $182 (E) $185

150$4

11080240 =+++ x , x = $170

: (B) :

GRE Math Set 3


14. The 10 households on a certain street have household incomes that range from $34,000 to

$150,000 and an average (arithmetic mean) household income of $60,000. If the household

with the highest income and the one with the lowest income are excluded, what is the average

household income for the remaining 8 households?

(A) $41,600 (B) $47,000 (C) $52,000 (D) $61,000 (E) $75,000

000,60$10

000,150$)8(000,34$=

++ households,

8 households income = $416,000, 8 $52,000. ! : (C) :

15. The graph above shows the distribution of the 100 scores in a certain competition. What is

the median score for the competition? (A) 7.5 (B) 7.75 (C) 8.0 (D) 8.25 (E) 8.5

median() ( nx ), scores, frequency() . frequency 100 median 50 . score 10.0 frequency 2(10.0) + 4(9.5) + 5(9.0) + 25(8.5) 36 score 8.0 frequency 20 50 frequency score 8.0 .

: (C) :

16. A certain elevator has a safe weight limit of 2,000 pounds. What is the greatest possible

number of people who can safely ride on the elevator at one time with the average (arithmetic

mean) weight of half the riders being 180 pounds and the average weight of the others being

215 pounds?

(A) 7 (B) 8 (C ) 9 (D) 10 (E) 11

R

poundsR

1802

+ poundsR

2152

2,000 pounds,

R 10 .

GRE Math Set 3


: (D) : 17. For the month of April, the average (arithmetic mean) of the daily high temperatures

recorded at a certain weather station was x degrees. If the average for the first 13 days of

the month was )10( -x degrees, what was the average, in degrees, for the remaining 17 days of the month?

(A) 10+x (B) 13017 +x (C)x

x 17130 + (D)

17

1017 +x (E)

17

13017 +x

17 y :

xyx

=+-

30

17)10(13, in terms of x y .

: (E) :

18. A statistician calculated the average (arithmetic mean) for 20 measurements to be 50. On

rechecking the calculations, the statistician found that a measurement of 78 had been treated

as 18 and a measurement of 40 had been treated as 50. If no other errors were made,

what is the correct average for the 20 measurements?

(A) 47.5 (B) 50.0 (C) 51.0 (D) 52.5 (E) 53.5

20 50 . 78 18 40 50 . . 78 18 +60 40 50 10 . 50 20 2.5 50 52.5 .

: (D). :

19. If k is an integer and 2515

GRE Math Set 3


20. In performing a sequence of experiments, a scientist made 20 measurements. The average

(arithmetic mean) of these measurements was 34. For security reasons the scientist coded the

data by multiplying each of the measurements by 10 and then adding 40 to each product.

What is the average of the coded measurements?

(A) 1,140 (B) 380 (C) 342 (D) 57 (E) 19

20 34, )(3420

averagesum

= 2034 = 680.

measurement 10 680 10 6800 . measurement 40 4020 = 800 6800 800 20 7600 20 = 380. .

: (B) :

21. The average (arithmetic mean) of n numbers is equal to 3

1 the sum of the n numbers.

If 2

1 the sum of the numbers is 15, what is the average of the n numbers?

(A) 3 (B) 5 (C) 10 (D) 30 (E) 45

Average = sumn

sum

3

1= , n 3 .

2

1 the sum of the numbers is

15 the sum of number = 30, n 103

30=

: (C) :

22. The average (arithmetic mean) of ten numbers on a list is m . If the numbers 10 and 24 are added to the list, the average (arithmetic mean) of these twelve numbers on the list is also m . What is the value of m ?

(A) 3

111 (B) 14 (C) 17 (D) 20


12

34

10

+==

summ

sum, sum m

jsum = 10m, ksum = 12m 34 10m = 12m - 34 m = 17

: (C) :

GRE Math Set 3


23. The average (arithmetic mean) price of the 5 houses on a certain street is $60,000. If the

price of the least expensive house is $55,000 which of the following could NOT be the price of

the most expensive house?

(A) $82,000 (B) $78,000 (C) $72,000 (D) $65,000 (E) $62,500

000,60$5

5=

houses, sum $300,00.

