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GRE Math Set 1
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 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
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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 2
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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 3
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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 4
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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 5
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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 6
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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 7
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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 8
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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 9
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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 10
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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 11
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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 12
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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 13
. 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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 14
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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 15
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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 16
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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 17
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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 18
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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 19
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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 20
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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 21
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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 22
$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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 23
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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 24
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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 25
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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 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
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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
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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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 4
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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 5
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
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 6
16. Which of the following must be true?
I. If caba +
GRE Math Set 2-1
, http://www.vstudy.co.kr, [email protected], 02-538-5999, Page 7
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
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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
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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
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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
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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
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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.
(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
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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
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5. If ba, , and c are three consecutive odd integers such that 2010
GRE Math Set 2-1
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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
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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
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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
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16. Which of the following must be true?
I. If caba +
GRE Math Set 2-1
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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
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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
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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
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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
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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
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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
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(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
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Category 2 Number Properties
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
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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
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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
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10. If 10
GRE Math Set 2-2
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Category 2 Number Properties
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
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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
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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
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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19. If k is an integer and 2515
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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
(E) It cannot be determined from the information given.
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
(E) It cannot be determined from the information given.
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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
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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
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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) .
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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
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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) :
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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
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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
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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 .
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: (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
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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
(E) It cannot be determined from the information given.
12
34
10
+==
summ
sum, sum m
jsum = 10m, ksum = 12m 34 10m = 12m - 34 m = 17
: (C) :
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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
(E) It cannot be determined from the information given.
.
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
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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) :
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
(E) It cannot be determined from the information given.
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
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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
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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
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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
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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) :
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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-
nu