$55,000 . $55,000 4 $55,000 = $220,000. $300,00 $220,000 $80,000 . $80,000 .

: (A) :

24. If the average (arithmetic mean) of x and y is 60 and the average (arithmetic mean) of y

and z is 80, what is the value of xz - ? (A) 70 (B) 40 (C) 20 (D) 10


.

802

,602

=+

=+ zyyx

160,120 =+=+ zyyx

xz - = 40

: (B) :

Mon Tue Wed Thu Fri

2

12+

4

11-

8

1-

0 8

31+

25. The table above shows the net change, in dollars, in the price of a share of a certain stock each

day last week. What was the average (arithmetic mean) daily net change, in dollars, for the 5-

day week?

(A) 2

1- (B)

4

1+ (C)

2

1+ (D)

4

5+ (E)

2

5+

. 5 net change 5 .

: (C) :

GRE Math Set 3


26. If the average (arithmetic mean) of 5 positive temperatures is x degrees Fahrenheit, then the sum of the 3 greatest of these temperatures, in degree Fahrenheit, could be

(A) x6 (B) x4 (C) 3

5x (D)

2

3x (E)

5

3x

xedcbaxedcba

5)(5)( =++++=++++ cba ++ .

d e 0(, 0 0 ) xedcba 5)( =++++ cba ++ <

x5 . cba ++ x5 . x5 (B). : (B) :

Score Number of

Students

83 5

70 6

92 3

5

64 1

27. The incomplete table above shows a distribution of scores for a class of 20 students. If the

average (arithmetic mean) score for the class is 78, what score is missing from the table?

(A) 73 (B) 75 (C) 77 (D) 79 (E) 81

20 students 78( average) = 1560 = 835 + 706 + 923 + x 5 + 641

773855 == xx

: (C) :

28. If x is the average (arithmetic mean) of 5 consecutive even integers, which of the following must be true?

I. x is an even integer. II. x is a nonzero integer. III. x is a multiple of 5.

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

(E) I, II, and III

even integer n 45

)8()6()4()2(+=

++++++++= n

nnnnnx

4+= nx , n + = I true. II n = -4 false.

: (A) :

GRE Math Set 3


29. The average (arithmetic mean) of 4 positive integers is 50. If the average of 2 of these

integers is 45, what is the greatest possible value that one of the other 2 integers can have?

(A) 55

(B) 65

(C) 100 (D) 109

(E) 115

4 50 200. 45 90 . 200 90 110 0 1 . 110 - 1 = 109

(D)

30. The trade name for a product has 6 letters , and the length of the package for the product is

l centimeters. The trade name of the product is printed lengthwise on one side of the package with a margin of m centimeters on each end of the side. If the space between each letter is considered negligible, what is the average (arithmetic mean) width, in centimeters, of

each letter in the trade name?

(A) m2-l (B) m-l (C) 8l (D) 6)( - ml (E) 6)2( - ml

m m

l

width = ( ml 2- ) 6 (E)

GRE Math Set 4

, http://www.vstudy.co.kr , [email protected] , 02-538-5999, Page 1

Chapter 4 Ratio, Proportion, and Fraction

1. A certain club has 35 male members and 45 female members. Which of the following equals

the ratio of the number of male members to the number of female members?

(A) 16:7

(B) 9:5

(C) 7:5

(D) 4:3 (E) 9:7

2. What is the ratio of 4

3 to the product

43

4 ?

(A) 4

1

(B) 3

1

(C) 9

4

(D) 4

9

(E) 4

3. If 7

2 of the 210 students in a class attended graduate school, what was the total number of

students in the class who did not attend graduate school?

(A) 42

(B) 60

(C) 105

(D) 142

(E) 150

GRE Math Set 4


4. A certain test consists of 8 sections with 25 questions numbered from 1 to 25, in each section.

If a student answered all of the even-numbered questions correctly and 4

3 of the odd-

numbered questions correctly, what was the total number of questions he answered

correctly?

(A) 150

(B) 172

(C) 174

(D) 175

(E) 176

5. Two cities that are geographically 380 miles apart are 2 inches apart on a certain map. If

the map is drawn to scale, how many miles apart are two cities that are 2

17 inches apart on

the map?

(A) 665

(B) 1,330

(C) 1,425

(D) 2,090 (E) 2,850

6. Lou and Selma were hired to paint a room for a total of $72. They completed the job with

Lou working 3 hours and 20 minutes and Selma working 2 hours and 40 minutes. If they

decided to split the $72 in proportion to the amount of time each spent on the job, how much

did Lou receive?

(A) $32.00

(B) $36.00

(C) $40.00

(D) $41.14

(E) $43.20

GRE Math Set 4


7. Kathy bought 4 times as many shares in Company X as Carl, and Carl bought 3 times as many shares in the same company as Tom. Which of the following is the ratio of the

numbers of shares bought by Kathy to the number of shares bought by Tom?

(A) 4

3

(B) 3

4

(C) 1

3

(D) 1

4

(E) 1

12

8. Of the 40 schools invited to participate in a research study, 8

7 agreed to participate. If 60

questionnaires were sent to each of the participating schools and 5

4 of these questionnaires

were completed and returned, what was the total number of questionnaires completed and

returned?

(A) 28

(B) 1,344

(C) 1,680

(D) 1,920

(E) 2,100

9. If a printer can 2 pages of text per second, then, at this rate, approximately how many minutes

will it take to print 5,000 pages of text?

(A) 4

(B) 25

(C) 42

(D) 250

(E) 417

GRE Math Set 4


10. If $160 is divided among 3 people in the ratio 8:5:3 , what is the least amount received by any one person?

(A) $80

(B) $50

(C) $30

(D) $20

(E) $10

11. Which three of the following fractions are equivalent?

80

5=v , 08.0

05.0=w , 0.8

5.0=x , 8.0

05.0=y , 008.0

05.0=z

(A) v , w , and x (B) v , x , and y (C) w , x , and y

(D) w , y , and z

(E) x , y , and z

12. A study based on a random sample revealed that, on average, 2 out of 5 adults have high blood

pressure. If these results hold true for the 580,000 adults in City A , approximately how many adults in City A have high blood pressure? (A) 116,000

(B) 145,000

(C) 232,000

(D) 250,000

(E) 290,000

GRE Math Set 4


13. Two boxes of pudding mix cost a total of $0.73 and 1 box of cake mix costs twice as much as

two boxes of pudding mix. At these rates, what is the cost of 3 boxes of pudding mix and 3

boxes of cake mix?

(A) $1.10

(B) $3.29

(C) $4.38 (D) $5.48

(E) $6.57

14. Out of their annual net income, a couple spent 25 percent for food, 13.5 percent for

entertainment, 20 percent for housing, 8 percent for car expenses, 15 percent for clothing, and

saved the rest. What was the ratio of the amount saved to the amount spent for

entertainment?

(A) 27

19

(B) 5

6

(C) 27

37

(D) 9

19

(E) 3

7

15. Approximately how many minutes will it take a secretary who types at an average rate of 60

words per minute to type a 5-page report with 250 words per page? (A) 4

(B) 12

(C) 21

(D) 30

(E) 75

GRE Math Set 4


16. If the speed of an airplane is 360 kilometers per hour, how many meters will it travel in one

second? (1 kilometer = 1,000 meters)

(A) 100

(B) 360

(C) 1,000

(D) 3,600 (E) 6,000

17. The total time required to transport rail freight from City X to City Y is 4 days, 15 hours.

If, by eliminating certain intermediate stops, the total time were reduced by 3

1, the total

transport time required would be

(A) 1 day, 3

15 hours

(B) 1 day, 13 hours

(C) 2 days, 3

210 hours

(D) 2 days, 3

215 hours

(E) 3 days, 2 hours

18. A small firm spent 4

1 of its annual budget during the first 2 months of its fiscal year and

spent 2

1 of its remaining budget during the next 5 months. What fraction of the annual

budget remained unspent at the end of the first 7 months?

(A) 16

3

(B) 4

1

(C) 16

5

(D) 8

3

(E) 12

5

GRE Math Set 4


19. Members of a social club met to address 280 newsletters. If they addressed 4

1 of the

newsletters during the first hour and 5

2 of the remaining newsletters during the second hour,

how many newsletters did they address during the second hour?

(A) 28 (B) 42 (C) 63 (D) 84 (E) 112

20. In a certain used-car lot, there are 4 times as many blue cars as green cars and 3

1 as many

green cars as red cars. If there are q4 red cars in the lot, how many blue cars are in the lot?

(A) q12

(B) 3

16q

(C) q3

(D) 3

4q

(E) 4

3q

21. When Inez and Fernando purchased a property for $20,000. Fernando contributed 3

1 the

amount contributed by Inez for the purchase. If they sold the property for $40,000 and

shared this amount in proportion to their respective contributions, how much was Fernandos

share? (A) $5,000

(B) $10,000

(C) $13,333

(D) $20,000

(E) $26,666

GRE Math Set 4


22. Of Amys monthly salary, 4

1 goes to rent and

5

1 to food. If the remainder of her monthly

salary amounts to $990, what is Amys monthly salary? (A) $1,042

(B) $1,273

(C) $1,800

(D) $1,980

(E) $2,200

23. When 9

2 of the votes on a certain resolution have been counted,

4

3 of those counted are in

favor of the resolution. What fraction of the remaining votes must be against the resolution

so that the total count will result in a vote of 2 to 1 against the resolution?

(A) 14

11

(B) 18

13

(C) 7

4

(D) 7

3

(E) 14

3

24. In a certain population, there are 3 times as many people aged twenty-one or under as there

are people over twenty-one. The ratio of those twenty-one or under to the total population is

(A) 1 to 2

(B) 1 to 3

(C) 1 to 4

(D) 2 to 3

(E) 3 to 4

GRE Math Set 4


25. Last year Department Store X had a sales total for December that was 4 times the average (arithmetic mean) of the monthly sales totals for January through November. The sales total

for December was what fraction of the sales total for the year?

(A) 4

1

(B) 15

4

(C) 3

1

(D) 11

4

(E) 5

4

26. Millie and Rick addressed 150 invitations. If Millie addressed 2

11 times as many as Rick

addressed, how many of the invitations did Rick address?

(A) 30

(B) 60

(C) 75

(D) 90

(E) 100

27. In 1990 a total of x earthquakes occurred worldwide, some but not all of which occurred in Asia. If m of these earthquakes occurred in Asia, which of the following represents the ratio of the number of earthquakes that occurred in Asia to the number that did not occur in

Asia?

(A) m

x

(B) x

m

(C) mx

m

-

(D) mx

x

-

(E) x

m-1

GRE Math Set 4


28. The ratio of two quantities is 3 to 4. If each of the quantities is increased by 5, what is the

ratio of these two new quantities?

(A) 4

3

(B) 9

8

(C) 19

18

(D) 24

23


29. Of the people who responded to a market survey, 120 preferred Brand X and the rest preferred Brand Y . If the respondents indicated a preference for Brand X over Brand Y by a ratio of 3 to 1, how many people responded to the survey? (A) 80

(B) 160

(C) 240

(D) 360

(E) 480

30. Orange juice of normal strength is prepared by adding three 6-ounce cans of water to the

contents of a 6-ounce can of orange juice concentrate. If 1 ounce of the can of concentrate

was already used for some other purpose, how many 6-ounce cans of water should be added to

the remainder of the concentrate to prepare juice of normal strength?

(A) 3

21

(B) 2

(C) 3

12

(D) 2

12

(E) 6

52

GRE Math Set 4


31. A coffee merchant makes a house blend of coffee using 4 pounds of Arabian beans costing $3

per pound, 2 pounds of Java beans costing $4 per pound, and 4 pounds of Brazilian beans

costing $1 per pound. If the merchant sold all of the blend for 50 percent more than the total

cost of the beans used, at what price per pound was the blend sold?

(A) $1.20

(B) $2.40 (C) $3.00

(D) $3.60

(E) $4.50

32. The current ratio of men to women on a certain board of trustees is 2 to 5. If 4 men were

added to the board, the ratio of men to women would be 2 to 3. How many men are currently

on the board?

(A) 2

(B) 4

(C) 5

(D) 6

(E) 8

33. A custom blend of paint is made by mixing red, yellow, and black paints in the ratio of 12 to 11

to 1. If all of the paints are sold only in pint-sized containers, how many containers of yellow

are used in preparing 6 gallons of the blend? (8 pints = 1 gallon)

(A) 2

(B) 3

(C) 11

(D) 22

(E) 24

GRE Math Set 4


34. A rectangular office floor is 28 meters by 20 meters. A custodian, working at a constant rate,

waxes a section of the floor that is 5 meters by 20 meters in 4

3 hour. How many hours does

it take the custodian, working at this same constant rate, to wax the entire floor?

(A) 2.52

(B) 3

(C) 3.45

(D) 4.2

(E) 7

35. A glucose solution contains 15 grams of glucose per 100 cubic centimeters of solution. If 45

cubic centimeters of the solution were poured into an empty container, how many grams of

glucose would be in the container?

(A) 3.00

(B) 5.00

(C) 5.50

(D) 6.50

(E) 6.75

36. Dan bought x pencils, some at 10 cents each and the rest at 6 cents each. For the same amount of money that he spent on these pencils, he could have bought x pencils at 9 cents each. What is the ratio of the number of 10-cent pencils purchased to the number of 6-cent

pencils purchased?

(A) 1 : 1

(B) 2 : 1

(C) 3 : 1

(D) 4 : 1

(E) 5 : 1

37. A certain shade of gray paint is obtained by mixing 3 parts of white paint with 5 parts of black

paint. If 2 gallons of the mixture is needed and the individual colors can be purchased only in

one-gallon or half-gallon cans, what is the least amount of paint, in gallons, that must be

purchased in order to measure out the portions needed for the mixture?

(A) 2

(B) 2

12

(C) 3

(D) 2

13

(E) 4

GRE Math Set 4


38. Working alone, printers X , Y , and Z can do a certain printing job, consisting of a large number of pages, in 12, 15, and 18 hours, respectively. What is the ratio of the time it takes

printer X to do the job, working alone at its rate, to the time it takes printers Y and Z to do the job, working together at their individual rates?

(A) 11

4

(B) 2

1

(C) 22

15

(D) 15

22

(E) 4

11

39. The ratio, by volume, of soap to alcohol to water in a certain solution is 2:50:100. The

solution will be altered so that the ratio of soap to alcohol is doubled while the ratio of soap to

water is halved. If the altered solution will contain 100 cubic centimeters of alcohol, how

many cubic centimeters of water will it contain?

(A) 50 (B) 200 (C) 400 (D) 625 (E) 800

40. Tom and Carlos begin to play a series of four games with 400 chips each. At the end of each

game, there is a loser who must surrender half of his chips to the winner. If Tom wins only

the first and third games, how many chips does he have after the pay-off at the end of the

fourth game?

(A) 200

(B) 275

(C) 400

(D) 525

(E) 550

STOP

GRE Math Set 4


Category 4 Ratio and Proportion

1. A certain club has 35 male members and 45 female members. Which of the following equals the ratio of the number of male members to the number of female members?

(A) 16:7

(B) 9:5

(C) 7:5

(D) 4:3 (E) 9:7

35 : 45 = 57: 59 = 9:79

7

95

75==

: (E) :

2. What is the ratio of 4

3 to the product

4

34 ?

(A) 4

1

(B) 3

1

(C) 9

4

(D) 4

9

(E) 4

4

3 :

43

4 = 1: 4 = 4

1

: (A) :

GRE Math Set 4


3. If 7

2 of the 210 students in a class attended graduate school, what was the total number of

students in the class who did not attend graduate school? (A) 42 (B) 60 (C) 105 (D) 142 (E) 150

210 2/7 , 5/7 .

210 7

5= 150,

: (E) :

4. A certain test consists of 8 sections with 25 questions numbered from 1 to 25, in each section.

If a student answered all of the even-numbered questions correctly and 4

3 of the odd-

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GRE Mathematics